Results of Confluent Hypergeometric Functions I: Difference between revisions

Latest revision as of 12:59, 22 May 2021

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
13.2.E1	${\displaystyle{\displaystyle z\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+(b-z% )\frac{\mathrm{d}w}{\mathrm{d}z}-aw=0}}$ `z\deriv[2]{w}{z}+(b-z)\deriv{w}{z}-aw = 0`		zdiff(w, [z$(2)])+(b - z)diff(w, z)- a*w = 0	zD[w, {z, 2}]+(b - z)D[w, z]- a*w == 0	Failure	Failure	Failed [300 / 300] Result: 1.299038106+.7500000000I Test Values: {a = -3/2, b = -3/2, w = 1/23^(1/2)+1/2I, z = 1/23^(1/2)+1/2I} Result: 1.299038106+.7500000000I Test Values: {a = -3/2, b = -3/2, w = 1/23^(1/2)+1/2I, z = -1/2+1/2I3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[1.299038105676658, 0.7499999999999999] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.299038105676658, 0.7499999999999999] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.2.E2	${\displaystyle{\displaystyle M\left(a,b,z\right)=\sum_{s=0}^{\infty}\frac{{% \left(a\right)_{s}}}{{\left(b\right)_{s}}s!}z^{s}}}$ `\KummerconfhyperM@{a}{b}{z} = \sum_{s=0}^{\infty}\frac{\Pochhammersym{a}{s}}{\Pochhammersym{b}{s}s!}z^{s}`		KummerM(a, b, z) = sum((pochhammer(a, s))/(pochhammer(b, s)factorial(s))(z)^(s), s = 0..infinity)	Hypergeometric1F1[a, b, z] == Sum[Divide[Pochhammer[a, s],Pochhammer[b, s](s)!](z)^(s), {s, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 252]
13.2.E3	${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=\sum_{s=0}^{\infty}% \frac{{\left(a\right)_{s}}}{\Gamma\left(b+s\right)s!}z^{s}}}$ `\OlverconfhyperM@{a}{b}{z} = \sum_{s=0}^{\infty}\frac{\Pochhammersym{a}{s}}{\EulerGamma@{b+s}s!}z^{s}`	${\displaystyle{\displaystyle\Re(b+s)>0}}$	KummerM(a, b, z)/GAMMA(b) = sum((pochhammer(a, s))/(GAMMA(b + s)factorial(s))(z)^(s), s = 0..infinity)	Hypergeometric1F1Regularized[a, b, z] == Sum[Divide[Pochhammer[a, s],Gamma[b + s](s)!](z)^(s), {s, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Failed [35 / 252] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.2.E4	${\displaystyle{\displaystyle M\left(a,b,z\right)=\Gamma\left(b\right){\mathbf{% M}}\left(a,b,z\right)}}$ `\KummerconfhyperM@{a}{b}{z} = \EulerGamma@{b}\OlverconfhyperM@{a}{b}{z}`	${\displaystyle{\displaystyle\Re b>0,\Re(b+s)>0}}$	KummerM(a, b, z) = GAMMA(b)*KummerM(a, b, z)/GAMMA(b)	Hypergeometric1F1[a, b, z] == Gamma[b]*Hypergeometric1F1Regularized[a, b, z]	Successful	Successful	-	Successful [Tested: 126]
13.2.E5	${\displaystyle{\displaystyle\lim_{b\to-n}\frac{M\left(a,b,z\right)}{\Gamma% \left(b\right)}={\mathbf{M}}\left(a,-n,z\right)}}$ `\lim_{b\to-n}\frac{\KummerconfhyperM@{a}{b}{z}}{\EulerGamma@{b}} = \OlverconfhyperM@{a}{-n}{z}`	${\displaystyle{\displaystyle\Re b>0,\Re((-n)+s)>0}}$	limit((KummerM(a, b, z))/(GAMMA(b)), b = - n) = KummerM(a, - n, z)/GAMMA(- n)	Limit[Divide[Hypergeometric1F1[a, b, z],Gamma[b]], b -> - n, GenerateConditions->None] == Hypergeometric1F1Regularized[a, - n, z]	Failure	Successful	Successful [Tested: 0]	Failed [112 / 126] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
13.2.E5	${\displaystyle{\displaystyle{\mathbf{M}}\left(a,-n,z\right)=\frac{{\left(a% \right)_{n+1}}}{(n+1)!}z^{n+1}M\left(a+n+1,n+2,z\right)}}$ `\OlverconfhyperM@{a}{-n}{z} = \frac{\Pochhammersym{a}{n+1}}{(n+1)!}z^{n+1}\KummerconfhyperM@{a+n+1}{n+2}{z}`	${\displaystyle{\displaystyle\Re b>0,\Re((-n)+s)>0}}$	KummerM(a, - n, z)/GAMMA(- n) = (pochhammer(a, n + 1))/(factorial(n + 1))(z)^(n + 1) KummerM(a + n + 1, n + 2, z)	Hypergeometric1F1Regularized[a, - n, z] == Divide[Pochhammer[a, n + 1],(n + 1)!](z)^(n + 1) Hypergeometric1F1[a + n + 1, n + 2, z]	Failure	Failure	Error	Successful [Tested: 126]
13.2.E7	${\displaystyle{\displaystyle U\left(-m,b,z\right)=(-1)^{m}{\left(b\right)_{m}}% M\left(-m,b,z\right)}}$ `\KummerconfhyperU@{-m}{b}{z} = (-1)^{m}\Pochhammersym{b}{m}\KummerconfhyperM@{-m}{b}{z}`		KummerU(- m, b, z) = (- 1)^(m)* pochhammer(b, m)*KummerM(- m, b, z)	HypergeometricU[- m, b, z] == (- 1)^(m)* Pochhammer[b, m]*Hypergeometric1F1[- m, b, z]	Failure	Failure	Error	Failed [7 / 126] Result: Indeterminate Test Values: {Rule[b, -2], Rule[m, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[b, -2], Rule[m, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.2.E7	${\displaystyle{\displaystyle(-1)^{m}{\left(b\right)_{m}}M\left(-m,b,z\right)=(% -1)^{m}\sum_{s=0}^{m}\genfrac{(}{)}{0.0pt}{}{m}{s}{\left(b+s\right)_{m-s}}(-z)% ^{s}}}$ `(-1)^{m}\Pochhammersym{b}{m}\KummerconfhyperM@{-m}{b}{z} = (-1)^{m}\sum_{s=0}^{m}\binom{m}{s}\Pochhammersym{b+s}{m-s}(-z)^{s}`		(- 1)^(m)* pochhammer(b, m)KummerM(- m, b, z) = (- 1)^(m) sum(binomial(m,s)pochhammer(b + s, m - s)(- z)^(s), s = 0..m)	(- 1)^(m)* Pochhammer[b, m]Hypergeometric1F1[- m, b, z] == (- 1)^(m) Sum[Binomial[m,s]Pochhammer[b + s, m - s](- z)^(s), {s, 0, m}, GenerateConditions->None]	Successful	Successful	Skip - symbolical successful subtest	Failed [21 / 126] Result: Indeterminate Test Values: {Rule[b, -2], Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[b, -2], Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
13.2.E8	${\displaystyle{\displaystyle U\left(a,a+n+1,z\right)=\frac{(-1)^{n}{\left(1-a-% n\right)_{n}}}{z^{a+n}}M\left(-n,1-a-n,z\right)}}$ `\KummerconfhyperU@{a}{a+n+1}{z} = \frac{(-1)^{n}\Pochhammersym{1-a-n}{n}}{z^{a+n}}\KummerconfhyperM@{-n}{1-a-n}{z}`		KummerU(a, a + n + 1, z) = ((- 1)^(n)* pochhammer(1 - a - n, n))/((z)^(a + n))*KummerM(- n, 1 - a - n, z)	HypergeometricU[a, a + n + 1, z] == Divide[(- 1)^(n)* Pochhammer[1 - a - n, n],(z)^(a + n)]*Hypergeometric1F1[- n, 1 - a - n, z]	Failure	Failure	Error	Failed [7 / 126] Result: Indeterminate Test Values: {Rule[a, -2], Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -2], Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.2.E8	${\displaystyle{\displaystyle\frac{(-1)^{n}{\left(1-a-n\right)_{n}}}{z^{a+n}}M% \left(-n,1-a-n,z\right)=z^{-a}\sum_{s=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{s}{% \left(a\right)_{s}}z^{-s}}}$ `\frac{(-1)^{n}\Pochhammersym{1-a-n}{n}}{z^{a+n}}\KummerconfhyperM@{-n}{1-a-n}{z} = z^{-a}\sum_{s=0}^{n}\binom{n}{s}\Pochhammersym{a}{s}z^{-s}`		((- 1)^(n)* pochhammer(1 - a - n, n))/((z)^(a + n))KummerM(- n, 1 - a - n, z) = (z)^(- a) sum(binomial(n,s)pochhammer(a, s)(z)^(- s), s = 0..n)	Divide[(- 1)^(n)* Pochhammer[1 - a - n, n],(z)^(a + n)]Hypergeometric1F1[- n, 1 - a - n, z] == (z)^(- a) Sum[Binomial[n,s]Pochhammer[a, s](z)^(- s), {s, 0, n}, GenerateConditions->None]	Failure	Failure	Error	Failed [7 / 126] Result: Indeterminate Test Values: {Rule[a, -2], Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -2], Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.2.E9	${\displaystyle{\displaystyle U\left(a,n+1,z\right)=\frac{(-1)^{n+1}}{n!\Gamma% \left(a-n\right)}\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}}{{\left(n+1% \right)_{k}}k!}z^{k}\left(\ln z+\psi\left(a+k\right)-\psi\left(1+k\right)-\psi% \left(n+k+1\right)\right)+\frac{1}{\Gamma\left(a\right)}\sum_{k=1}^{n}\frac{(k% -1)!{\left(1-a+k\right)_{n-k}}}{(n-k)!}z^{-k}}}$ `\KummerconfhyperU@{a}{n+1}{z} = \frac{(-1)^{n+1}}{n!\EulerGamma@{a-n}}\sum_{k=0}^{\infty}\frac{\Pochhammersym{a}{k}}{\Pochhammersym{n+1}{k}k!}z^{k}\left(\ln@@{z}+\digamma@{a+k}-\digamma@{1+k}-\digamma@{n+k+1}\right)+\frac{1}{\EulerGamma@{a}}\sum_{k=1}^{n}\frac{(k-1)!\Pochhammersym{1-a+k}{n-k}}{(n-k)!}z^{-k}`	${\displaystyle{\displaystyle\Re(a-n)>0,\Re a>0}}$	KummerU(a, n + 1, z) = ((- 1)^(n + 1))/(factorial(n)GAMMA(a - n))sum((pochhammer(a, k))/(pochhammer(n + 1, k)factorial(k))(z)^(k)(ln(z)+ Psi(a + k)- Psi(1 + k)- Psi(n + k + 1)), k = 0..infinity)+(1)/(GAMMA(a))sum((factorial(k - 1)pochhammer(1 - a + k, n - k))/(factorial(n - k))(z)^(- k), k = 1..n)	HypergeometricU[a, n + 1, z] == Divide[(- 1)^(n + 1),(n)!Gamma[a - n]]Sum[Divide[Pochhammer[a, k],Pochhammer[n + 1, k](k)!](z)^(k)(Log[z]+ PolyGamma[a + k]- PolyGamma[1 + k]- PolyGamma[n + k + 1]), {k, 0, Infinity}, GenerateConditions->None]+Divide[1,Gamma[a]]Sum[Divide[(k - 1)!Pochhammer[1 - a + k, n - k],(n - k)!](z)^(- k), {k, 1, n}, GenerateConditions->None]	Aborted	Aborted	Failed [7 / 14] Result: Float(undefined)+Float(undefined)I Test Values: {a = 2, z = 1/23^(1/2)+1/2I, n = 1} Result: Float(undefined)+Float(undefined)I Test Values: {a = 2, z = -1/2+1/2I3^(1/2), n = 1} ... skip entries to safe data	Skipped - Because timed out
13.2.E10	${\displaystyle{\displaystyle U\left(-m,n+1,z\right)=(-1)^{m}{\left(n+1\right)_% {m}}M\left(-m,n+1,z\right)}}$ `\KummerconfhyperU@{-m}{n+1}{z} = (-1)^{m}\Pochhammersym{n+1}{m}\KummerconfhyperM@{-m}{n+1}{z}`		KummerU(- m, n + 1, z) = (- 1)^(m)* pochhammer(n + 1, m)*KummerM(- m, n + 1, z)	HypergeometricU[- m, n + 1, z] == (- 1)^(m)* Pochhammer[n + 1, m]*Hypergeometric1F1[- m, n + 1, z]	Failure	Failure	Successful [Tested: 63]	Successful [Tested: 63]
13.2.E10	${\displaystyle{\displaystyle(-1)^{m}{\left(n+1\right)_{m}}M\left(-m,n+1,z% \right)=(-1)^{m}\sum_{s=0}^{m}\genfrac{(}{)}{0.0pt}{}{m}{s}{\left(n+s+1\right)% _{m-s}}(-z)^{s}}}$ `(-1)^{m}\Pochhammersym{n+1}{m}\KummerconfhyperM@{-m}{n+1}{z} = (-1)^{m}\sum_{s=0}^{m}\binom{m}{s}\Pochhammersym{n+s+1}{m-s}(-z)^{s}`		(- 1)^(m)* pochhammer(n + 1, m)KummerM(- m, n + 1, z) = (- 1)^(m) sum(binomial(m,s)pochhammer(n + s + 1, m - s)(- z)^(s), s = 0..m)	(- 1)^(m)* Pochhammer[n + 1, m]Hypergeometric1F1[- m, n + 1, z] == (- 1)^(m) Sum[Binomial[m,s]Pochhammer[n + s + 1, m - s](- z)^(s), {s, 0, m}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 63]	Successful [Tested: 63]
13.2.E11	${\displaystyle{\displaystyle U\left(a,-n,z\right)=z^{n+1}U\left(a+n+1,n+2,z% \right)}}$ `\KummerconfhyperU@{a}{-n}{z} = z^{n+1}\KummerconfhyperU@{a+n+1}{n+2}{z}`		KummerU(a, - n, z) = (z)^(n + 1)* KummerU(a + n + 1, n + 2, z)	HypergeometricU[a, - n, z] == (z)^(n + 1)* HypergeometricU[a + n + 1, n + 2, z]	Failure	Successful	Successful [Tested: 126]	Successful [Tested: 126]
13.2.E12	${\displaystyle{\displaystyle U\left(a,b,ze^{2\pi\mathrm{i}m}\right)=\frac{2\pi% \mathrm{i}e^{-\pi\mathrm{i}bm}\sin\left(\pi bm\right)}{\Gamma\left(1+a-b\right% )\sin\left(\pi b\right)}{\mathbf{M}}\left(a,b,z\right)+e^{-2\pi\mathrm{i}bm}U% \left(a,b,z\right)}}$ `\KummerconfhyperU@{a}{b}{ze^{2\pi\iunit m}} = \frac{2\pi\iunit e^{-\pi\iunit bm}\sin@{\pi bm}}{\EulerGamma@{1+a-b}\sin@{\pi b}}\OlverconfhyperM@{a}{b}{z}+e^{-2\pi\iunit bm}\KummerconfhyperU@{a}{b}{z}`	${\displaystyle{\displaystyle\Re(1+a-b)>0,\Re(b+s)>0}}$	KummerU(a, b, zexp(2PiIm)) = (2PiIexp(- PiIbm)sin(Pibm))/(GAMMA(1 + a - b)sin(Pib))KummerM(a, b, z)/GAMMA(b)+ exp(- 2PiIbm)*KummerU(a, b, z)	HypergeometricU[a, b, zExp[2PiIm]] == Divide[2PiIExp[- PiIbm]Sin[Pibm],Gamma[1 + a - b]Sin[Pib]]Hypergeometric1F1Regularized[a, b, z]+ Exp[- 2PiIbm]*HypergeometricU[a, b, z]	Failure	Failure	Failed [230 / 300] Result: -.101548209-1.031304846I Test Values: {a = -3/2, b = -3/2, z = 1/23^(1/2)+1/2I, m = 1} Result: -.101548218-1.031304823I Test Values: {a = -3/2, b = -3/2, z = 1/23^(1/2)+1/2I, m = 3} ... skip entries to safe data	Failed [230 / 300] Result: Complex[-0.10154820915393259, -1.0313048488210503] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.1015482091539317, -1.03130484882105] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[m, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
13.2.E33	${\displaystyle{\displaystyle\mathscr{W}\left\{{\mathbf{M}}\left(a,b,z\right),z% ^{1-b}{\mathbf{M}}\left(a-b+1,2-b,z\right)\right\}=\sin\left(\pi b\right)z^{-b% }e^{z}/\pi}}$ `\Wronskian@{\OlverconfhyperM@{a}{b}{z},z^{1-b}\OlverconfhyperM@{a-b+1}{2-b}{z}} = \sin@{\pi b}z^{-b}e^{z}/\pi`	${\displaystyle{\displaystyle\Re(b+s)>0,\Re((2-b)+s)>0}}$	(KummerM(a, b, z)/GAMMA(b))diff((z)^(1 - b) KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b), z)-diff(KummerM(a, b, z)/GAMMA(b), z)((z)^(1 - b) KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b)) = sin(Pib)(z)^(- b)* exp(z)/Pi	Wronskian[{Hypergeometric1F1Regularized[a, b, z], (z)^(1 - b)* Hypergeometric1F1Regularized[a - b + 1, 2 - b, z]}, z] == Sin[Pib](z)^(- b)* Exp[z]/Pi	Failure	Failure	Error	Successful [Tested: 252]
13.2.E34	${\displaystyle{\displaystyle\mathscr{W}\left\{{\mathbf{M}}\left(a,b,z\right),U% \left(a,b,z\right)\right\}=-\ifrac{z^{-b}e^{z}}{\Gamma\left(a\right)}}}$ `\Wronskian@{\OlverconfhyperM@{a}{b}{z},\KummerconfhyperU@{a}{b}{z}} = -\ifrac{z^{-b}e^{z}}{\EulerGamma@{a}}`	${\displaystyle{\displaystyle\Re a>0,\Re(b+s)>0}}$	(KummerM(a, b, z)/GAMMA(b))diff(KummerU(a, b, z), z)-diff(KummerM(a, b, z)/GAMMA(b), z)(KummerU(a, b, z)) = -((z)^(- b)* exp(z))/(GAMMA(a))	Wronskian[{Hypergeometric1F1Regularized[a, b, z], HypergeometricU[a, b, z]}, z] == -Divide[(z)^(- b)* Exp[z],Gamma[a]]	Failure	Failure	Error	Successful [Tested: 126]
13.2.E35	${\displaystyle{\displaystyle\mathscr{W}\left\{{\mathbf{M}}\left(a,b,z\right),e% ^{z}U\left(b-a,b,e^{+\pi\mathrm{i}}z\right)\right\}=\ifrac{e^{-b\pi\mathrm{i}}% z^{-b}e^{z}}{\Gamma\left(b-a\right)}}}$ `\Wronskian@{\OlverconfhyperM@{a}{b}{z},e^{z}\KummerconfhyperU@{b-a}{b}{e^{+\pi\iunit}z}} = \ifrac{e^{- b\pi\iunit}z^{-b}e^{z}}{\EulerGamma@{b-a}}`	${\displaystyle{\displaystyle\Re(b-a)>0,\Re(b+s)>0}}$	(KummerM(a, b, z)/GAMMA(b))diff(exp(z)KummerU(b - a, b, exp(+ PiI)z), z)-diff(KummerM(a, b, z)/GAMMA(b), z)(exp(z)KummerU(b - a, b, exp(+ PiI)z)) = (exp(- bPiI)(z)^(- b) exp(z))/(GAMMA(b - a))	Wronskian[{Hypergeometric1F1Regularized[a, b, z], Exp[z]HypergeometricU[b - a, b, Exp[+ PiI]z]}, z] == Divide[Exp[- bPiI](z)^(- b)* Exp[z],Gamma[b - a]]	Failure	Failure	Failed [23 / 105] Result: -.6693440963-2.281274239I Test Values: {a = -3/2, b = 3/2, z = 1/23^(1/2)+1/2I} Result: -.4620307839+.3929465556I Test Values: {a = -3/2, b = 3/2, z = -1/2+1/2I3^(1/2)} ... skip entries to safe data	Failed [20 / 105] Result: Complex[-0.6693440961046373, -2.2812742393329124] Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.46203078407110554, 0.39294655583435506] Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.2.E35	${\displaystyle{\displaystyle\mathscr{W}\left\{{\mathbf{M}}\left(a,b,z\right),e% ^{z}U\left(b-a,b,e^{-\pi\mathrm{i}}z\right)\right\}=\ifrac{e^{+b\pi\mathrm{i}}% z^{-b}e^{z}}{\Gamma\left(b-a\right)}}}$ `\Wronskian@{\OlverconfhyperM@{a}{b}{z},e^{z}\KummerconfhyperU@{b-a}{b}{e^{-\pi\iunit}z}} = \ifrac{e^{+ b\pi\iunit}z^{-b}e^{z}}{\EulerGamma@{b-a}}`	${\displaystyle{\displaystyle\Re(b-a)>0,\Re(b+s)>0}}$	(KummerM(a, b, z)/GAMMA(b))diff(exp(z)KummerU(b - a, b, exp(- PiI)z), z)-diff(KummerM(a, b, z)/GAMMA(b), z)(exp(z)KummerU(b - a, b, exp(- PiI)z)) = (exp(+ bPiI)(z)^(- b) exp(z))/(GAMMA(b - a))	Wronskian[{Hypergeometric1F1Regularized[a, b, z], Exp[z]HypergeometricU[b - a, b, Exp[- PiI]z]}, z] == Divide[Exp[+ bPiI](z)^(- b)* Exp[z],Gamma[b - a]]	Failure	Failure	Failed [53 / 105] Result: -1.068139482+1.255929884I Test Values: {a = -3/2, b = 3/2, z = 1/2-1/2I3^(1/2)} Result: .1184211651-.4036057902I Test Values: {a = -3/2, b = 3/2, z = -1/23^(1/2)-1/2I} ... skip entries to safe data	Failed [50 / 105] Result: Complex[-1.0681394822800954, 1.2559298845291709] Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} Result: Complex[0.11842116492450601, -0.40360579036441874] Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} ... skip entries to safe data
13.2.E36	${\displaystyle{\displaystyle\mathscr{W}\left\{z^{1-b}{\mathbf{M}}\left(a-b+1,2% -b,z\right),U\left(a,b,z\right)\right\}=-\ifrac{z^{-b}e^{z}}{\Gamma\left(a-b+1% \right)}}}$ `\Wronskian@{z^{1-b}\OlverconfhyperM@{a-b+1}{2-b}{z},\KummerconfhyperU@{a}{b}{z}} = -\ifrac{z^{-b}e^{z}}{\EulerGamma@{a-b+1}}`	${\displaystyle{\displaystyle\Re(a-b+1)>0,\Re((2-b)+s)>0}}$	((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b))diff(KummerU(a, b, z), z)-diff((z)^(1 - b) KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b), z)(KummerU(a, b, z)) = -((z)^(- b) exp(z))/(GAMMA(a - b + 1))	Wronskian[{(z)^(1 - b)* Hypergeometric1F1Regularized[a - b + 1, 2 - b, z], HypergeometricU[a, b, z]}, z] == -Divide[(z)^(- b)* Exp[z],Gamma[a - b + 1]]	Failure	Failure	Error	Successful [Tested: 161]
13.2.E37	${\displaystyle{\displaystyle\mathscr{W}\left\{z^{1-b}{\mathbf{M}}\left(a-b+1,2% -b,z\right),e^{z}U\left(b-a,b,e^{+\pi\mathrm{i}}z\right)\right\}=-\ifrac{z^{-b% }e^{z}}{\Gamma\left(1-a\right)}}}$ `\Wronskian@{z^{1-b}\OlverconfhyperM@{a-b+1}{2-b}{z},e^{z}\KummerconfhyperU@{b-a}{b}{e^{+\pi\iunit}z}} = -\ifrac{z^{-b}e^{z}}{\EulerGamma@{1-a}}`	${\displaystyle{\displaystyle\Re(1-a)>0,\Re((2-b)+s)>0}}$	((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b))diff(exp(z)KummerU(b - a, b, exp(+ PiI)z), z)-diff((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b), z)(exp(z)KummerU(b - a, b, exp(+ PiI)z)) = -((z)^(- b)* exp(z))/(GAMMA(1 - a))	Wronskian[{(z)^(1 - b)* Hypergeometric1F1Regularized[a - b + 1, 2 - b, z], Exp[z]HypergeometricU[b - a, b, Exp[+ PiI]z]}, z] == -Divide[(z)^(- b) Exp[z],Gamma[1 - a]]	Failure	Aborted	Error	Successful [Tested: 168]
13.2.E37	${\displaystyle{\displaystyle\mathscr{W}\left\{z^{1-b}{\mathbf{M}}\left(a-b+1,2% -b,z\right),e^{z}U\left(b-a,b,e^{-\pi\mathrm{i}}z\right)\right\}=-\ifrac{z^{-b% }e^{z}}{\Gamma\left(1-a\right)}}}$ `\Wronskian@{z^{1-b}\OlverconfhyperM@{a-b+1}{2-b}{z},e^{z}\KummerconfhyperU@{b-a}{b}{e^{-\pi\iunit}z}} = -\ifrac{z^{-b}e^{z}}{\EulerGamma@{1-a}}`	${\displaystyle{\displaystyle\Re(1-a)>0,\Re((2-b)+s)>0}}$	((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b))diff(exp(z)KummerU(b - a, b, exp(- PiI)z), z)-diff((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b), z)(exp(z)KummerU(b - a, b, exp(- PiI)z)) = -((z)^(- b)* exp(z))/(GAMMA(1 - a))	Wronskian[{(z)^(1 - b)* Hypergeometric1F1Regularized[a - b + 1, 2 - b, z], Exp[z]HypergeometricU[b - a, b, Exp[- PiI]z]}, z] == -Divide[(z)^(- b) Exp[z],Gamma[1 - a]]	Failure	Aborted	Error	Successful [Tested: 168]
13.2.E38	${\displaystyle{\displaystyle\mathscr{W}\left\{U\left(a,b,z\right),e^{z}U\left(% b-a,b,e^{+\pi\mathrm{i}}z\right)\right\}=e^{+(a-b)\pi\mathrm{i}}z^{-b}e^{z}}}$ `\Wronskian@{\KummerconfhyperU@{a}{b}{z},e^{z}\KummerconfhyperU@{b-a}{b}{e^{+\pi\iunit}z}} = e^{+(a-b)\pi\iunit}z^{-b}e^{z}`		(KummerU(a, b, z))diff(exp(z)KummerU(b - a, b, exp(+ PiI)z), z)-diff(KummerU(a, b, z), z)(exp(z)KummerU(b - a, b, exp(+ PiI)z)) = exp(+(a - b)PiI)(z)^(- b) exp(z)	Wronskian[{HypergeometricU[a, b, z], Exp[z]HypergeometricU[b - a, b, Exp[+ PiI]z]}, z] == Exp[+(a - b)PiI](z)^(- b)* Exp[z]	Failure	Aborted	Failed [38 / 252] Result: 4.753561418-.1121990572I Test Values: {a = -3/2, b = -2, z = 1/23^(1/2)+1/2I} Result: -1.142634185-.4073142366I Test Values: {a = -3/2, b = -2, z = -1/2+1/2I3^(1/2)} ... skip entries to safe data	Failed [32 / 252] Result: Complex[4.753561408836843, -0.1121990577209182] Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.1426341834354088, -0.40731423683768475] Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.2.E38	${\displaystyle{\displaystyle\mathscr{W}\left\{U\left(a,b,z\right),e^{z}U\left(% b-a,b,e^{-\pi\mathrm{i}}z\right)\right\}=e^{-(a-b)\pi\mathrm{i}}z^{-b}e^{z}}}$ `\Wronskian@{\KummerconfhyperU@{a}{b}{z},e^{z}\KummerconfhyperU@{b-a}{b}{e^{-\pi\iunit}z}} = e^{-(a-b)\pi\iunit}z^{-b}e^{z}`		(KummerU(a, b, z))diff(exp(z)KummerU(b - a, b, exp(- PiI)z), z)-diff(KummerU(a, b, z), z)(exp(z)KummerU(b - a, b, exp(- PiI)z)) = exp(-(a - b)PiI)(z)^(- b) exp(z)	Wronskian[{HypergeometricU[a, b, z], Exp[z]HypergeometricU[b - a, b, Exp[- PiI]z]}, z] == Exp[-(a - b)PiI](z)^(- b)* Exp[z]	Failure	Aborted	Failed [80 / 252] Result: .5941419621-3.243473855I Test Values: {a = -3/2, b = -2, z = 1/2-1/2I3^(1/2)} Result: -.4376938533+.7184072077I Test Values: {a = -3/2, b = -2, z = -1/23^(1/2)-1/2I} ... skip entries to safe data	Failed [80 / 252] Result: Complex[0.5941419683502733, -3.243473853028733] Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} Result: Complex[-0.4376938536795689, 0.7184072074542298] Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} ... skip entries to safe data
13.2.E39	${\displaystyle{\displaystyle M\left(a,b,z\right)=e^{z}M\left(b-a,b,-z\right)}}$ `\KummerconfhyperM@{a}{b}{z} = e^{z}\KummerconfhyperM@{b-a}{b}{-z}`		KummerM(a, b, z) = exp(z)*KummerM(b - a, b, - z)	Hypergeometric1F1[a, b, z] == Exp[z]*Hypergeometric1F1[b - a, b, - z]	Failure	Successful	Error	Failed [42 / 252] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.2.E40	${\displaystyle{\displaystyle U\left(a,b,z\right)=z^{1-b}U\left(a-b+1,2-b,z% \right)}}$ `\KummerconfhyperU@{a}{b}{z} = z^{1-b}\KummerconfhyperU@{a-b+1}{2-b}{z}`		KummerU(a, b, z) = (z)^(1 - b)* KummerU(a - b + 1, 2 - b, z)	HypergeometricU[a, b, z] == (z)^(1 - b)* HypergeometricU[a - b + 1, 2 - b, z]	Successful	Successful	-	Successful [Tested: 252]
13.2.E41	${\displaystyle{\displaystyle\frac{1}{\Gamma\left(b\right)}M\left(a,b,z\right)=% \frac{e^{-a\pi\mathrm{i}}}{\Gamma\left(b-a\right)}U\left(a,b,z\right)+\frac{e^% {+(b-a)\pi\mathrm{i}}}{\Gamma\left(a\right)}e^{z}U\left(b-a,b,e^{+\pi\mathrm{i% }}z\right)}}$ `\frac{1}{\EulerGamma@{b}}\KummerconfhyperM@{a}{b}{z} = \frac{e^{- a\pi\iunit}}{\EulerGamma@{b-a}}\KummerconfhyperU@{a}{b}{z}+\frac{e^{+(b-a)\pi\iunit}}{\EulerGamma@{a}}e^{z}\KummerconfhyperU@{b-a}{b}{e^{+\pi\iunit}z}`	${\displaystyle{\displaystyle\Re b>0,\Re(b-a)>0,\Re a>0}}$	(1)/(GAMMA(b))KummerM(a, b, z) = (exp(- aPiI))/(GAMMA(b - a))KummerU(a, b, z)+(exp(+(b - a)PiI))/(GAMMA(a))exp(z)KummerU(b - a, b, exp(+ PiI)z)	Divide[1,Gamma[b]]Hypergeometric1F1[a, b, z] == Divide[Exp[- aPiI],Gamma[b - a]]HypergeometricU[a, b, z]+Divide[Exp[+(b - a)PiI],Gamma[a]]Exp[z]HypergeometricU[b - a, b, Exp[+ PiI]z]	Failure	Failure	Failed [6 / 21] Result: 3.583210384+1.512741910I Test Values: {a = 3/2, b = 2, z = 1/23^(1/2)+1/2I} Result: 1.096602540+.7868998856I Test Values: {a = 3/2, b = 2, z = -1/2+1/2I3^(1/2)} ... skip entries to safe data	Failed [6 / 21] Result: Complex[3.583210382577498, 1.512741908514331] Test Values: {Rule[a, 1.5], Rule[b, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.096602539454242, 0.7868998849931845] Test Values: {Rule[a, 1.5], Rule[b, 2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.2.E41	${\displaystyle{\displaystyle\frac{1}{\Gamma\left(b\right)}M\left(a,b,z\right)=% \frac{e^{+a\pi\mathrm{i}}}{\Gamma\left(b-a\right)}U\left(a,b,z\right)+\frac{e^% {-(b-a)\pi\mathrm{i}}}{\Gamma\left(a\right)}e^{z}U\left(b-a,b,e^{-\pi\mathrm{i% }}z\right)}}$ `\frac{1}{\EulerGamma@{b}}\KummerconfhyperM@{a}{b}{z} = \frac{e^{+ a\pi\iunit}}{\EulerGamma@{b-a}}\KummerconfhyperU@{a}{b}{z}+\frac{e^{-(b-a)\pi\iunit}}{\EulerGamma@{a}}e^{z}\KummerconfhyperU@{b-a}{b}{e^{-\pi\iunit}z}`	${\displaystyle{\displaystyle\Re b>0,\Re(b-a)>0,\Re a>0}}$	(1)/(GAMMA(b))KummerM(a, b, z) = (exp(+ aPiI))/(GAMMA(b - a))KummerU(a, b, z)+(exp(-(b - a)PiI))/(GAMMA(a))exp(z)KummerU(b - a, b, exp(- PiI)z)	Divide[1,Gamma[b]]Hypergeometric1F1[a, b, z] == Divide[Exp[+ aPiI],Gamma[b - a]]HypergeometricU[a, b, z]+Divide[Exp[-(b - a)PiI],Gamma[a]]Exp[z]HypergeometricU[b - a, b, Exp[- PiI]z]	Failure	Failure	Failed [15 / 21] Result: 2.239690726-1.798422043I Test Values: {a = 3/2, b = 2, z = 1/2-1/2I3^(1/2)} Result: .9984283068-.3592011980I Test Values: {a = 3/2, b = 2, z = -1/23^(1/2)-1/2I} ... skip entries to safe data	Failed [15 / 21] Result: Complex[2.239690726834086, -1.7984220417127512] Test Values: {Rule[a, 1.5], Rule[b, 2], Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} Result: Complex[0.9984283065924617, -0.35920119796837185] Test Values: {Rule[a, 1.5], Rule[b, 2], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} ... skip entries to safe data
13.2.E42	${\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{\Gamma\left(1-b\right)}% {\Gamma\left(a-b+1\right)}M\left(a,b,z\right)+\frac{\Gamma\left(b-1\right)}{% \Gamma\left(a\right)}z^{1-b}M\left(a-b+1,2-b,z\right)}}$ `\KummerconfhyperU@{a}{b}{z} = \frac{\EulerGamma@{1-b}}{\EulerGamma@{a-b+1}}\KummerconfhyperM@{a}{b}{z}+\frac{\EulerGamma@{b-1}}{\EulerGamma@{a}}z^{1-b}\KummerconfhyperM@{a-b+1}{2-b}{z}`	${\displaystyle{\displaystyle\Re(1-b)>0,\Re(a-b+1)>0,\Re(b-1)>0,\Re a>0}}$	KummerU(a, b, z) = (GAMMA(1 - b))/(GAMMA(a - b + 1))KummerM(a, b, z)+(GAMMA(b - 1))/(GAMMA(a))(z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)	HypergeometricU[a, b, z] == Divide[Gamma[1 - b],Gamma[a - b + 1]]Hypergeometric1F1[a, b, z]+Divide[Gamma[b - 1],Gamma[a]](z)^(1 - b)* Hypergeometric1F1[a - b + 1, 2 - b, z]	Successful	Successful	-	-
13.3.E1	${\displaystyle{\displaystyle(b-a)M\left(a-1,b,z\right)+(2a-b+z)M\left(a,b,z% \right)-aM\left(a+1,b,z\right)=0}}$ `(b-a)\KummerconfhyperM@{a-1}{b}{z}+(2a-b+z)\KummerconfhyperM@{a}{b}{z}-a\KummerconfhyperM@{a+1}{b}{z} = 0`		(b - a)KummerM(a - 1, b, z)+(2a - b + z)KummerM(a, b, z)- aKummerM(a + 1, b, z) = 0	(b - a)Hypergeometric1F1[a - 1, b, z]+(2a - b + z)Hypergeometric1F1[a, b, z]- aHypergeometric1F1[a + 1, b, z] == 0	Successful	Successful	-	Failed [42 / 252] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.3.E2	${\displaystyle{\displaystyle b(b-1)M\left(a,b-1,z\right)+b(1-b-z)M\left(a,b,z% \right)+z(b-a)M\left(a,b+1,z\right)=0}}$ `b(b-1)\KummerconfhyperM@{a}{b-1}{z}+b(1-b-z)\KummerconfhyperM@{a}{b}{z}+z(b-a)\KummerconfhyperM@{a}{b+1}{z} = 0`		b(b - 1)KummerM(a, b - 1, z)+ b(1 - b - z)KummerM(a, b, z)+ z(b - a)KummerM(a, b + 1, z) = 0	b(b - 1)Hypergeometric1F1[a, b - 1, z]+ b(1 - b - z)Hypergeometric1F1[a, b, z]+ z(b - a)Hypergeometric1F1[a, b + 1, z] == 0	Successful	Successful	-	Failed [42 / 252] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.3.E3	${\displaystyle{\displaystyle(a-b+1)M\left(a,b,z\right)-aM\left(a+1,b,z\right)+% (b-1)M\left(a,b-1,z\right)=0}}$ `(a-b+1)\KummerconfhyperM@{a}{b}{z}-a\KummerconfhyperM@{a+1}{b}{z}+(b-1)\KummerconfhyperM@{a}{b-1}{z} = 0`		(a - b + 1)KummerM(a, b, z)- aKummerM(a + 1, b, z)+(b - 1)*KummerM(a, b - 1, z) = 0	(a - b + 1)Hypergeometric1F1[a, b, z]- aHypergeometric1F1[a + 1, b, z]+(b - 1)*Hypergeometric1F1[a, b - 1, z] == 0	Successful	Successful	-	Failed [35 / 252] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.3.E4	${\displaystyle{\displaystyle bM\left(a,b,z\right)-bM\left(a-1,b,z\right)-zM% \left(a,b+1,z\right)=0}}$ `b\KummerconfhyperM@{a}{b}{z}-b\KummerconfhyperM@{a-1}{b}{z}-z\KummerconfhyperM@{a}{b+1}{z} = 0`		bKummerM(a, b, z)- bKummerM(a - 1, b, z)- z*KummerM(a, b + 1, z) = 0	bHypergeometric1F1[a, b, z]- bHypergeometric1F1[a - 1, b, z]- z*Hypergeometric1F1[a, b + 1, z] == 0	Successful	Successful	-	Failed [42 / 252] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.3.E5	${\displaystyle{\displaystyle b(a+z)M\left(a,b,z\right)+z(a-b)M\left(a,b+1,z% \right)-abM\left(a+1,b,z\right)=0}}$ `b(a+z)\KummerconfhyperM@{a}{b}{z}+z(a-b)\KummerconfhyperM@{a}{b+1}{z}-ab\KummerconfhyperM@{a+1}{b}{z} = 0`		b(a + z)KummerM(a, b, z)+ z(a - b)KummerM(a, b + 1, z)- abKummerM(a + 1, b, z) = 0	b(a + z)Hypergeometric1F1[a, b, z]+ z(a - b)Hypergeometric1F1[a, b + 1, z]- abHypergeometric1F1[a + 1, b, z] == 0	Successful	Successful	-	Failed [42 / 252] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.3.E6	${\displaystyle{\displaystyle(a-1+z)M\left(a,b,z\right)+(b-a)M\left(a-1,b,z% \right)+(1-b)M\left(a,b-1,z\right)=0}}$ `(a-1+z)\KummerconfhyperM@{a}{b}{z}+(b-a)\KummerconfhyperM@{a-1}{b}{z}+(1-b)\KummerconfhyperM@{a}{b-1}{z} = 0`		(a - 1 + z)KummerM(a, b, z)+(b - a)KummerM(a - 1, b, z)+(1 - b)*KummerM(a, b - 1, z) = 0	(a - 1 + z)Hypergeometric1F1[a, b, z]+(b - a)Hypergeometric1F1[a - 1, b, z]+(1 - b)*Hypergeometric1F1[a, b - 1, z] == 0	Successful	Successful	-	Failed [42 / 252] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.3.E7	${\displaystyle{\displaystyle U\left(a-1,b,z\right)+(b-2a-z)U\left(a,b,z\right)% +a(a-b+1)U\left(a+1,b,z\right)=0}}$ `\KummerconfhyperU@{a-1}{b}{z}+(b-2a-z)\KummerconfhyperU@{a}{b}{z}+a(a-b+1)\KummerconfhyperU@{a+1}{b}{z} = 0`		KummerU(a - 1, b, z)+(b - 2a - z)KummerU(a, b, z)+ a(a - b + 1)KummerU(a + 1, b, z) = 0	HypergeometricU[a - 1, b, z]+(b - 2a - z)HypergeometricU[a, b, z]+ a(a - b + 1)HypergeometricU[a + 1, b, z] == 0	Successful	Successful	-	Successful [Tested: 252]
13.3.E8	${\displaystyle{\displaystyle(b-a-1)U\left(a,b-1,z\right)+(1-b-z)U\left(a,b,z% \right)+zU\left(a,b+1,z\right)=0}}$ `(b-a-1)\KummerconfhyperU@{a}{b-1}{z}+(1-b-z)\KummerconfhyperU@{a}{b}{z}+z\KummerconfhyperU@{a}{b+1}{z} = 0`		(b - a - 1)KummerU(a, b - 1, z)+(1 - b - z)KummerU(a, b, z)+ z*KummerU(a, b + 1, z) = 0	(b - a - 1)HypergeometricU[a, b - 1, z]+(1 - b - z)HypergeometricU[a, b, z]+ z*HypergeometricU[a, b + 1, z] == 0	Successful	Successful	-	Successful [Tested: 252]
13.3.E9	${\displaystyle{\displaystyle U\left(a,b,z\right)-aU\left(a+1,b,z\right)-U\left% (a,b-1,z\right)=0}}$ `\KummerconfhyperU@{a}{b}{z}-a\KummerconfhyperU@{a+1}{b}{z}-\KummerconfhyperU@{a}{b-1}{z} = 0`		KummerU(a, b, z)- a*KummerU(a + 1, b, z)- KummerU(a, b - 1, z) = 0	HypergeometricU[a, b, z]- a*HypergeometricU[a + 1, b, z]- HypergeometricU[a, b - 1, z] == 0	Successful	Successful	-	Successful [Tested: 252]
13.3.E10	${\displaystyle{\displaystyle(b-a)U\left(a,b,z\right)+U\left(a-1,b,z\right)-zU% \left(a,b+1,z\right)=0}}$ `(b-a)\KummerconfhyperU@{a}{b}{z}+\KummerconfhyperU@{a-1}{b}{z}-z\KummerconfhyperU@{a}{b+1}{z} = 0`		(b - a)KummerU(a, b, z)+ KummerU(a - 1, b, z)- zKummerU(a, b + 1, z) = 0	(b - a)HypergeometricU[a, b, z]+ HypergeometricU[a - 1, b, z]- zHypergeometricU[a, b + 1, z] == 0	Successful	Successful	-	Successful [Tested: 252]
13.3.E11	${\displaystyle{\displaystyle(a+z)U\left(a,b,z\right)-zU\left(a,b+1,z\right)+a(% b-a-1)U\left(a+1,b,z\right)=0}}$ `(a+z)\KummerconfhyperU@{a}{b}{z}-z\KummerconfhyperU@{a}{b+1}{z}+a(b-a-1)\KummerconfhyperU@{a+1}{b}{z} = 0`		(a + z)KummerU(a, b, z)- zKummerU(a, b + 1, z)+ a(b - a - 1)KummerU(a + 1, b, z) = 0	(a + z)HypergeometricU[a, b, z]- zHypergeometricU[a, b + 1, z]+ a(b - a - 1)HypergeometricU[a + 1, b, z] == 0	Successful	Successful	-	Successful [Tested: 252]
13.3.E12	${\displaystyle{\displaystyle(a-1+z)U\left(a,b,z\right)-U\left(a-1,b,z\right)+(% a-b+1)U\left(a,b-1,z\right)=0}}$ `(a-1+z)\KummerconfhyperU@{a}{b}{z}-\KummerconfhyperU@{a-1}{b}{z}+(a-b+1)\KummerconfhyperU@{a}{b-1}{z} = 0`		(a - 1 + z)KummerU(a, b, z)- KummerU(a - 1, b, z)+(a - b + 1)KummerU(a, b - 1, z) = 0	(a - 1 + z)HypergeometricU[a, b, z]- HypergeometricU[a - 1, b, z]+(a - b + 1)HypergeometricU[a, b - 1, z] == 0	Successful	Successful	-	Successful [Tested: 252]
13.3.E13	${\displaystyle{\displaystyle(a+1)zM\left(a+2,b+2,z\right)+(b+1)(b-z)M\left(a+1% ,b+1,z\right)-b(b+1)M\left(a,b,z\right)=0}}$ `(a+1)z\KummerconfhyperM@{a+2}{b+2}{z}+(b+1)(b-z)\KummerconfhyperM@{a+1}{b+1}{z}-b(b+1)\KummerconfhyperM@{a}{b}{z} = 0`		(a + 1)zKummerM(a + 2, b + 2, z)+(b + 1)(b - z)KummerM(a + 1, b + 1, z)- b(b + 1)KummerM(a, b, z) = 0	(a + 1)zHypergeometric1F1[a + 2, b + 2, z]+(b + 1)(b - z)Hypergeometric1F1[a + 1, b + 1, z]- b(b + 1)Hypergeometric1F1[a, b, z] == 0	Successful	Successful	-	Failed [35 / 252] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.3.E14	${\displaystyle{\displaystyle(a+1)zU\left(a+2,b+2,z\right)+(z-b)U\left(a+1,b+1,% z\right)-U\left(a,b,z\right)=0}}$ `(a+1)z\KummerconfhyperU@{a+2}{b+2}{z}+(z-b)\KummerconfhyperU@{a+1}{b+1}{z}-\KummerconfhyperU@{a}{b}{z} = 0`		(a + 1)zKummerU(a + 2, b + 2, z)+(z - b)*KummerU(a + 1, b + 1, z)- KummerU(a, b, z) = 0	(a + 1)zHypergeometricU[a + 2, b + 2, z]+(z - b)*HypergeometricU[a + 1, b + 1, z]- HypergeometricU[a, b, z] == 0	Successful	Successful	-	Successful [Tested: 252]
13.3.E15	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}M\left(a,b,z\right)=% \frac{a}{b}M\left(a+1,b+1,z\right)}}$ `\deriv{}{z}\KummerconfhyperM@{a}{b}{z} = \frac{a}{b}\KummerconfhyperM@{a+1}{b+1}{z}`		diff(KummerM(a, b, z), z) = (a)/(b)*KummerM(a + 1, b + 1, z)	D[Hypergeometric1F1[a, b, z], z] == Divide[a,b]*Hypergeometric1F1[a + 1, b + 1, z]	Successful	Successful	-	Successful [Tested: 252]
13.3.E16	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}M\left(a% ,b,z\right)=\frac{{\left(a\right)_{n}}}{{\left(b\right)_{n}}}M\left(a+n,b+n,z% \right)}}$ `\deriv[n]{}{z}\KummerconfhyperM@{a}{b}{z} = \frac{\Pochhammersym{a}{n}}{\Pochhammersym{b}{n}}\KummerconfhyperM@{a+n}{b+n}{z}`		diff(KummerM(a, b, z), [z$(n)]) = (pochhammer(a, n))/(pochhammer(b, n))*KummerM(a + n, b + n, z)	D[Hypergeometric1F1[a, b, z], {z, n}] == Divide[Pochhammer[a, n],Pochhammer[b, n]]*Hypergeometric1F1[a + n, b + n, z]	Successful	Failure	-	Failed [42 / 300] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
13.3.E17	${\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}% \left(z^{a-1}M\left(a,b,z\right)\right)={\left(a\right)_{n}}z^{a+n-1}M\left(a+% n,b,z\right)}}$ `\left(z\deriv{}{z}z\right)^{n}\left(z^{a-1}\KummerconfhyperM@{a}{b}{z}\right) = \Pochhammersym{a}{n}z^{a+n-1}\KummerconfhyperM@{a+n}{b}{z}`		(zdiff(z, z))^(n)((z)^(a - 1)* KummerM(a, b, z)) = pochhammer(a, n)(z)^(a + n - 1) KummerM(a + n, b, z)	(zD[z, z])^(n)((z)^(a - 1)* Hypergeometric1F1[a, b, z]) == Pochhammer[a, n](z)^(a + n - 1) Hypergeometric1F1[a + n, b, z]	Failure	Failure	Failed [300 / 300] Result: 3.392872106-2.234328368I Test Values: {a = -3/2, b = -3/2, z = 1/23^(1/2)+1/2I, n = 1} Result: 1.628540387-.5000628115I Test Values: {a = -3/2, b = -3/2, z = 1/23^(1/2)+1/2I, n = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[3.392872106018638, -2.234328368828302] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.628540387739978, -0.5000628109822313] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
13.3.E18	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^% {b-1}M\left(a,b,z\right)\right)={\left(b-n\right)_{n}}z^{b-n-1}M\left(a,b-n,z% \right)}}$ `\deriv[n]{}{z}\left(z^{b-1}\KummerconfhyperM@{a}{b}{z}\right) = \Pochhammersym{b-n}{n}z^{b-n-1}\KummerconfhyperM@{a}{b-n}{z}`		diff((z)^(b - 1)* KummerM(a, b, z), [z$(n)]) = pochhammer(b - n, n)(z)^(b - n - 1) KummerM(a, b - n, z)	D[(z)^(b - 1)* Hypergeometric1F1[a, b, z], {z, n}] == Pochhammer[b - n, n](z)^(b - n - 1) Hypergeometric1F1[a, b - n, z]	Failure	Failure	Error	Failed [300 / 300] Result: Plus[Complex[-0.23854907479223686, -4.055477620017901], Times[Complex[-0.2588190451025206, -0.9659258262890682], DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[, -1.5], Plus[-1, Times[-1, ], 1], Plus[Times[-1, ], 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], []], Times[-1, Plus[-1, Times[-1, ], 1], Plus[Power[, 2], Power[, 3], Times[2, , -1.5], Times[2, Power[, 2], -1.5], Power[-1.5, 2], Times[, Power[-1.5, 2]], Times[-1, , 1], Times[-1, Power[, 2], 1], Times[-1, -1.5, 1], Times[-1, , -1.5, 1], Times[-1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-2, Power[, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, , 1, Pow<syntaxhighlight lang=mathematica>Result: Plus[Complex[7.020632087540109, 10.129888243360973], Times[Complex[-1.4142135623730947, -1.414213562373095], DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[, -1.5], Plus[-1, Times[-1, ], 2], Plus[Times[-1, ], 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], []], Times[-1, Plus[-1, Times[-1, ], 2], Plus[Power[, 2], Power[, 3], Times[2, , -1.5], Times[2, Power[, 2], -1.5], Power[-1.5, 2], Times[, Power[-1.5, 2]], Times[-1, , 2], Times[-1, Power[, 2], 2], Times[-1, -1.5, 2], Times[-1, , -1.5, 2], Times[-1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-2, Power[, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, , 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1.5, 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[1, ]]], Times[Plus[1, ], Plus[, -1.5, Times[-1, 2]], Plus[-2, Times[-4, ], Times[-2, Power[, 2]], Times[-3, -1.5], Times[-2, , -1.5], Times[2, 2], Times[2, , 2], Times[-1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Times[, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[2, ]]], Times[Plus[1, ], Plus[2, ], Plus[, -1.5, Times[-1, 2]], Plus[1, , -1.5, Times[-1, 2]], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Times[Binomial[Plus[-1, -1.5], 2], Hypergeometric1F1[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[2], Times[Binomial[Plus[-1, -1.5], 2], Plus[Hypergeometric1F1[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1.5, 2, Power[Plus[Power[-1.5, 2], Times[-1, -1.5, 2]], -1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Hypergeometric1F1[Plus[1, -1.5], Plus[1, -1.5], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][3.0]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
13.3.E19	${\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}% \left(z^{b-a-1}e^{-z}M\left(a,b,z\right)\right)={\left(b-a\right)_{n}}z^{b-a+n% -1}e^{-z}M\left(a-n,b,z\right)}}$ `\left(z\deriv{}{z}z\right)^{n}\left(z^{b-a-1}e^{-z}\KummerconfhyperM@{a}{b}{z}\right) = \Pochhammersym{b-a}{n}z^{b-a+n-1}e^{-z}\KummerconfhyperM@{a-n}{b}{z}`		(zdiff(z, z))^(n)((z)^(b - a - 1)* exp(- z)KummerM(a, b, z)) = pochhammer(b - a, n)(z)^(b - a + n - 1)* exp(- z)*KummerM(a - n, b, z)	(zD[z, z])^(n)((z)^(b - a - 1)* Exp[- z]Hypergeometric1F1[a, b, z]) == Pochhammer[b - a, n](z)^(b - a + n - 1)* Exp[- z]*Hypergeometric1F1[a - n, b, z]	Failure	Failure	Failed [298 / 300] Result: 1.000000000-.649969050e-10I Test Values: {a = -3/2, b = -3/2, z = 1/23^(1/2)+1/2I, n = 1} Result: .8660254040+.4999999999I Test Values: {a = -3/2, b = -3/2, z = 1/23^(1/2)+1/2I, n = 2} ... skip entries to safe data	Failed [298 / 300] Result: Complex[1.0, -5.551115123125783*^-17] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.8660254037844388, 0.49999999999999983] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
13.3.E20	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^% {-z}M\left(a,b,z\right)\right)=(-1)^{n}\frac{{\left(b-a\right)_{n}}}{{\left(b% \right)_{n}}}e^{-z}M\left(a,b+n,z\right)}}$ `\deriv[n]{}{z}\left(e^{-z}\KummerconfhyperM@{a}{b}{z}\right) = (-1)^{n}\frac{\Pochhammersym{b-a}{n}}{\Pochhammersym{b}{n}}e^{-z}\KummerconfhyperM@{a}{b+n}{z}`		diff(exp(- z)KummerM(a, b, z), [z$(n)]) = (- 1)^(n)(pochhammer(b - a, n))/(pochhammer(b, n))exp(- z)KummerM(a, b + n, z)	D[Exp[- z]Hypergeometric1F1[a, b, z], {z, n}] == (- 1)^(n)Divide[Pochhammer[b - a, n],Pochhammer[b, n]]Exp[- z]Hypergeometric1F1[a, b + n, z]	Failure	Failure	Error	Failed [300 / 300] Result: Plus[Complex[0.0, 0.0], Times[Complex[-0.36912880004696536, 0.20165598253870784], DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[, -1.5], Plus[-1, Times[-1, ], 1], Plus[Times[-1, ], 1], []], Times[Plus[-1, Times[-1, ], 1], Plus[, Times[2, Power[, 2]], Times[, -1.5], -1.5, Times[, -1.5], Times[-1, , 1], Times[-1, -1.5, 1], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[1, ]]], Times[-1, Plus[1, ], Plus[Times[-1, ], Times[-1, Power[, 2]], Times[-1, -1.5], Times[-1, , -1.5], Times[, 1], Times[-1.5, 1], Times[3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, 1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[2, ]]], Times[Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Tim<syntaxhighlight lang=mathematica>Result: Plus[Complex[0.0, 0.0], Times[Complex[0.7382576000939307, -0.4033119650774157], DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[, -1.5], Plus[-1, Times[-1, ], 2], Plus[Times[-1, ], 2], []], Times[Plus[-1, Times[-1, ], 2], Plus[, Times[2, Power[, 2]], Times[, -1.5], -1.5, Times[, -1.5], Times[-1, , 2], Times[-1, -1.5, 2], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[1, ]]], Times[-1, Plus[1, ], Plus[Times[-1, ], Times[-1, Power[, 2]], Times[-1, -1.5], Times[-1, , -1.5], Times[, 2], Times[-1.5, 2], Times[3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[2, ]]], Times[Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Times[Power[Factorial[2], -1], Hypergeometric1F1[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[2], Times[Power[-1.5, -1], Power[Factorial[2], -1], Plus[Times[-1.5, Hypergeometric1F1[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, 2, Hypergeometric1F1[Plus[1, -1.5], Plus[1, -1.5], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][3.0]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
13.3.E21	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^% {b-1}e^{-z}M\left(a,b,z\right)\right)={\left(b-n\right)_{n}}z^{b-n-1}e^{-z}M% \left(a-n,b-n,z\right)}}$ `\deriv[n]{}{z}\left(z^{b-1}e^{-z}\KummerconfhyperM@{a}{b}{z}\right) = \Pochhammersym{b-n}{n}z^{b-n-1}e^{-z}\KummerconfhyperM@{a-n}{b-n}{z}`		diff((z)^(b - 1)* exp(- z)KummerM(a, b, z), [z$(n)]) = pochhammer(b - n, n)(z)^(b - n - 1)* exp(- z)*KummerM(a - n, b - n, z)	D[(z)^(b - 1)* Exp[- z]Hypergeometric1F1[a, b, z], {z, n}] == Pochhammer[b - n, n](z)^(b - n - 1)* Exp[- z]*Hypergeometric1F1[a - n, b - n, z]	Failure	Aborted	Error	Failed [300 / 300] Result: Plus[Complex[-0.6470476127563014, -2.4148145657226703], DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[1, , Times[-1, -1.5]], []], Times[-1, Plus[1, ], Plus[-2, Times[-1, ], -1.5, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[1, ]]], Times[Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[-1, -1.5]], Hypergeometric1F1[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[Power[-1.5, -1], Power[E, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[-2, -1.5]], Plus[Times[-1, -1.5, Hypergeometric1F1[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[Power[-1.5, 2], Hypergeometric1F1[-1.5, -1.5, Pow<syntaxhighlight lang=mathematica>Result: Plus[Complex[6.187184335382289, 6.187184335382291], Times[2.0, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[1, , Times[-1, -1.5]], []], Times[-1, Plus[1, ], Plus[-2, Times[-1, ], -1.5, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[1, ]]], Times[Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[-1, -1.5]], Hypergeometric1F1[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[Power[-1.5, -1], Power[E, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[-2, -1.5]], Plus[Times[-1, -1.5, Hypergeometric1F1[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[Power[-1.5, 2], Hypergeometric1F1[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Hypergeometric1F1[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Hypergeometric1F1[Plus[1, -1.5], Plus[1, -1.5], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][2.0]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
13.3.E22	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}U\left(a,b,z\right)=% -aU\left(a+1,b+1,z\right)}}$ `\deriv{}{z}\KummerconfhyperU@{a}{b}{z} = -a\KummerconfhyperU@{a+1}{b+1}{z}`		diff(KummerU(a, b, z), z) = - a*KummerU(a + 1, b + 1, z)	D[HypergeometricU[a, b, z], z] == - a*HypergeometricU[a + 1, b + 1, z]	Successful	Successful	-	Successful [Tested: 252]
13.3.E23	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}U\left(a% ,b,z\right)=(-1)^{n}{\left(a\right)_{n}}U\left(a+n,b+n,z\right)}}$ `\deriv[n]{}{z}\KummerconfhyperU@{a}{b}{z} = (-1)^{n}\Pochhammersym{a}{n}\KummerconfhyperU@{a+n}{b+n}{z}`		diff(KummerU(a, b, z), [z$(n)]) = (- 1)^(n)* pochhammer(a, n)*KummerU(a + n, b + n, z)	D[HypergeometricU[a, b, z], {z, n}] == (- 1)^(n)* Pochhammer[a, n]*HypergeometricU[a + n, b + n, z]	Failure	Successful	Error	Successful [Tested: 300]
13.3.E24	${\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}% \left(z^{a-1}U\left(a,b,z\right)\right)={\left(a\right)_{n}}{\left(a-b+1\right% )_{n}}z^{a+n-1}U\left(a+n,b,z\right)}}$ `\left(z\deriv{}{z}z\right)^{n}\left(z^{a-1}\KummerconfhyperU@{a}{b}{z}\right) = \Pochhammersym{a}{n}\Pochhammersym{a-b+1}{n}z^{a+n-1}\KummerconfhyperU@{a+n}{b}{z}`		(zdiff(z, z))^(n)((z)^(a - 1)* KummerU(a, b, z)) = pochhammer(a, n)pochhammer(a - b + 1, n)(z)^(a + n - 1)* KummerU(a + n, b, z)	(zD[z, z])^(n)((z)^(a - 1)* HypergeometricU[a, b, z]) == Pochhammer[a, n]Pochhammer[a - b + 1, n](z)^(a + n - 1)* HypergeometricU[a + n, b, z]	Failure	Failure	Failed [295 / 300] Result: 4.557501915-2.807038782I Test Values: {a = -3/2, b = -3/2, z = 1/23^(1/2)+1/2I, n = 1} Result: 2.124956377+.5363245788I Test Values: {a = -3/2, b = -3/2, z = 1/23^(1/2)+1/2I, n = 2} ... skip entries to safe data	Failed [295 / 300] Result: Complex[4.557501914022213, -2.807038783226017] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[2.124956376243804, 0.5363245787128816] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
13.3.E25	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^% {b-1}U\left(a,b,z\right)\right)=(-1)^{n}{\left(a-b+1\right)_{n}}z^{b-n-1}U% \left(a,b-n,z\right)}}$ `\deriv[n]{}{z}\left(z^{b-1}\KummerconfhyperU@{a}{b}{z}\right) = (-1)^{n}\Pochhammersym{a-b+1}{n}z^{b-n-1}\KummerconfhyperU@{a}{b-n}{z}`		diff((z)^(b - 1)* KummerU(a, b, z), [z$(n)]) = (- 1)^(n)* pochhammer(a - b + 1, n)(z)^(b - n - 1) KummerU(a, b - n, z)	D[(z)^(b - 1)* HypergeometricU[a, b, z], {z, n}] == (- 1)^(n)* Pochhammer[a - b + 1, n](z)^(b - n - 1) HypergeometricU[a, b - n, z]	Failure	Aborted	Error	Failed [300 / 300] Result: Plus[Complex[-0.1522159386707833, -5.3504318269524465], Times[Complex[-0.2588190451025206, -0.9659258262890682], DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[, -1.5], Plus[-1, Times[-1, ], 1], Plus[Times[-1, ], 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], []], Times[-1, Plus[-1, Times[-1, ], 1], Plus[Power[, 2], Power[, 3], Times[2, , -1.5], Times[2, Power[, 2], -1.5], Power[-1.5, 2], Times[, Power[-1.5, 2]], Times[-1, , 1], Times[-1, Power[, 2], 1], Times[-1, -1.5, 1], Times[-1, , -1.5, 1], Times[-1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-2, Power[, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, , 1, Pow<syntaxhighlight lang=mathematica>Result: Plus[Complex[9.411642901699432, 13.489513219804685], Times[Complex[-1.4142135623730947, -1.414213562373095], DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[, -1.5], Plus[-1, Times[-1, ], 2], Plus[Times[-1, ], 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], []], Times[-1, Plus[-1, Times[-1, ], 2], Plus[Power[, 2], Power[, 3], Times[2, , -1.5], Times[2, Power[, 2], -1.5], Power[-1.5, 2], Times[, Power[-1.5, 2]], Times[-1, , 2], Times[-1, Power[, 2], 2], Times[-1, -1.5, 2], Times[-1, , -1.5, 2], Times[-1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-2, Power[, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, , 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1.5, 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[1, ]]], Times[Plus[1, ], Plus[, -1.5, Times[-1, 2]], Plus[-2, Times[-4, ], Times[-2, Power[, 2]], Times[-3, -1.5], Times[-2, , -1.5], Times[2, 2], Times[2, , 2], Times[-1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Times[, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[2, ]]], Times[Plus[1, ], Plus[2, ], Plus[, -1.5, Times[-1, 2]], Plus[1, , -1.5, Times[-1, 2]], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Times[Binomial[Plus[-1, -1.5], 2], HypergeometricU[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[2], Times[Binomial[Plus[-1, -1.5], 2], Plus[HypergeometricU[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, -1.5, Power[Plus[-1.5, Times[-1, 2]], -1], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], HypergeometricU[Plus[1, -1.5], Plus[1, -1.5], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][3.0]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
13.3.E26	${\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}% \left(z^{b-a-1}e^{-z}U\left(a,b,z\right)\right)=(-1)^{n}z^{b-a+n-1}e^{-z}U% \left(a-n,b,z\right)}}$ `\left(z\deriv{}{z}z\right)^{n}\left(z^{b-a-1}e^{-z}\KummerconfhyperU@{a}{b}{z}\right) = (-1)^{n}z^{b-a+n-1}e^{-z}\KummerconfhyperU@{a-n}{b}{z}`		(zdiff(z, z))^(n)((z)^(b - a - 1)* exp(- z)KummerU(a, b, z)) = (- 1)^(n) (z)^(b - a + n - 1)* exp(- z)*KummerU(a - n, b, z)	(zD[z, z])^(n)((z)^(b - a - 1)* Exp[- z]HypergeometricU[a, b, z]) == (- 1)^(n) (z)^(b - a + n - 1)* Exp[- z]*HypergeometricU[a - n, b, z]	Failure	Failure	Failed [298 / 300] Result: 1.496936093+.1242553737I Test Values: {a = -3/2, b = -3/2, z = 1/23^(1/2)+1/2I, n = 1} Result: 1.600796058+1.474329192I Test Values: {a = -3/2, b = -3/2, z = 1/23^(1/2)+1/2I, n = 2} ... skip entries to safe data	Failed [298 / 300] Result: Complex[1.4969360926980415, 0.12425537363460365] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.6007960572551263, 1.4743291911897365] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
13.3.E27	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^% {-z}U\left(a,b,z\right)\right)=(-1)^{n}e^{-z}U\left(a,b+n,z\right)}}$ `\deriv[n]{}{z}\left(e^{-z}\KummerconfhyperU@{a}{b}{z}\right) = (-1)^{n}e^{-z}\KummerconfhyperU@{a}{b+n}{z}`		diff(exp(- z)KummerU(a, b, z), [z$(n)]) = (- 1)^(n) exp(- z)*KummerU(a, b + n, z)	D[Exp[- z]HypergeometricU[a, b, z], {z, n}] == (- 1)^(n) Exp[- z]*HypergeometricU[a, b + n, z]	Failure	Failure	Error	Failed [300 / 300] Result: Plus[Complex[0.40360579036441874, 0.11842116492450602], Times[Complex[-0.36912880004696536, 0.20165598253870784], DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[, -1.5], Plus[-1, Times[-1, ], 1], Plus[Times[-1, ], 1], []], Times[Plus[-1, Times[-1, ], 1], Plus[, Times[2, Power[, 2]], Times[, -1.5], -1.5, Times[, -1.5], Times[-1, , 1], Times[-1, -1.5, 1], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[1, ]]], Times[-1, Plus[1, ], Plus[Times[-1, ], Times[-1, Power[, 2]], Times[-1, -1.5], Times[-1, , -1.5], Times[, 1], Times[-1.5, 1], Times[3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, 1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[2, ]]], Times[Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[3, ]]]], 0],<syntaxhighlight lang=mathematica>Result: Plus[Complex[0.20950938468408564, -0.2672919019422666], Times[Complex[0.7382576000939307, -0.4033119650774157], DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[, -1.5], Plus[-1, Times[-1, ], 2], Plus[Times[-1, ], 2], []], Times[Plus[-1, Times[-1, ], 2], Plus[, Times[2, Power[, 2]], Times[, -1.5], -1.5, Times[, -1.5], Times[-1, , 2], Times[-1, -1.5, 2], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[1, ]]], Times[-1, Plus[1, ], Plus[Times[-1, ], Times[-1, Power[, 2]], Times[-1, -1.5], Times[-1, , -1.5], Times[, 2], Times[-1.5, 2], Times[3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[2, ]]], Times[Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Times[Power[Factorial[2], -1], HypergeometricU[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[2], Times[Power[Factorial[2], -1], Plus[HypergeometricU[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1.5, 2, HypergeometricU[Plus[1, -1.5], Plus[1, -1.5], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][3.0]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
13.3.E28	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^% {b-1}e^{-z}U\left(a,b,z\right)\right)=(-1)^{n}z^{b-n-1}e^{-z}U\left(a-n,b-n,z% \right)}}$ `\deriv[n]{}{z}\left(z^{b-1}e^{-z}\KummerconfhyperU@{a}{b}{z}\right) = (-1)^{n}z^{b-n-1}e^{-z}\KummerconfhyperU@{a-n}{b-n}{z}`		diff((z)^(b - 1)* exp(- z)KummerU(a, b, z), [z$(n)]) = (- 1)^(n) (z)^(b - n - 1)* exp(- z)*KummerU(a - n, b - n, z)	D[(z)^(b - 1)* Exp[- z]HypergeometricU[a, b, z], {z, n}] == (- 1)^(n) (z)^(b - n - 1)* Exp[- z]*HypergeometricU[a - n, b - n, z]	Failure	Aborted	Error	Failed [300 / 300] Result: Plus[Complex[-0.9968056293665363, -3.1564168178949528], DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[1, , Times[-1, -1.5]], []], Times[-1, Plus[1, ], Plus[-2, Times[-1, ], -1.5, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[1, ]]], Times[Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[-1, -1.5]], HypergeometricU[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[-1, Power[E, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[-2, -1.5]], Plus[HypergeometricU[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, -1.5, HypergeometricU[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi<syntaxhighlight lang=mathematica>Result: Plus[Complex[8.32628899631003, 8.182173774638818], Times[2.0, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[1, , Times[-1, -1.5]], []], Times[-1, Plus[1, ], Plus[-2, Times[-1, ], -1.5, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[1, ]]], Times[Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[-1, -1.5]], HypergeometricU[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[-1, Power[E, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[-2, -1.5]], Plus[HypergeometricU[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, -1.5, HypergeometricU[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], HypergeometricU[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], HypergeometricU[Plus[1, -1.5], Plus[1, -1.5], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][2.0]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
13.3.E29	${\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}=% z^{n}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}z^{n}}}$ `\left(z\deriv{}{z}z\right)^{n} = z^{n}\deriv[n]{}{z}z^{n}`		(zdiff(z, z))^(n) = (z)^(n) diff((z)^(n), [z$(n)])	(zD[z, z])^(n) == (z)^(n) D[(z)^(n), {z, n}]	Failure	Failure	Failed [7 / 7] Result: -.1616869430e-8-5.000000005I Test Values: {z = 1/23^(1/2)+1/2I, n = 3} Result: -5.000000005+.1616869430e-8I Test Values: {z = -1/2+1/2I3^(1/2), n = 3} ... skip entries to safe data	Failed [7 / 7] Result: Complex[0.0, -5.0] Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: -5.0 Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.4.E1	${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma% \left(a\right)\Gamma\left(b-a\right)}\int_{0}^{1}e^{zt}t^{a-1}(1-t)^{b-a-1}% \mathrm{d}t}}$ `\OlverconfhyperM@{a}{b}{z} = \frac{1}{\EulerGamma@{a}\EulerGamma@{b-a}}\int_{0}^{1}e^{zt}t^{a-1}(1-t)^{b-a-1}\diff{t}`	${\displaystyle{\displaystyle\Re b>\Re a,\Re a>0,\Re(b-a)>0,\Re(b+s)>0}}$	KummerM(a, b, z)/GAMMA(b) = (1)/(GAMMA(a)GAMMA(b - a))int(exp(zt)(t)^(a - 1)*(1 - t)^(b - a - 1), t = 0..1)	Hypergeometric1F1Regularized[a, b, z] == Divide[1,Gamma[a]Gamma[b - a]]Integrate[Exp[zt](t)^(a - 1)*(1 - t)^(b - a - 1), {t, 0, 1}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 21]
13.4.E2	${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma% \left(b-c\right)}\int_{0}^{1}{\mathbf{M}}\left(a,c,zt\right)t^{c-1}(1-t)^{b-c-% 1}\mathrm{d}t}}$ `\OlverconfhyperM@{a}{b}{z} = \frac{1}{\EulerGamma@{b-c}}\int_{0}^{1}\OlverconfhyperM@{a}{c}{zt}t^{c-1}(1-t)^{b-c-1}\diff{t}`	${\displaystyle{\displaystyle\Re b>\Re c,\Re c>0,\Re(b-c)>0,\Re(b+s)>0,\Re(c+s)% >0}}$	KummerM(a, b, z)/GAMMA(b) = (1)/(GAMMA(b - c))int(KummerM(a, c, zt)/GAMMA(c)(t)^(c - 1)(1 - t)^(b - c - 1), t = 0..1)	Hypergeometric1F1Regularized[a, b, z] == Divide[1,Gamma[b - c]]Integrate[Hypergeometric1F1Regularized[a, c, zt](t)^(c - 1)(1 - t)^(b - c - 1), {t, 0, 1}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 126]
13.4.E3	${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,-z\right)=\frac{z^{\frac{1}{% 2}-\frac{1}{2}b}}{\Gamma\left(a\right)}\int_{0}^{\infty}e^{-t}t^{a-\frac{1}{2}% b-\frac{1}{2}}J_{b-1}\left(2\sqrt{zt}\right)\mathrm{d}t}}$ `\OlverconfhyperM@{a}{b}{-z} = \frac{z^{\frac{1}{2}-\frac{1}{2}b}}{\EulerGamma@{a}}\int_{0}^{\infty}e^{-t}t^{a-\frac{1}{2}b-\frac{1}{2}}\BesselJ{b-1}@{2\sqrt{zt}}\diff{t}`	${\displaystyle{\displaystyle\Re a>0,\Re((b-1)+k+1)>0,\Re(b+s)>0}}$	KummerM(a, b, - z)/GAMMA(b) = ((z)^((1)/(2)-(1)/(2)b))/(GAMMA(a))int(exp(- t)(t)^(a -(1)/(2)b -(1)/(2))* BesselJ(b - 1, 2sqrt(zt)), t = 0..infinity)	Hypergeometric1F1Regularized[a, b, - z] == Divide[(z)^(Divide[1,2]-Divide[1,2]b),Gamma[a]]Integrate[Exp[- t](t)^(a -Divide[1,2]b -Divide[1,2])* BesselJ[b - 1, 2Sqrt[zt]], {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Error	Skipped - Because timed out
13.4.E4	${\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{1}{\Gamma\left(a\right)% }\int_{0}^{\infty}e^{-zt}t^{a-1}(1+t)^{b-a-1}\mathrm{d}t}}$ `\KummerconfhyperU@{a}{b}{z} = \frac{1}{\EulerGamma@{a}}\int_{0}^{\infty}e^{-zt}t^{a-1}(1+t)^{b-a-1}\diff{t}`	${\displaystyle{\displaystyle\Re a>0,\|\operatorname{ph}{z}\|<\frac{1}{2}\pi}}$	KummerU(a, b, z) = (1)/(GAMMA(a))int(exp(- zt)(t)^(a - 1)(1 + t)^(b - a - 1), t = 0..infinity)	HypergeometricU[a, b, z] == Divide[1,Gamma[a]]Integrate[Exp[- zt](t)^(a - 1)(1 + t)^(b - a - 1), {t, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 90]
13.4.E5	${\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{z^{1-a}}{\Gamma\left(a% \right)\Gamma\left(1+a-b\right)}\int_{0}^{\infty}\frac{U\left(b-a,b,t\right)e^% {-t}t^{a-1}}{t+z}\mathrm{d}t}}$ `\KummerconfhyperU@{a}{b}{z} = \frac{z^{1-a}}{\EulerGamma@{a}\EulerGamma@{1+a-b}}\int_{0}^{\infty}\frac{\KummerconfhyperU@{b-a}{b}{t}e^{-t}t^{a-1}}{t+z}\diff{t}`	${\displaystyle{\displaystyle\|\operatorname{ph}{z}\|<\pi,\Re a>\max\left(\Re b-1% ,\Re a>0,\Re(1+a-b)>0}\)\@add@PDF@RDFa@triples\end{document}}$	KummerU(a, b, z) = ((z)^(1 - a))/(GAMMA(a)GAMMA(1 + a - b))int((KummerU(b - a, b, t)exp(- t)(t)^(a - 1))/(t + z), t = 0..infinity)	HypergeometricU[a, b, z] == Divide[(z)^(1 - a),Gamma[a]Gamma[1 + a - b]]Integrate[Divide[HypergeometricU[b - a, b, t]Exp[- t](t)^(a - 1),t + z], {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
13.4.E6	${\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{(-1)^{n}z^{1-b-n}}{% \Gamma\left(1+a-b\right)}\int_{0}^{\infty}\frac{{\mathbf{M}}\left(b-a,b,t% \right)e^{-t}t^{b+n-1}}{t+z}\mathrm{d}t}}$ `\KummerconfhyperU@{a}{b}{z} = \frac{(-1)^{n}z^{1-b-n}}{\EulerGamma@{1+a-b}}\int_{0}^{\infty}\frac{\OlverconfhyperM@{b-a}{b}{t}e^{-t}t^{b+n-1}}{t+z}\diff{t}`	${\displaystyle{\displaystyle\left\|\operatorname{ph}z\right\|<\pi,-\Re b<n,n<1+% \Re\left(a-b\right),\Re(1+a-b)>0,\Re(b+s)>0}}$	KummerU(a, b, z) = ((- 1)^(n)* (z)^(1 - b - n))/(GAMMA(1 + a - b))int((KummerM(b - a, b, t)/GAMMA(b)exp(- t)*(t)^(b + n - 1))/(t + z), t = 0..infinity)	HypergeometricU[a, b, z] == Divide[(- 1)^(n)* (z)^(1 - b - n),Gamma[1 + a - b]]Integrate[Divide[Hypergeometric1F1Regularized[b - a, b, t]Exp[- t]*(t)^(b + n - 1),t + z], {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
13.4.E7	${\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{2z^{\frac{1}{2}-\frac{1% }{2}b}}{\Gamma\left(a\right)\Gamma\left(a-b+1\right)}\\int_{0}^{\infty}e^{-t}% t^{a-\frac{1}{2}b-\frac{1}{2}}K_{b-1}\left(2\sqrt{zt}\right)\mathrm{d}t}}$ `\KummerconfhyperU@{a}{b}{z} = \frac{2z^{\frac{1}{2}-\frac{1}{2}b}}{\EulerGamma@{a}\EulerGamma@{a-b+1}}\\int_{0}^{\infty}e^{-t}t^{a-\frac{1}{2}b-\frac{1}{2}}\modBesselK{b-1}@{2\sqrt{zt}}\diff{t}`	${\displaystyle{\displaystyle\Re a>\max\left(\Re b-1,\Re a>0,\Re(a-b+1)>0}\)\@add@PDF@RDFa@triples\end{document}}$	KummerU(a, b, z) = (2(z)^((1)/(2)-(1)/(2)b))/(GAMMA(a)GAMMA(a - b + 1)) int(exp(- t)(t)^(a -(1)/(2)b -(1)/(2))* BesselK(b - 1, 2sqrt(zt)), t = 0..infinity)	HypergeometricU[a, b, z] == Divide[2(z)^(Divide[1,2]-Divide[1,2]b),Gamma[a]Gamma[a - b + 1]] Integrate[Exp[- t](t)^(a -Divide[1,2]b -Divide[1,2])* BesselK[b - 1, 2Sqrt[zt]], {t, 0, Infinity}, GenerateConditions->None]	Successful	Aborted	-	Skipped - Because timed out
13.4.E8	${\displaystyle{\displaystyle U\left(a,b,z\right)=z^{c-a}\\int_{0}^{\infty}e^{% -zt}t^{c-1}{{}_{2}{\mathbf{F}}_{1}}\left(a,a-b+1;c;-t\right)\mathrm{d}t}}$ `\KummerconfhyperU@{a}{b}{z} = z^{c-a}\\int_{0}^{\infty}e^{-zt}t^{c-1}\genhyperOlverF{2}{1}@{a,a-b+1}{c}{-t}\diff{t}`	${\displaystyle{\displaystyle\|\operatorname{ph}{z}\|<\frac{1}{2}\pi}}$	KummerU(a, b, z) = (z)^(c - a)* int(exp(- zt)(t)^(c - 1)* hypergeom([a , a - b + 1], [c], - t), t = 0..infinity)	HypergeometricU[a, b, z] == (z)^(c - a)* Integrate[Exp[- zt](t)^(c - 1)* HypergeometricPFQRegularized[{a , a - b + 1}, {c}, - t], {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Failed [294 / 300] Result: Float(undefined)+Float(undefined)I Test Values: {a = -3/2, b = -3/2, c = -3/2, z = 1/23^(1/2)+1/2I} Result: Float(undefined)+Float(undefined)I Test Values: {a = -3/2, b = -3/2, c = -3/2, z = 1/2-1/2I3^(1/2)} ... skip entries to safe data	Skipped - Because timed out
13.4.E9	${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=\frac{\Gamma\left(1% +a-b\right)}{2\pi\mathrm{i}\Gamma\left(a\right)}\int_{0}^{(1+)}e^{zt}t^{a-1}{(% t-1)^{b-a-1}}\mathrm{d}t}}$ `\OlverconfhyperM@{a}{b}{z} = \frac{\EulerGamma@{1+a-b}}{2\pi\iunit\EulerGamma@{a}}\int_{0}^{(1+)}e^{zt}t^{a-1}{(t-1)^{b-a-1}}\diff{t}`	${\displaystyle{\displaystyle\Re a>0,\Re(1+a-b)>0,\Re(b+s)>0}}$	KummerM(a, b, z)/GAMMA(b) = (GAMMA(1 + a - b))/(2PiIGAMMA(a))int(exp(zt)(t)^(a - 1)*(t - 1)^(b - a - 1), t = 0..(1 +))	Hypergeometric1F1Regularized[a, b, z] == Divide[Gamma[1 + a - b],2PiIGamma[a]]Integrate[Exp[zt](t)^(a - 1)*(t - 1)^(b - a - 1), {t, 0, (1 +)}, GenerateConditions->None]	Error	Failure	-	Error
13.4.E10	${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=e^{-a\pi\mathrm{i}}% \frac{\Gamma\left(1-a\right)}{2\pi\mathrm{i}\Gamma\left(b-a\right)}\int_{1}^{(% 0+)}e^{zt}t^{a-1}{(1-t)^{b-a-1}}\mathrm{d}t}}$ `\OlverconfhyperM@{a}{b}{z} = e^{-a\pi\iunit}\frac{\EulerGamma@{1-a}}{2\pi\iunit\EulerGamma@{b-a}}\int_{1}^{(0+)}e^{zt}t^{a-1}{(1-t)^{b-a-1}}\diff{t}`	${\displaystyle{\displaystyle\Re\left(b-a\right)>0,\Re(1-a)>0,\Re(b-a)>0,\Re(b+% s)>0}}$	KummerM(a, b, z)/GAMMA(b) = exp(- aPiI)(GAMMA(1 - a))/(2PiIGAMMA(b - a))int(exp(zt)(t)^(a - 1)(1 - t)^(b - a - 1), t = 1..(0 +))	Hypergeometric1F1Regularized[a, b, z] == Exp[- aPiI]Divide[Gamma[1 - a],2PiIGamma[b - a]]Integrate[Exp[zt](t)^(a - 1)(1 - t)^(b - a - 1), {t, 1, (0 +)}, GenerateConditions->None]	Error	Failure	-	Error
13.4.E11	${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=e^{-b\pi\mathrm{i}}% \Gamma\left(1-a\right)\Gamma\left(1+a-b\right)\\frac{1}{4\pi^{2}}\int_{\alpha% }^{(0+,1+,0-,1-)}e^{zt}t^{a-1}{(1-t)^{b-a-1}}\mathrm{d}t}}$ `\OlverconfhyperM@{a}{b}{z} = e^{-b\pi\iunit}\EulerGamma@{1-a}\EulerGamma@{1+a-b}\\frac{1}{4\pi^{2}}\int_{\alpha}^{(0+,1+,0-,1-)}e^{zt}t^{a-1}{(1-t)^{b-a-1}}\diff{t}`	${\displaystyle{\displaystyle\Re(1-a)>0,\Re(1+a-b)>0,\Re(b+s)>0}}$	KummerM(a, b, z)/GAMMA(b) = exp(- bPiI)GAMMA(1 - a)GAMMA(1 + a - b)(1)/(4(Pi)^(2))int(exp(zt)(t)^(a - 1)(1 - t)^(b - a - 1), t = alpha..(0 + , 1 + , 0 - , 1 -))	Hypergeometric1F1Regularized[a, b, z] == Exp[- bPiI]Gamma[1 - a]Gamma[1 + a - b]Divide[1,4(Pi)^(2)]Integrate[Exp[zt](t)^(a - 1)(1 - t)^(b - a - 1), {t, \[Alpha], (0 + , 1 + , 0 - , 1 -)}, GenerateConditions->None]	Error	Failure	-	Error
13.4.E12	${\displaystyle{\displaystyle{\mathbf{M}}\left(a,c,z\right)=\frac{\Gamma\left(b% \right)}{2\pi\mathrm{i}}z^{1-b}\int_{-\infty}^{(0+,1+)}e^{zt}t^{-b}{{}_{2}{% \mathbf{F}}_{1}}\left(a,b;c;\ifrac{1}{t}\right)\mathrm{d}t}}$ `\OlverconfhyperM@{a}{c}{z} = \frac{\EulerGamma@{b}}{2\pi\iunit}z^{1-b}\int_{-\infty}^{(0+,1+)}e^{zt}t^{-b}\genhyperOlverF{2}{1}@{a,b}{c}{\ifrac{1}{t}}\diff{t}`	${\displaystyle{\displaystyle\left\|\operatorname{ph}z\right\|<\frac{1}{2}\pi,\Re b% >0,\Re(c+s)>0}}$	KummerM(a, c, z)/GAMMA(c) = (GAMMA(b))/(2PiI)(z)^(1 - b) int(exp(zt)(t)^(- b)* hypergeom([a , b], [c], (1)/(t)), t = - infinity..(0 + , 1 +))	Hypergeometric1F1Regularized[a, c, z] == Divide[Gamma[b],2PiI](z)^(1 - b) Integrate[Exp[zt](t)^(- b)* HypergeometricPFQRegularized[{a , b}, {c}, Divide[1,t]], {t, - Infinity, (0 + , 1 +)}, GenerateConditions->None]	Error	Failure	-	Error
13.4.E13	${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=\frac{z^{1-b}}{2\pi% \mathrm{i}}\int_{-\infty}^{(0+,1+)}e^{zt}t^{-b}\!\left(1-\frac{1}{t}\right)^{-% a}\mathrm{d}t}}$ `\OlverconfhyperM@{a}{b}{z} = \frac{z^{1-b}}{2\pi\iunit}\int_{-\infty}^{(0+,1+)}e^{zt}t^{-b}\!\left(1-\frac{1}{t}\right)^{-a}\diff{t}`	${\displaystyle{\displaystyle\|\operatorname{ph}{z}\|<\frac{1}{2}\pi,\Re(b+s)>0}}$	KummerM(a, b, z)/GAMMA(b) = ((z)^(1 - b))/(2PiI)int(exp(zt)(t)^(- b)(1 -(1)/(t))^(- a), t = - infinity..(0 + , 1 +))	Hypergeometric1F1Regularized[a, b, z] == Divide[(z)^(1 - b),2PiI]Integrate[Exp[zt](t)^(- b)(1 -Divide[1,t])^(- a), {t, - Infinity, (0 + , 1 +)}, GenerateConditions->None]	Error	Failure	-	Error
13.4.E14	${\displaystyle{\displaystyle U\left(a,b,z\right)=e^{-a\pi\mathrm{i}}\frac{% \Gamma\left(1-a\right)}{2\pi\mathrm{i}}\int_{\infty}^{(0+)}e^{-zt}t^{a-1}{(1+t% )^{b-a-1}}\mathrm{d}t}}$ `\KummerconfhyperU@{a}{b}{z} = e^{-a\pi\iunit}\frac{\EulerGamma@{1-a}}{2\pi\iunit}\int_{\infty}^{(0+)}e^{-zt}t^{a-1}{(1+t)^{b-a-1}}\diff{t}`	${\displaystyle{\displaystyle\left\|\operatorname{ph}z\right\|<\frac{1}{2}\pi,\Re% (1-a)>0}}$	KummerU(a, b, z) = exp(- aPiI)(GAMMA(1 - a))/(2PiI)int(exp(- zt)(t)^(a - 1)*(1 + t)^(b - a - 1), t = infinity..(0 +))	HypergeometricU[a, b, z] == Exp[- aPiI]Divide[Gamma[1 - a],2PiI]Integrate[Exp[- zt](t)^(a - 1)*(1 + t)^(b - a - 1), {t, Infinity, (0 +)}, GenerateConditions->None]	Error	Failure	-	Error
13.4.E15	${\displaystyle{\displaystyle\frac{U\left(a,b,z\right)}{\Gamma\left(c\right)% \Gamma\left(c-b+1\right)}=\frac{z^{1-c}}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e% ^{zt}t^{-c}{{}_{2}{\mathbf{F}}_{1}}\left(a,c;a+c-b+1;1-\frac{1}{t}\right)% \mathrm{d}t}}$ `\frac{\KummerconfhyperU@{a}{b}{z}}{\EulerGamma@{c}\EulerGamma@{c-b+1}} = \frac{z^{1-c}}{2\pi\iunit}\int_{-\infty}^{(0+)}e^{zt}t^{-c}\genhyperOlverF{2}{1}@{a,c}{a+c-b+1}{1-\frac{1}{t}}\diff{t}`	${\displaystyle{\displaystyle\left\|\operatorname{ph}z\right\|<\frac{1}{2}\pi,\Re c% >0,\Re(c-b+1)>0}}$	(KummerU(a, b, z))/(GAMMA(c)GAMMA(c - b + 1)) = ((z)^(1 - c))/(2PiI)int(exp(zt)(t)^(- c)* hypergeom([a , c], [a + c - b + 1], 1 -(1)/(t)), t = - infinity..(0 +))	Divide[HypergeometricU[a, b, z],Gamma[c]Gamma[c - b + 1]] == Divide[(z)^(1 - c),2PiI]Integrate[Exp[zt](t)^(- c)* HypergeometricPFQRegularized[{a , c}, {a + c - b + 1}, 1 -Divide[1,t]], {t, - Infinity, (0 +)}, GenerateConditions->None]	Error	Failure	-	Error
13.4.E16	${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,-z\right)=\frac{1}{2\pi% \mathrm{i}\Gamma\left(a\right)}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}% \frac{\Gamma\left(a+t\right)\Gamma\left(-t\right)}{\Gamma\left(b+t\right)}z^{t% }\mathrm{d}t}}$ `\OlverconfhyperM@{a}{b}{-z} = \frac{1}{2\pi\iunit\EulerGamma@{a}}\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{a+t}\EulerGamma@{-t}}{\EulerGamma@{b+t}}z^{t}\diff{t}`	${\displaystyle{\displaystyle\|\operatorname{ph}{z}\|<\tfrac{1}{2}\pi,\Re a>0,\Re% (a+t)>0,\Re(-t)>0,\Re(b+t)>0,\Re(b+s)>0}}$	KummerM(a, b, - z)/GAMMA(b) = (1)/(2PiIGAMMA(a))int((GAMMA(a + t)GAMMA(- t))/(GAMMA(b + t))(z)^(t), t = - Iinfinity..Iinfinity)	Hypergeometric1F1Regularized[a, b, - z] == Divide[1,2PiIGamma[a]]Integrate[Divide[Gamma[a + t]Gamma[- t],Gamma[b + t]](z)^(t), {t, - IInfinity, IInfinity}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
13.4.E17	${\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{z^{-a}}{2\pi\mathrm{i}}% \int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma\left(a+t\right)\Gamma% \left(1+a-b+t\right)\Gamma\left(-t\right)}{\Gamma\left(a\right)\Gamma\left(1+a% -b\right)}z^{-t}\mathrm{d}t}}$ `\KummerconfhyperU@{a}{b}{z} = \frac{z^{-a}}{2\pi\iunit}\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{a+t}\EulerGamma@{1+a-b+t}\EulerGamma@{-t}}{\EulerGamma@{a}\EulerGamma@{1+a-b}}z^{-t}\diff{t}`	${\displaystyle{\displaystyle\|\operatorname{ph}{z}\|<\tfrac{3}{2}\pi,\Re(a+t)>0,% \Re(1+a-b+t)>0,\Re(-t)>0,\Re a>0,\Re(1+a-b)>0}}$	KummerU(a, b, z) = ((z)^(- a))/(2PiI)int((GAMMA(a + t)GAMMA(1 + a - b + t)GAMMA(- t))/(GAMMA(a)GAMMA(1 + a - b))(z)^(- t), t = - Iinfinity..I*infinity)	HypergeometricU[a, b, z] == Divide[(z)^(- a),2PiI]Integrate[Divide[Gamma[a + t]Gamma[1 + a - b + t]Gamma[- t],Gamma[a]Gamma[1 + a - b]](z)^(- t), {t, - IInfinity, I*Infinity}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
13.4.E18	${\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{z^{1-b}e^{z}}{2\pi% \mathrm{i}}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma\left(b-1+t% \right)\Gamma\left(t\right)}{\Gamma\left(a+t\right)}z^{-t}\mathrm{d}t}}$ `\KummerconfhyperU@{a}{b}{z} = \frac{z^{1-b}e^{z}}{2\pi\iunit}\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{b-1+t}\EulerGamma@{t}}{\EulerGamma@{a+t}}z^{-t}\diff{t}`	${\displaystyle{\displaystyle\|\operatorname{ph}{z}\|<\tfrac{1}{2}\pi,\Re(b-1+t)>% 0,\Re t>0,\Re(a+t)>0}}$	KummerU(a, b, z) = ((z)^(1 - b)* exp(z))/(2PiI)int((GAMMA(b - 1 + t)GAMMA(t))/(GAMMA(a + t))(z)^(- t), t = - Iinfinity..I*infinity)	HypergeometricU[a, b, z] == Divide[(z)^(1 - b)* Exp[z],2PiI]Integrate[Divide[Gamma[b - 1 + t]Gamma[t],Gamma[a + t]](z)^(- t), {t, - IInfinity, I*Infinity}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
13.6.E1	${\displaystyle{\displaystyle M\left(a,a,z\right)=e^{z}}}$ `\KummerconfhyperM@{a}{a}{z} = e^{z}`		KummerM(a, a, z) = exp(z)	Hypergeometric1F1[a, a, z] == Exp[z]	Successful	Successful	-	Successful [Tested: 42]
13.6.E2	${\displaystyle{\displaystyle M\left(1,2,2z\right)=\frac{e^{z}}{z}\sinh z}}$ `\KummerconfhyperM@{1}{2}{2z} = \frac{e^{z}}{z}\sinh@@{z}`		KummerM(1, 2, 2z) = (exp(z))/(z)sinh(z)	Hypergeometric1F1[1, 2, 2z] == Divide[Exp[z],z]Sinh[z]	Successful	Successful	-	Successful [Tested: 7]
13.6.E3	${\displaystyle{\displaystyle M\left(0,b,z\right)=U\left(0,b,z\right)}}$ `\KummerconfhyperM@{0}{b}{z} = \KummerconfhyperU@{0}{b}{z}`		KummerM(0, b, z) = KummerU(0, b, z)	Hypergeometric1F1[0, b, z] == HypergeometricU[0, b, z]	Successful	Successful	-	Successful [Tested: 42]
13.6.E3	${\displaystyle{\displaystyle U\left(0,b,z\right)=1}}$ `\KummerconfhyperU@{0}{b}{z} = 1`		KummerU(0, b, z) = 1	HypergeometricU[0, b, z] == 1	Successful	Successful	-	Successful [Tested: 42]
13.6.E4	${\displaystyle{\displaystyle U\left(a,a+1,z\right)=z^{-a}}}$ `\KummerconfhyperU@{a}{a+1}{z} = z^{-a}`		KummerU(a, a + 1, z) = (z)^(- a)	HypergeometricU[a, a + 1, z] == (z)^(- a)	Failure	Successful	Successful [Tested: 42]	Successful [Tested: 42]
13.6.E5	${\displaystyle{\displaystyle M\left(a,a+1,-z\right)=e^{-z}M\left(1,a+1,z\right% )}}$ `\KummerconfhyperM@{a}{a+1}{-z} = e^{-z}\KummerconfhyperM@{1}{a+1}{z}`		KummerM(a, a + 1, - z) = exp(- z)*KummerM(1, a + 1, z)	Hypergeometric1F1[a, a + 1, - z] == Exp[- z]*Hypergeometric1F1[1, a + 1, z]	Successful	Successful	Skip - symbolical successful subtest	Failed [7 / 42] Result: Indeterminate Test Values: {Rule[a, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.6.E5	${\displaystyle{\displaystyle e^{-z}M\left(1,a+1,z\right)=az^{-a}\gamma\left(a,% z\right)}}$ `e^{-z}\KummerconfhyperM@{1}{a+1}{z} = az^{-a}\incgamma@{a}{z}`	${\displaystyle{\displaystyle\Re a>0}}$	exp(- z)KummerM(1, a + 1, z) = a(z)^(- a)* GAMMA(a)-GAMMA(a, z)	Exp[- z]Hypergeometric1F1[1, a + 1, z] == a(z)^(- a)* Gamma[a, 0, z]	Failure	Successful	Failed [21 / 21] Result: .1786149082+.5798847761I Test Values: {a = 3/2, z = 1/23^(1/2)+1/2I} Result: 4.103691021-1.156198608I Test Values: {a = 3/2, z = -1/2+1/2I3^(1/2)} ... skip entries to safe data	Successful [Tested: 21]
13.6.E6	${\displaystyle{\displaystyle U\left(a,a,z\right)=z^{1-a}U\left(1,2-a,z\right)}}$ `\KummerconfhyperU@{a}{a}{z} = z^{1-a}\KummerconfhyperU@{1}{2-a}{z}`		KummerU(a, a, z) = (z)^(1 - a)* KummerU(1, 2 - a, z)	HypergeometricU[a, a, z] == (z)^(1 - a)* HypergeometricU[1, 2 - a, z]	Successful	Successful	-	Successful [Tested: 42]
13.6.E6	${\displaystyle{\displaystyle z^{1-a}U\left(1,2-a,z\right)=z^{1-a}e^{z}E_{a}% \left(z\right)}}$ `z^{1-a}\KummerconfhyperU@{1}{2-a}{z} = z^{1-a}e^{z}\genexpintE{a}@{z}`		(z)^(1 - a)* KummerU(1, 2 - a, z) = (z)^(1 - a)* exp(z)*Ei(a, z)	(z)^(1 - a)* HypergeometricU[1, 2 - a, z] == (z)^(1 - a)* Exp[z]*ExpIntegralE[a, z]	Successful	Successful	-	Successful [Tested: 42]
13.6.E6	${\displaystyle{\displaystyle z^{1-a}e^{z}E_{a}\left(z\right)=e^{z}\Gamma\left(% 1-a,z\right)}}$ `z^{1-a}e^{z}\genexpintE{a}@{z} = e^{z}\incGamma@{1-a}{z}`		(z)^(1 - a)* exp(z)Ei(a, z) = exp(z)GAMMA(1 - a, z)	(z)^(1 - a)* Exp[z]ExpIntegralE[a, z] == Exp[z]Gamma[1 - a, z]	Successful	Successful	-	Successful [Tested: 42]
13.6.E7	${\displaystyle{\displaystyle M\left(\tfrac{1}{2},\tfrac{3}{2},-z^{2}\right)=% \frac{\sqrt{\pi}}{2z}\operatorname{erf}\left(z\right)}}$ `\KummerconfhyperM@{\tfrac{1}{2}}{\tfrac{3}{2}}{-z^{2}} = \frac{\sqrt{\pi}}{2z}\erf@{z}`		KummerM((1)/(2), (3)/(2), - (z)^(2)) = (sqrt(Pi))/(2z)erf(z)	Hypergeometric1F1[Divide[1,2], Divide[3,2], - (z)^(2)] == Divide[Sqrt[Pi],2z]Erf[z]	Successful	Successful	-	Successful [Tested: 7]
13.6.E8	${\displaystyle{\displaystyle U\left(\tfrac{1}{2},\tfrac{1}{2},z^{2}\right)=% \sqrt{\pi}e^{z^{2}}\operatorname{erfc}\left(z\right)}}$ `\KummerconfhyperU@{\tfrac{1}{2}}{\tfrac{1}{2}}{z^{2}} = \sqrt{\pi}e^{z^{2}}\erfc@{z}`		KummerU((1)/(2), (1)/(2), (z)^(2)) = sqrt(Pi)exp((z)^(2))erfc(z)	HypergeometricU[Divide[1,2], Divide[1,2], (z)^(2)] == Sqrt[Pi]Exp[(z)^(2)]Erfc[z]	Failure	Failure	Failed [2 / 7] Result: .418096912e-1+2.795226389I Test Values: {z = -1/2+1/2I3^(1/2)} Result: -2.288685714-4.974950146I Test Values: {z = -1/23^(1/2)-1/2I}	Failed [2 / 7] Result: Complex[0.041809690497868646, 2.7952263885381483] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-2.28868571442365, -4.974950145988551] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
13.6.E9	${\displaystyle{\displaystyle M\left(\nu+\tfrac{1}{2},2\nu+1,2z\right)=\Gamma% \left(1+\nu\right)e^{z}\left(\ifrac{z}{2}\right)^{-\nu}I_{\nu}\left(z\right)}}$ `\KummerconfhyperM@{\nu+\tfrac{1}{2}}{2\nu+1}{2z} = \EulerGamma@{1+\nu}e^{z}\left(\ifrac{z}{2}\right)^{-\nu}\modBesselI{\nu}@{z}`	${\displaystyle{\displaystyle\Re(1+\nu)>0,\Re(\nu+k+1)>0}}$	KummerM(nu +(1)/(2), 2nu + 1, 2z) = GAMMA(1 + nu)exp(z)((z)/(2))^(- nu)* BesselI(nu, z)	Hypergeometric1F1[\[Nu]+Divide[1,2], 2\[Nu]+ 1, 2z] == Gamma[1 + \[Nu]]Exp[z](Divide[z,2])^(- \[Nu])* BesselI[\[Nu], z]	Successful	Successful	-	Failed [7 / 56] Result: Complex[-1.026957443693084, -2.3780953180269115] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -0.5]} Result: Complex[0.5295327248436391, -0.1815534052901876] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, -0.5]} ... skip entries to safe data
13.6.E10	${\displaystyle{\displaystyle U\left(\nu+\tfrac{1}{2},2\nu+1,2z\right)=\frac{1}% {\sqrt{\pi}}e^{z}\left(2z\right)^{-\nu}K_{\nu}\left(z\right)}}$ `\KummerconfhyperU@{\nu+\tfrac{1}{2}}{2\nu+1}{2z} = \frac{1}{\sqrt{\pi}}e^{z}\left(2z\right)^{-\nu}\modBesselK{\nu}@{z}`		KummerU(nu +(1)/(2), 2nu + 1, 2z) = (1)/(sqrt(Pi))exp(z)(2z)^(- nu) BesselK(nu, z)	HypergeometricU[\[Nu]+Divide[1,2], 2\[Nu]+ 1, 2z] == Divide[1,Sqrt[Pi]]Exp[z](2z)^(- \[Nu]) BesselK[\[Nu], z]	Successful	Successful	-	Successful [Tested: 70]
13.6.E11	${\displaystyle{\displaystyle U\left(\tfrac{5}{6},\tfrac{5}{3},\tfrac{4}{3}z^{3% /2}\right)=\sqrt{\pi}\frac{3^{5/6}\exp\left(\tfrac{2}{3}z^{3/2}\right)}{2^{2/3% }z}\mathrm{Ai}\left(z\right)}}$ `\KummerconfhyperU@{\tfrac{5}{6}}{\tfrac{5}{3}}{\tfrac{4}{3}z^{3/2}} = \sqrt{\pi}\frac{3^{5/6}\exp@{\tfrac{2}{3}z^{3/2}}}{2^{2/3}z}\AiryAi@{z}`		KummerU((5)/(6), (5)/(3), (4)/(3)(z)^(3/2)) = sqrt(Pi)((3)^(5/6)* exp((2)/(3)(z)^(3/2)))/((2)^(2/3) z)*AiryAi(z)	HypergeometricU[Divide[5,6], Divide[5,3], Divide[4,3](z)^(3/2)] == Sqrt[Pi]Divide[(3)^(5/6)* Exp[Divide[2,3](z)^(3/2)],(2)^(2/3) z]*AiryAi[z]	Failure	Failure	Failed [1 / 7] Result: .7957982359-.7292249892I Test Values: {z = -1/23^(1/2)-1/2*I}	Failed [1 / 7] Result: Complex[0.7957982355202466, -0.7292249896477329] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
13.6.E12	${\displaystyle{\displaystyle U\left(\tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2},% \tfrac{1}{2}z^{2}\right)=2^{\frac{1}{2}a+\frac{1}{4}}e^{\frac{1}{4}z^{2}}U% \left(a,z\right)}}$ `\KummerconfhyperU@{\tfrac{1}{2}a+\tfrac{1}{4}}{\tfrac{1}{2}}{\tfrac{1}{2}z^{2}} = 2^{\frac{1}{2}a+\frac{1}{4}}e^{\frac{1}{4}z^{2}}\paraU@{a}{z}`		KummerU((1)/(2)a +(1)/(4), (1)/(2), (1)/(2)(z)^(2)) = (2)^((1)/(2)a +(1)/(4)) exp((1)/(4)(z)^(2))CylinderU(a, z)	HypergeometricU[Divide[1,2]a +Divide[1,4], Divide[1,2], Divide[1,2](z)^(2)] == (2)^(Divide[1,2]a +Divide[1,4]) Exp[Divide[1,4](z)^(2)]ParabolicCylinderD[- 1/2 -(a), z]	Failure	Failure	Failed [10 / 42] Result: .7071067808-1.224744871I Test Values: {a = -3/2, z = -1/2+1/2I3^(1/2)} Result: 1.224744871+.7071067810I Test Values: {a = -3/2, z = -1/23^(1/2)-1/2I} ... skip entries to safe data	Failed [10 / 42] Result: Complex[0.7071067811865475, -1.224744871391589] Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[1.224744871391589, 0.7071067811865475] Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} ... skip entries to safe data
13.6.E13	${\displaystyle{\displaystyle U\left(\tfrac{1}{2}a+\tfrac{3}{4},\tfrac{3}{2},% \tfrac{1}{2}z^{2}\right)=2^{\frac{1}{2}a+\frac{3}{4}}\frac{e^{\frac{1}{4}z^{2}% }}{z}U\left(a,z\right)}}$ `\KummerconfhyperU@{\tfrac{1}{2}a+\tfrac{3}{4}}{\tfrac{3}{2}}{\tfrac{1}{2}z^{2}} = 2^{\frac{1}{2}a+\frac{3}{4}}\frac{e^{\frac{1}{4}z^{2}}}{z}\paraU@{a}{z}`		KummerU((1)/(2)a +(3)/(4), (3)/(2), (1)/(2)(z)^(2)) = (2)^((1)/(2)a +(3)/(4))(exp((1)/(4)(z)^(2)))/(z)CylinderU(a, z)	HypergeometricU[Divide[1,2]a +Divide[3,4], Divide[3,2], Divide[1,2](z)^(2)] == (2)^(Divide[1,2]a +Divide[3,4])Divide[Exp[Divide[1,4](z)^(2)],z]ParabolicCylinderD[- 1/2 -(a), z]	Failure	Failure	Failed [10 / 42] Result: 3.981039608+.280376847I Test Values: {a = 3/2, z = -1/2+1/2I3^(1/2)} Result: 9.425210776+2.041008108I Test Values: {a = 3/2, z = -1/23^(1/2)-1/2I} ... skip entries to safe data	Failed [10 / 42] Result: Complex[3.9810396073031904, 0.2803768494018799] Test Values: {Rule[a, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[9.42521077642933, 2.0410081046172346] Test Values: {Rule[a, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} ... skip entries to safe data
13.6.E14	${\displaystyle{\displaystyle M\left(\tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2},% \tfrac{1}{2}z^{2}\right)=\frac{2^{\frac{1}{2}a-\frac{3}{4}}\Gamma\left(\tfrac{% 1}{2}a+\tfrac{3}{4}\right)e^{\frac{1}{4}z^{2}}}{\sqrt{\pi}}\\left(U\left(a,z% \right)+U\left(a,-z\right)\right)}}$ `\KummerconfhyperM@{\tfrac{1}{2}a+\tfrac{1}{4}}{\tfrac{1}{2}}{\tfrac{1}{2}z^{2}} = \frac{2^{\frac{1}{2}a-\frac{3}{4}}\EulerGamma@{\tfrac{1}{2}a+\tfrac{3}{4}}e^{\frac{1}{4}z^{2}}}{\sqrt{\pi}}\\left(\paraU@{a}{z}+\paraU@{a}{-z}\right)`	${\displaystyle{\displaystyle\Re(\tfrac{1}{2}a+\tfrac{3}{4})>0}}$	KummerM((1)/(2)a +(1)/(4), (1)/(2), (1)/(2)(z)^(2)) = ((2)^((1)/(2)a -(3)/(4)) GAMMA((1)/(2)a +(3)/(4))exp((1)/(4)(z)^(2)))/(sqrt(Pi))(CylinderU(a, z)+ CylinderU(a, - z))	Hypergeometric1F1[Divide[1,2]a +Divide[1,4], Divide[1,2], Divide[1,2](z)^(2)] == Divide[(2)^(Divide[1,2]a -Divide[3,4]) Gamma[Divide[1,2]a +Divide[3,4]]Exp[Divide[1,4](z)^(2)],Sqrt[Pi]](ParabolicCylinderD[- 1/2 -(a), z]+ ParabolicCylinderD[- 1/2 -(a), - z])	Successful	Successful	-	Successful [Tested: 28]
13.6.E15	${\displaystyle{\displaystyle M\left(\tfrac{1}{2}a+\tfrac{3}{4},\tfrac{3}{2},% \tfrac{1}{2}z^{2}\right)=\frac{2^{\frac{1}{2}a-\frac{5}{4}}\Gamma\left(\tfrac{% 1}{2}a+\tfrac{1}{4}\right)e^{\frac{1}{4}z^{2}}}{z\sqrt{\pi}}\\left(U\left(a,-% z\right)-U\left(a,z\right)\right)}}$ `\KummerconfhyperM@{\tfrac{1}{2}a+\tfrac{3}{4}}{\tfrac{3}{2}}{\tfrac{1}{2}z^{2}} = \frac{2^{\frac{1}{2}a-\frac{5}{4}}\EulerGamma@{\tfrac{1}{2}a+\tfrac{1}{4}}e^{\frac{1}{4}z^{2}}}{z\sqrt{\pi}}\\left(\paraU@{a}{-z}-\paraU@{a}{z}\right)`	${\displaystyle{\displaystyle\Re(\tfrac{1}{2}a+\tfrac{1}{4})>0}}$	KummerM((1)/(2)a +(3)/(4), (3)/(2), (1)/(2)(z)^(2)) = ((2)^((1)/(2)a -(5)/(4)) GAMMA((1)/(2)a +(1)/(4))exp((1)/(4)(z)^(2)))/(zsqrt(Pi))*(CylinderU(a, - z)- CylinderU(a, z))	Hypergeometric1F1[Divide[1,2]a +Divide[3,4], Divide[3,2], Divide[1,2](z)^(2)] == Divide[(2)^(Divide[1,2]a -Divide[5,4]) Gamma[Divide[1,2]a +Divide[1,4]]Exp[Divide[1,4](z)^(2)],zSqrt[Pi]]*(ParabolicCylinderD[- 1/2 -(a), - z]- ParabolicCylinderD[- 1/2 -(a), z])	Successful	Successful	-	Successful [Tested: 21]
13.6.E16	${\displaystyle{\displaystyle M\left(-n,\tfrac{1}{2},z^{2}\right)=(-1)^{n}\frac% {n!}{(2n)!}H_{2n}\left(z\right)}}$ `\KummerconfhyperM@{-n}{\tfrac{1}{2}}{z^{2}} = (-1)^{n}\frac{n!}{(2n)!}\HermitepolyH{2n}@{z}`		KummerM(- n, (1)/(2), (z)^(2)) = (- 1)^(n)(factorial(n))/(factorial(2n))HermiteH(2n, z)	Hypergeometric1F1[- n, Divide[1,2], (z)^(2)] == (- 1)^(n)Divide[(n)!,(2n)!]HermiteH[2n, z]	Failure	Failure	Successful [Tested: 21]	Successful [Tested: 21]
13.6.E17	${\displaystyle{\displaystyle M\left(-n,\tfrac{3}{2},z^{2}\right)=(-1)^{n}\frac% {n!}{(2n+1)!2z}H_{2n+1}\left(z\right)}}$ `\KummerconfhyperM@{-n}{\tfrac{3}{2}}{z^{2}} = (-1)^{n}\frac{n!}{(2n+1)!2z}\HermitepolyH{2n+1}@{z}`		KummerM(- n, (3)/(2), (z)^(2)) = (- 1)^(n)(factorial(n))/(factorial(2n + 1)2z)HermiteH(2n + 1, z)	Hypergeometric1F1[- n, Divide[3,2], (z)^(2)] == (- 1)^(n)Divide[(n)!,(2n + 1)!2z]HermiteH[2n + 1, z]	Failure	Failure	Successful [Tested: 21]	Successful [Tested: 21]
13.6.E18	${\displaystyle{\displaystyle U\left(\tfrac{1}{2}-\tfrac{1}{2}n,\tfrac{3}{2},z^% {2}\right)=2^{-n}z^{-1}H_{n}\left(z\right)}}$ `\KummerconfhyperU@{\tfrac{1}{2}-\tfrac{1}{2}n}{\tfrac{3}{2}}{z^{2}} = 2^{-n}z^{-1}\HermitepolyH{n}@{z}`		KummerU((1)/(2)-(1)/(2)n, (3)/(2), (z)^(2)) = (2)^(- n) (z)^(- 1)* HermiteH(n, z)	HypergeometricU[Divide[1,2]-Divide[1,2]n, Divide[3,2], (z)^(2)] == (2)^(- n) (z)^(- 1)* HermiteH[n, z]	Failure	Failure	Failed [2 / 21] Result: .5000000003-2.598076212I Test Values: {z = -1/2+1/2I3^(1/2), n = 2} Result: .8660254044+1.500000000I Test Values: {z = -1/23^(1/2)-1/2I, n = 2}	Failed [2 / 21] Result: Complex[0.5000000000000006, -2.5980762113533156] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[0.8660254037844388, 1.5] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
13.6.E19	${\displaystyle{\displaystyle U\left(-n,\alpha+1,z\right)=(-1)^{n}{\left(\alpha% +1\right)_{n}}M\left(-n,\alpha+1,z\right)}}$ `\KummerconfhyperU@{-n}{\alpha+1}{z} = (-1)^{n}\Pochhammersym{\alpha+1}{n}\KummerconfhyperM@{-n}{\alpha+1}{z}`		KummerU(- n, alpha + 1, z) = (- 1)^(n)* pochhammer(alpha + 1, n)*KummerM(- n, alpha + 1, z)	HypergeometricU[- n, \[Alpha]+ 1, z] == (- 1)^(n)* Pochhammer[\[Alpha]+ 1, n]*Hypergeometric1F1[- n, \[Alpha]+ 1, z]	Failure	Failure	Successful [Tested: 63]	Successful [Tested: 63]
13.6.E19	${\displaystyle{\displaystyle(-1)^{n}{\left(\alpha+1\right)_{n}}M\left(-n,% \alpha+1,z\right)=(-1)^{n}n!L^{(\alpha)}_{n}\left(z\right)}}$ `(-1)^{n}\Pochhammersym{\alpha+1}{n}\KummerconfhyperM@{-n}{\alpha+1}{z} = (-1)^{n}n!\LaguerrepolyL[\alpha]{n}@{z}`		(- 1)^(n)* pochhammer(alpha + 1, n)KummerM(- n, alpha + 1, z) = (- 1)^(n) factorial(n)*LaguerreL(n, alpha, z)	(- 1)^(n)* Pochhammer[\[Alpha]+ 1, n]Hypergeometric1F1[- n, \[Alpha]+ 1, z] == (- 1)^(n) (n)!*LaguerreL[n, \[Alpha], z]	Missing Macro Error	Successful	Skip - symbolical successful subtest	Successful [Tested: 63]
13.6.E20	${\displaystyle{\displaystyle U\left(-n,z-n+1,a\right)={\left(-z\right)_{n}}M% \left(-n,z-n+1,a\right)}}$ `\KummerconfhyperU@{-n}{z-n+1}{a} = \Pochhammersym{-z}{n}\KummerconfhyperM@{-n}{z-n+1}{a}`		KummerU(- n, z - n + 1, a) = pochhammer(- z, n)*KummerM(- n, z - n + 1, a)	HypergeometricU[- n, z - n + 1, a] == Pochhammer[- z, n]*Hypergeometric1F1[- n, z - n + 1, a]	Failure	Failure	Error	Failed [6 / 126] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[n, 3], Rule[z, 2]} Result: Indeterminate Test Values: {Rule[a, 1.5], Rule[n, 3], Rule[z, 2]} ... skip entries to safe data
13.6.E20	${\displaystyle{\displaystyle{\left(-z\right)_{n}}M\left(-n,z-n+1,a\right)=a^{n% }C_{n}\left(z;a\right)}}$ `\Pochhammersym{-z}{n}\KummerconfhyperM@{-n}{z-n+1}{a} = a^{n}\CharlierpolyC{n}@{z}{a}`		Error	Pochhammer[- z, n]Hypergeometric1F1[- n, z - n + 1, a] == (a)^(n) HypergeometricPFQ[{-(n), -(z)}, {}, -Divide[1,a]]	Missing Macro Error	Missing Macro Error	Skip - symbolical successful subtest	Skip - symbolical successful subtest
13.6.E21	${\displaystyle{\displaystyle U\left(a,b,z\right)=z^{-a}{{}_{2}F_{0}}\left(a,a-% b+1;-;-z^{-1}\right)}}$ `\KummerconfhyperU@{a}{b}{z} = z^{-a}\genhyperF{2}{0}@{a,a-b+1}{-}{-z^{-1}}`		KummerU(a, b, z) = (z)^(- a)* hypergeom([a , a - b + 1], [-], - (z)^(- 1))	HypergeometricU[a, b, z] == (z)^(- a)* HypergeometricPFQ[{a , a - b + 1}, {-}, - (z)^(- 1)]	Error	Failure	-	Error
13.7.E4	${\displaystyle{\displaystyle U\left(a,b,z\right)=z^{-a}\sum_{s=0}^{n-1}\frac{{% \left(a\right)_{s}}{\left(a-b+1\right)_{s}}}{s!}(-z)^{-s}+\varepsilon_{n}(z)}}$ `\KummerconfhyperU@{a}{b}{z} = z^{-a}\sum_{s=0}^{n-1}\frac{\Pochhammersym{a}{s}\Pochhammersym{a-b+1}{s}}{s!}(-z)^{-s}+\varepsilon_{n}(z)`		KummerU(a, b, z) = (z)^(- a)* sum((pochhammer(a, s)pochhammer(a - b + 1, s))/(factorial(s))(- z)^(- s), s = 0..n - 1)+ varepsilon[n](z)	HypergeometricU[a, b, z] == (z)^(- a)* Sum[Divide[Pochhammer[a, s]Pochhammer[a - b + 1, s],(s)!](- z)^(- s), {s, 0, n - 1}, GenerateConditions->None]+ Subscript[\[CurlyEpsilon], n][z]	Failure	Failure	Failed [300 / 300] Result: 1.515657870-.5735934827I Test Values: {a = -3/2, b = -3/2, z = 1/23^(1/2)+1/2I, varepsilon[n] = 1/23^(1/2)+1/2I, n = 1, varepsilon = 1} Result: 1.515657870-.5735934827I Test Values: {a = -3/2, b = -3/2, z = 1/23^(1/2)+1/2I, varepsilon[n] = 1/23^(1/2)+1/2I, n = 1, varepsilon = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[1.515657869456145, -0.5735934817267648] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ε, 1], Rule[Subscript[ε, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.515657869456145, -0.5735934817267648] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ε, 2], Rule[Subscript[ε, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
13.7.E10	${\displaystyle{\displaystyle U\left(a,b,z\right)=z^{-a}\sum_{s=0}^{n-1}\frac{{% \left(a\right)_{s}}{\left(a-b+1\right)_{s}}}{s!}(-z)^{-s}+R_{n}(a,b,z)}}$ `\KummerconfhyperU@{a}{b}{z} = z^{-a}\sum_{s=0}^{n-1}\frac{\Pochhammersym{a}{s}\Pochhammersym{a-b+1}{s}}{s!}(-z)^{-s}+R_{n}(a,b,z)`	${\displaystyle{\displaystyle\Re a>0,\Re(a-b+1)>0}}$	KummerU(a, b, z) = (z)^(- a)* sum((pochhammer(a, s)pochhammer(a - b + 1, s))/(factorial(s))(- z)^(- s)+(((- 1)^(n)* 2Pi(z)^(a - b))/(GAMMA(a)GAMMA(a - b + 1))(sum((pochhammer(1 - a, s)pochhammer(b - a, s))/(factorial(s))(- z)^(- s)* G[n + 2a - b - s](z), s = 0..m - 1)+ pochhammer(1 - a, m)pochhammer(b - a, m)*R[m , n](a , b , z))), s = 0..n - 1)	HypergeometricU[a, b, z] == (z)^(- a)* Sum[Divide[Pochhammer[a, s]Pochhammer[a - b + 1, s],(s)!](- z)^(- s)+(Divide[(- 1)^(n)* 2Pi(z)^(a - b),Gamma[a]Gamma[a - b + 1]](Sum[Divide[Pochhammer[1 - a, s]Pochhammer[b - a, s],(s)!](- z)^(- s)* Subscript[G, n + 2a - b - s][z], {s, 0, m - 1}, GenerateConditions->None]+ Pochhammer[1 - a, m]Pochhammer[b - a, m]*Subscript[R, m , n][a , b , z])), {s, 0, n - 1}, GenerateConditions->None]	Failure	Failure	Failed [300 / 300] Result: .1211969897-.2855680854e-1I+(-.7071067811+.7071067809I)(2.023326709-.5908179514I+(.8862269255-1.534990063I)(1.500000000, -1.500000000, .8660254040+.5000000000I)) Test Values: {a = 3/2, b = -3/2, z = 1/23^(1/2)+1/2I, G[n+2a-b-s] = 1/23^(1/2)+1/2I, R[m,n] = 1/23^(1/2)+1/2I, m = 1, n = 1} Result: .1211969897-.2855680854e-1I+(-.7071067811+.7071067809I)(-6.242805838+4.181635900I+(-1.772453851+3.069980127I)(1.500000000, -1.500000000, .8660254040+.5000000000I)) Test Values: {a = 3/2, b = -3/2, z = 1/23^(1/2)+1/2I, G[n+2a-b-s] = 1/23^(1/2)+1/2I, R[m,n] = 1/23^(1/2)+1/2I, m = 1, n = 2} ... skip entries to safe data	Error
13.8.E3	${\displaystyle{\displaystyle\left(e^{t}-1\right)^{a-1}\exp\left(t+z(1-e^{-t})% \right)=\sum_{s=0}^{\infty}q_{s}(z,a)t^{s+a-1}}}$ `\left(e^{t}-1\right)^{a-1}\exp@{t+z(1-e^{-t})} = \sum_{s=0}^{\infty}q_{s}(z,a)t^{s+a-1}`		(exp(t)- 1)^(a - 1)* exp(t + z(1 - exp(- t))) = sum(q[s](z , a) (t)^(s + a - 1), s = 0..infinity)	(Exp[t]- 1)^(a - 1)* Exp[t + z(1 - Exp[- t])] == Sum[Subscript[q, s][z , a] (t)^(s + a - 1), {s, 0, Infinity}, GenerateConditions->None]	Error	Failure	-	Error
13.8#Ex1	${\displaystyle{\displaystyle p_{k}(z)=\sum_{s=0}^{k}\genfrac{(}{)}{0.0pt}{}{k}% {s}{\left(1-b+s\right)_{k-s}}z^{s}c_{k+s}(z)}}$ `p_{k}(z) = \sum_{s=0}^{k}\binom{k}{s}\Pochhammersym{1-b+s}{k-s}z^{s}c_{k+s}(z)`		p[k](z) = sum(binomial(k,s)pochhammer(1 - b + s, k - s)(z)^(s)* c[k + s](z), s = 0..k)	Subscript[p, k][z] == Sum[Binomial[k,s]Pochhammer[1 - b + s, k - s](z)^(s)* Subscript[c, k + s][z], {s, 0, k}, GenerateConditions->None]	Failure	Failure	Failed [300 / 300] Result: -.7500000009-2.299038107I Test Values: {b = -3/2, z = 1/23^(1/2)+1/2I, c[k+s] = 1/23^(1/2)+1/2I, p[k] = 1/23^(1/2)+1/2I, k = 1} Result: -3.375000005-14.57772229I Test Values: {b = -3/2, z = 1/23^(1/2)+1/2I, c[k+s] = 1/23^(1/2)+1/2I, p[k] = 1/23^(1/2)+1/2I, k = 2} ... skip entries to safe data	Skipped - Because timed out
13.8#Ex2	${\displaystyle{\displaystyle q_{k}(z)=\sum_{s=0}^{k}\genfrac{(}{)}{0.0pt}{}{k}% {s}{\left(2-b+s\right)_{k-s}}z^{s}c_{k+s+1}(z)}}$ `q_{k}(z) = \sum_{s=0}^{k}\binom{k}{s}\Pochhammersym{2-b+s}{k-s}z^{s}c_{k+s+1}(z)`		q[k](z) = sum(binomial(k,s)pochhammer(2 - b + s, k - s)(z)^(s)* c[k + s + 1](z), s = 0..k)	Subscript[q, k][z] == Sum[Binomial[k,s]Pochhammer[2 - b + s, k - s](z)^(s)* Subscript[c, k + s + 1][z], {s, 0, k}, GenerateConditions->None]	Failure	Failure	Failed [300 / 300] Result: -1.250000001-3.165063511I Test Values: {b = -3/2, z = 1/23^(1/2)+1/2I, c[k+s+1] = 1/23^(1/2)+1/2I, q[k] = 1/23^(1/2)+1/2I, k = 1} Result: -6.875000009-22.63990012I Test Values: {b = -3/2, z = 1/23^(1/2)+1/2I, c[k+s+1] = 1/23^(1/2)+1/2I, q[k] = 1/23^(1/2)+1/2I, k = 2} ... skip entries to safe data	Skipped - Because timed out
13.8.E16	${\displaystyle{\displaystyle(k+1)c_{k+1}(z)+\sum_{s=0}^{k}\left(\frac{bB_{s+1}% }{(s+1)!}+\frac{z(s+1)B_{s+2}}{(s+2)!}\right)c_{k-s}(z)=0}}$ `(k+1)c_{k+1}(z)+\sum_{s=0}^{k}\left(\frac{b\BernoullinumberB{s+1}}{(s+1)!}+\frac{z(s+1)\BernoullinumberB{s+2}}{(s+2)!}\right)c_{k-s}(z) = 0`		(k + 1)c[k + 1](z)+ sum(((bbernoulli(s + 1))/(factorial(s + 1))+(z(s + 1)bernoulli(s + 2))/(factorial(s + 2)))*c[k - s](z), s = 0..k) = 0	(k + 1)Subscript[c, k + 1][z]+ Sum[(Divide[bBernoulliB[s + 1],(s + 1)!]+Divide[z(s + 1)BernoulliB[s + 2],(s + 2)!])*Subscript[c, k - s][z], {s, 0, k}, GenerateConditions->None] == 0	Failure	Failure	Failed [300 / 300] Result: 2.313541668+4.086338379I Test Values: {b = -3/2, z = 1/23^(1/2)+1/2I, c[1+k] = 1/23^(1/2)+1/2I, c[k-s] = 1/23^(1/2)+1/2I, k = 3} Result: 1.377763239+3.777643283I Test Values: {b = -3/2, z = 1/23^(1/2)+1/2I, c[1+k] = 1/23^(1/2)+1/2I, c[k-s] = -1/2+1/2I3^(1/2), k = 3} ... skip entries to safe data	Skipped - Because timed out
13.8#Ex3	${\displaystyle{\displaystyle\frac{\partial f}{\partial t}=\left(b\left(\frac{1% }{t}-\frac{1}{e^{t}-1}\right)-z\left(\frac{1}{t^{2}}-\frac{e^{t}}{\left(e^{t}-% 1\right)^{2}}\right)\right)f}}$ `\pderiv{f}{t} = \left(b\left(\frac{1}{t}-\frac{1}{e^{t}-1}\right)-z\left(\frac{1}{t^{2}}-\frac{e^{t}}{\left(e^{t}-1\right)^{2}}\right)\right)f`		diff(f, t) = (b((1)/(t)-(1)/(exp(t)- 1))- z((1)/((t)^(2))-(exp(t))/((exp(t)- 1)^(2))))*f	D[f, t] == (b(Divide[1,t]-Divide[1,Exp[t]- 1])- z(Divide[1,(t)^(2)]-Divide[Exp[t],(Exp[t]- 1)^(2)]))*f	Failure	Failure	Failed [300 / 300] Result: .8434854075+.5301342049I Test Values: {b = -3/2, f = 1/23^(1/2)+1/2I, t = -3/2, z = 1/23^(1/2)+1/2I} Result: .7413969054+.5027796732I Test Values: {b = -3/2, f = 1/23^(1/2)+1/2I, t = -3/2, z = -1/2+1/2I3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.8434854065788572, 0.5301342044541701] Test Values: {Rule[b, -1.5], Rule[f, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.7413969045334019, 0.5027796727745873] Test Values: {Rule[b, -1.5], Rule[f, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.9.E1	${\displaystyle{\displaystyle p(a,b)=\left\lceil-a\right\rceil}}$ `p(a,b) = \ceiling{-a}`	${\displaystyle{\displaystyle a<0,b\geq 0}}$	p(a , b) = ceil(- a)	p[a , b] == Ceiling[- a]	Failure	Failure	Failed [90 / 90] Result: (.8660254040+.5000000000I)(-1.500000000, 1.500000000)-2. Test Values: {a = -3/2, b = 3/2, p = 1/23^(1/2)+1/2I} Result: (-.5000000000+.8660254040I)(-1.500000000, 1.500000000)-2. Test Values: {a = -3/2, b = 3/2, p = -1/2+1/2I3^(1/2)} ... skip entries to safe data	Error
13.9.E2	${\displaystyle{\displaystyle p(a,b)=0}}$ `p(a,b) = 0`	${\displaystyle{\displaystyle a\geq 0,b\geq 0}}$	p(a , b) = 0	p[a , b] == 0	Skipped - no semantic math	Skipped - no semantic math	-	-
13.9.E3	${\displaystyle{\displaystyle p(a,b)=1}}$ `p(a,b) = 1`	${\displaystyle{\displaystyle a\geq 0,-1<b,b<0}}$	p(a , b) = 1	p[a , b] == 1	Skipped - no semantic math	Skipped - no semantic math	-	-
13.9.E4	${\displaystyle{\displaystyle p(a,b)=\left\lfloor-\tfrac{1}{2}b\right\rfloor-% \left\lfloor-\tfrac{1}{2}(b+1)\right\rfloor}}$ `p(a,b) = \floor{-\tfrac{1}{2}b}-\floor{-\tfrac{1}{2}(b+1)}`	${\displaystyle{\displaystyle a\geq 0,b\leq-1}}$	p(a , b) = floor(-(1)/(2)b)- floor(-(1)/(2)(b + 1))	p[a , b] == Floor[-Divide[1,2]b]- Floor[-Divide[1,2](b + 1)]	Failure	Failure	Error	Error
13.9.E5	${\displaystyle{\displaystyle p(a,b)=\left\lceil-a\right\rceil-\left\lceil-b% \right\rceil}}$ `p(a,b) = \ceiling{-a}-\ceiling{-b}`	${\displaystyle{\displaystyle\left\lceil-a\right\rceil\geq\left\lceil-b\right% \rceil,a<0,b<0}}$	p(a , b) = ceil(- a)- ceil(- b)	p[a , b] == Ceiling[- a]- Ceiling[- b]	Failure	Failure	Error	Error
13.9.E6	${\displaystyle{\displaystyle p(a,b)=\left\lfloor\tfrac{1}{2}\left(\left\lceil-% b\right\rceil-\left\lceil-a\right\rceil+1\right)\right\rfloor-\left\lfloor% \tfrac{1}{2}\left(\left\lceil-b\right\rceil-\left\lceil-a\right\rceil\right)% \right\rfloor}}$ `p(a,b) = \floor{\tfrac{1}{2}\left(\ceiling{-b}-\ceiling{-a}+1\right)}-\floor{\tfrac{1}{2}\left(\ceiling{-b}-\ceiling{-a}\right)}`	${\displaystyle{\displaystyle\left\lceil-b\right\rceil>\left\lceil-a\right% \rceil,\left\lceil-a\right\rceil>0}}$	p(a , b) = floor((1)/(2)(ceil(- b)- ceil(- a)+ 1))- floor((1)/(2)(ceil(- b)- ceil(- a)))	p[a , b] == Floor[Divide[1,2](Ceiling[- b]- Ceiling[- a]+ 1)]- Floor[Divide[1,2](Ceiling[- b]- Ceiling[- a])]	Failure	Failure	Failed [20 / 20] Result: (.8660254040+.5000000000I)(-.5000000000, -1.500000000)-1. Test Values: {a = -1/2, b = -3/2, p = 1/23^(1/2)+1/2I} Result: (-.5000000000+.8660254040I)(-.5000000000, -1.500000000)-1. Test Values: {a = -1/2, b = -3/2, p = -1/2+1/2I3^(1/2)} ... skip entries to safe data	Error
13.9.E11	${\displaystyle{\displaystyle T(a,b)=\left\lfloor-a\right\rfloor+1}}$ `T(a,b) = \floor{-a}+1`	${\displaystyle{\displaystyle a<0,\Gamma\left(a\right)\Gamma\left(a-b+1\right)>% 0}}$	T(a , b) = floor(- a)+ 1	T[a , b] == Floor[- a]+ 1	Failure	Failure	Error	Error
13.9.E12	${\displaystyle{\displaystyle T(a,b)=\left\lfloor-a\right\rfloor}}$ `T(a,b) = \floor{-a}`	${\displaystyle{\displaystyle a<0,\Gamma\left(a\right)\Gamma\left(a-b+1\right)<% 0}}$	T(a , b) = floor(- a)	T[a , b] == Floor[- a]	Failure	Failure	Error	Error
13.9.E13	${\displaystyle{\displaystyle T(a,b)=0}}$ `T(a,b) = 0`	${\displaystyle{\displaystyle a>0}}$	T(a , b) = 0	T[a , b] == 0	Skipped - no semantic math	Skipped - no semantic math	-	-
13.9.E14	${\displaystyle{\displaystyle P(a,b)=\left\lceil b-a-1\right\rceil}}$ `P(a,b) = \ceiling{b-a-1}`	${\displaystyle{\displaystyle a+1<b}}$	P(a , b) = ceil(b - a - 1)	P[a , b] == Ceiling[b - a - 1]	Failure	Failure	Failed [100 / 100] Result: (.8660254040+.5000000000I)(-1.500000000, 1.500000000)-2. Test Values: {P = 1/23^(1/2)+1/2I, a = -3/2, b = 3/2} Result: (.8660254040+.5000000000I)(-1.500000000, .5000000000)-1. Test Values: {P = 1/23^(1/2)+1/2I, a = -3/2, b = 1/2} ... skip entries to safe data	Error
13.9.E15	${\displaystyle{\displaystyle P(a,b)=0}}$ `P(a,b) = 0`	${\displaystyle{\displaystyle a+1\geq b}}$	P(a , b) = 0	P[a , b] == 0	Skipped - no semantic math	Skipped - no semantic math	-	-
13.10.E1	${\displaystyle{\displaystyle\int{\mathbf{M}}\left(a,b,z\right)\mathrm{d}z=% \frac{1}{a-1}{\mathbf{M}}\left(a-1,b-1,z\right)}}$ `\int\OlverconfhyperM@{a}{b}{z}\diff{z} = \frac{1}{a-1}\OlverconfhyperM@{a-1}{b-1}{z}`	${\displaystyle{\displaystyle\Re(b+s)>0,\Re((b-1)+s)>0}}$	int(KummerM(a, b, z)/GAMMA(b), z) = (1)/(a - 1)*KummerM(a - 1, b - 1, z)/GAMMA(b - 1)	Integrate[Hypergeometric1F1Regularized[a, b, z], z, GenerateConditions->None] == Divide[1,a - 1]*Hypergeometric1F1Regularized[a - 1, b - 1, z]	Successful	Failure	-	Failed [252 / 252] Result: Complex[-0.4231421876608173, 0.0] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.42314218766081735, 0.0] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.10.E2	${\displaystyle{\displaystyle\int U\left(a,b,z\right)\mathrm{d}z=-\frac{1}{a-1}% U\left(a-1,b-1,z\right)}}$ `\int\KummerconfhyperU@{a}{b}{z}\diff{z} = -\frac{1}{a-1}\KummerconfhyperU@{a-1}{b-1}{z}`		int(KummerU(a, b, z), z) = -(1)/(a - 1)*KummerU(a - 1, b - 1, z)	Integrate[HypergeometricU[a, b, z], z, GenerateConditions->None] == -Divide[1,a - 1]*HypergeometricU[a - 1, b - 1, z]	Successful	Successful	-	Successful [Tested: 252]
13.10.E3	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-zt}t^{b-1}{\mathbf{M}}\left(a% ,c,kt\right)\mathrm{d}t=\Gamma\left(b\right)z^{-b}{{}_{2}{\mathbf{F}}_{1}}% \left(a,b;c;\ifrac{k}{z}\right)}}$ `\int_{0}^{\infty}e^{-zt}t^{b-1}\OlverconfhyperM@{a}{c}{kt}\diff{t} = \EulerGamma@{b}z^{-b}\genhyperOlverF{2}{1}@{a,b}{c}{\ifrac{k}{z}}`	${\displaystyle{\displaystyle\Re b>0,\Re z>\max\left(\Re k,\Re(c+s)>0}\)\@add@PDF@RDFa@triples\end{document}}$	int(exp(- zt)(t)^(b - 1)* KummerM(a, c, kt)/GAMMA(c), t = 0..infinity) = GAMMA(b)(z)^(- b)* hypergeom([a , b], [c], (k)/(z))	Integrate[Exp[- zt](t)^(b - 1)* Hypergeometric1F1Regularized[a, c, kt], {t, 0, Infinity}, GenerateConditions->None] == Gamma[b](z)^(- b)* HypergeometricPFQRegularized[{a , b}, {c}, Divide[k,z]]	Failure	Aborted	Failed [300 / 300] Result: Float(undefined)+Float(undefined)I Test Values: {a = -3/2, b = -3/2, c = -3/2, z = 1/23^(1/2)+1/2I, k = 1} Result: Float(undefined)+Float(undefined)I Test Values: {a = -3/2, b = -3/2, c = -3/2, z = 1/23^(1/2)+1/2I, k = 2} ... skip entries to safe data	Skipped - Because timed out
13.10.E4	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-zt}t^{b-1}{\mathbf{M}}\left(a% ,b,t\right)\mathrm{d}t=z^{-b}\left(1-\frac{1}{z}\right)^{-a}}}$ `\int_{0}^{\infty}e^{-zt}t^{b-1}\OlverconfhyperM@{a}{b}{t}\diff{t} = z^{-b}\left(1-\frac{1}{z}\right)^{-a}`	${\displaystyle{\displaystyle\Re b>0,\Re z>1,\Re(b+s)>0}}$	int(exp(- zt)(t)^(b - 1)* KummerM(a, b, t)/GAMMA(b), t = 0..infinity) = (z)^(- b)*(1 -(1)/(z))^(- a)	Integrate[Exp[- zt](t)^(b - 1)* Hypergeometric1F1Regularized[a, b, t], {t, 0, Infinity}, GenerateConditions->None] == (z)^(- b)*(1 -Divide[1,z])^(- a)	Failure	Aborted	Failed [24 / 36] Result: -.2095131204 Test Values: {a = -3/2, b = 3/2, z = 3/2} Result: -.2500000000 Test Values: {a = -3/2, b = 3/2, z = 2} ... skip entries to safe data	Skipped - Because timed out
13.10.E5	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-t}t^{b-1}{\mathbf{M}}\left(a,% c,t\right)\mathrm{d}t=\frac{\Gamma\left(b\right)\Gamma\left(c-a-b\right)}{% \Gamma\left(c-a\right)\Gamma\left(c-b\right)}}}$ `\int_{0}^{\infty}e^{-t}t^{b-1}\OlverconfhyperM@{a}{c}{t}\diff{t} = \frac{\EulerGamma@{b}\EulerGamma@{c-a-b}}{\EulerGamma@{c-a}\EulerGamma@{c-b}}`	${\displaystyle{\displaystyle\Re\left(c-a\right)>\Re b,\Re b>0,\Re(c-a-b)>0,\Re% (c-a)>0,\Re(c-b)>0,\Re(c+s)>0}}$	int(exp(- t)(t)^(b - 1) KummerM(a, c, t)/GAMMA(c), t = 0..infinity) = (GAMMA(b)GAMMA(c - a - b))/(GAMMA(c - a)GAMMA(c - b))	Integrate[Exp[- t](t)^(b - 1) Hypergeometric1F1Regularized[a, c, t], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[b]Gamma[c - a - b],Gamma[c - a]Gamma[c - b]]	Successful	Aborted	-	Skipped - Because timed out
13.10.E6	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-zt-t^{2}}t^{2b-2}{\mathbf{M}}% \left(a,b,t^{2}\right)\mathrm{d}t=\tfrac{1}{2}\pi^{-\frac{1}{2}}\Gamma\left(b-% \tfrac{1}{2}\right)U\left(b-\tfrac{1}{2},a+\tfrac{1}{2},\tfrac{1}{4}z^{2}% \right)}}$ `\int_{0}^{\infty}e^{-zt-t^{2}}t^{2b-2}\OlverconfhyperM@{a}{b}{t^{2}}\diff{t} = \tfrac{1}{2}\pi^{-\frac{1}{2}}\EulerGamma@{b-\tfrac{1}{2}}\KummerconfhyperU@{b-\tfrac{1}{2}}{a+\tfrac{1}{2}}{\tfrac{1}{4}z^{2}}`	${\displaystyle{\displaystyle\Re b>\tfrac{1}{2},\Re z>0,\Re(b-\tfrac{1}{2})>0,% \Re(b+s)>0}}$	int(exp(- zt - (t)^(2))(t)^(2b - 2) KummerM(a, b, (t)^(2))/GAMMA(b), t = 0..infinity) = (1)/(2)(Pi)^(-(1)/(2)) GAMMA(b -(1)/(2))KummerU(b -(1)/(2), a +(1)/(2), (1)/(4)(z)^(2))	Integrate[Exp[- zt - (t)^(2)](t)^(2b - 2) Hypergeometric1F1Regularized[a, b, (t)^(2)], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,2](Pi)^(-Divide[1,2]) Gamma[b -Divide[1,2]]HypergeometricU[b -Divide[1,2], a +Divide[1,2], Divide[1,4](z)^(2)]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
13.10.E7	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-zt}t^{b-1}U\left(a,c,t\right)% \mathrm{d}t=\Gamma\left(b\right)\Gamma\left(b-c+1\right)\z^{-b}{{}_{2}{% \mathbf{F}}_{1}}\left(a,b;a+b-c+1;1-\frac{1}{z}\right)}}$ `\int_{0}^{\infty}e^{-zt}t^{b-1}\KummerconfhyperU@{a}{c}{t}\diff{t} = \EulerGamma@{b}\EulerGamma@{b-c+1}\z^{-b}\genhyperOlverF{2}{1}@{a,b}{a+b-c+1}{1-\frac{1}{z}}`	${\displaystyle{\displaystyle\Re b>\max\left(\Re c-1,\Re z>0,\Re b>0,\Re(b-c+1)% >0}\)\@add@PDF@RDFa@triples\end{document}}$	int(exp(- zt)(t)^(b - 1)* KummerU(a, c, t), t = 0..infinity) = GAMMA(b)GAMMA(b - c + 1) (z)^(- b)* hypergeom([a , b], [a + b - c + 1], 1 -(1)/(z))	Integrate[Exp[- zt](t)^(b - 1)* HypergeometricU[a, c, t], {t, 0, Infinity}, GenerateConditions->None] == Gamma[b]Gamma[b - c + 1] (z)^(- b)* HypergeometricPFQRegularized[{a , b}, {a + b - c + 1}, 1 -Divide[1,z]]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
13.10.E8	${\displaystyle{\displaystyle\frac{1}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{tz% }t^{-a}{\mathbf{M}}\left(a,b,\ifrac{y}{t}\right)\mathrm{d}t=\frac{1}{\Gamma% \left(a\right)}z^{\frac{1}{2}(2a-b-1)}y^{\frac{1}{2}(1-b)}I_{b-1}\left(2\sqrt{% zy}\right)}}$ `\frac{1}{2\pi\iunit}\int_{-\infty}^{(0+)}e^{tz}t^{-a}\OlverconfhyperM@{a}{b}{\ifrac{y}{t}}\diff{t} = \frac{1}{\EulerGamma@{a}}z^{\frac{1}{2}(2a-b-1)}y^{\frac{1}{2}(1-b)}\modBesselI{b-1}@{2\sqrt{zy}}`	${\displaystyle{\displaystyle\Re z>0,\Re a>0,\Re(b+s)>0,\Re((b-1)+k+1)>0}}$	(1)/(2PiI)int(exp(t(x + yI))(t)^(- a)* KummerM(a, b, (y)/(t))/GAMMA(b), t = - infinity..(0 +)) = (1)/(GAMMA(a))(x + yI)^((1)/(2)(2a - b - 1))* (y)^((1)/(2)(1 - b)) BesselI(b - 1, 2sqrt((x + yI)*y))	Divide[1,2PiI]Integrate[Exp[t(x + yI)](t)^(- a)* Hypergeometric1F1Regularized[a, b, Divide[y,t]], {t, - Infinity, (0 +)}, GenerateConditions->None] == Divide[1,Gamma[a]](x + yI)^(Divide[1,2](2a - b - 1))* (y)^(Divide[1,2](1 - b)) BesselI[b - 1, 2Sqrt[(x + yI)*y]]	Error	Failure	-	Error
13.10.E9	${\displaystyle{\displaystyle\frac{1}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{tz% }t^{-a}U\left(a,b,\ifrac{y}{t}\right)\mathrm{d}t=\frac{2z^{\frac{1}{2}(2a-b-1)% }y^{\frac{1}{2}(1-b)}}{\Gamma\left(a\right)\Gamma\left(a-b+1\right)}K_{b-1}% \left(2\sqrt{zy}\right)}}$ `\frac{1}{2\pi\iunit}\int_{-\infty}^{(0+)}e^{tz}t^{-a}\KummerconfhyperU@{a}{b}{\ifrac{y}{t}}\diff{t} = \frac{2z^{\frac{1}{2}(2a-b-1)}y^{\frac{1}{2}(1-b)}}{\EulerGamma@{a}\EulerGamma@{a-b+1}}\modBesselK{b-1}@{2\sqrt{zy}}`	${\displaystyle{\displaystyle\Re z>0,\Re a>0,\Re(a-b+1)>0}}$	(1)/(2PiI)int(exp(t(x + yI))(t)^(- a)* KummerU(a, b, (y)/(t)), t = - infinity..(0 +)) = (2(x + yI)^((1)/(2)(2a - b - 1))* (y)^((1)/(2)(1 - b)))/(GAMMA(a)GAMMA(a - b + 1))BesselK(b - 1, 2sqrt((x + yI)y))	Divide[1,2PiI]Integrate[Exp[t(x + yI)](t)^(- a)* HypergeometricU[a, b, Divide[y,t]], {t, - Infinity, (0 +)}, GenerateConditions->None] == Divide[2(x + yI)^(Divide[1,2](2a - b - 1))* (y)^(Divide[1,2](1 - b)),Gamma[a]Gamma[a - b + 1]]BesselK[b - 1, 2Sqrt[(x + yI)y]]	Error	Failure	-	Error
13.10.E10	${\displaystyle{\displaystyle\int_{0}^{\infty}t^{\lambda-1}{\mathbf{M}}\left(a,% b,-t\right)\mathrm{d}t=\frac{\Gamma\left(\lambda\right)\Gamma\left(a-\lambda% \right)}{\Gamma\left(a\right)\Gamma\left(b-\lambda\right)}}}$ `\int_{0}^{\infty}t^{\lambda-1}\OlverconfhyperM@{a}{b}{-t}\diff{t} = \frac{\EulerGamma@{\lambda}\EulerGamma@{a-\lambda}}{\EulerGamma@{a}\EulerGamma@{b-\lambda}}`	${\displaystyle{\displaystyle 0<\Re\lambda,\Re\lambda<\Re a,\Re(\lambda)>0,\Re(% a-\lambda)>0,\Re a>0,\Re(b-\lambda)>0,\Re(b+s)>0}}$	int((t)^(lambda - 1)* KummerM(a, b, - t)/GAMMA(b), t = 0..infinity) = (GAMMA(lambda)GAMMA(a - lambda))/(GAMMA(a)GAMMA(b - lambda))	Integrate[(t)^(\[Lambda]- 1)* Hypergeometric1F1Regularized[a, b, - t], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[\[Lambda]]Gamma[a - \[Lambda]],Gamma[a]Gamma[b - \[Lambda]]]	Successful	Aborted	-	Skipped - Because timed out
13.10.E11	${\displaystyle{\displaystyle\int_{0}^{\infty}t^{\lambda-1}U\left(a,b,t\right)% \mathrm{d}t=\frac{\Gamma\left(\lambda\right)\Gamma\left(a-\lambda\right)\Gamma% \left(\lambda-b+1\right)}{\Gamma\left(a\right)\Gamma\left(a-b+1\right)}}}$ `\int_{0}^{\infty}t^{\lambda-1}\KummerconfhyperU@{a}{b}{t}\diff{t} = \frac{\EulerGamma@{\lambda}\EulerGamma@{a-\lambda}\EulerGamma@{\lambda-b+1}}{\EulerGamma@{a}\EulerGamma@{a-b+1}}`	${\displaystyle{\displaystyle\max\left(\Re b-1<\Re\lambda,0\right)<\Re\lambda,% \Re\lambda<\Re a,\Re(\lambda)>0,\Re(a-\lambda)>0,\Re(\lambda-b+1)>0,\Re a>0,% \Re(a-b+1)>0}}$	int((t)^(lambda - 1)* KummerU(a, b, t), t = 0..infinity) = (GAMMA(lambda)GAMMA(a - lambda)GAMMA(lambda - b + 1))/(GAMMA(a)*GAMMA(a - b + 1))	Integrate[(t)^(\[Lambda]- 1)* HypergeometricU[a, b, t], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[\[Lambda]]Gamma[a - \[Lambda]]Gamma[\[Lambda]- b + 1],Gamma[a]*Gamma[a - b + 1]]	Successful	Successful	-	Successful [Tested: 300]
13.10.E12	${\displaystyle{\displaystyle\int_{0}^{\infty}\cos\left(2xt\right){\mathbf{M}}% \left(a,b,-t^{2}\right)\mathrm{d}t=\frac{\sqrt{\pi}}{2\Gamma\left(a\right)}x^{% 2a-1}e^{-x^{2}}U\left(b-\tfrac{1}{2},a+\tfrac{1}{2},x^{2}\right)}}$ `\int_{0}^{\infty}\cos@{2xt}\OlverconfhyperM@{a}{b}{-t^{2}}\diff{t} = \frac{\sqrt{\pi}}{2\EulerGamma@{a}}x^{2a-1}e^{-x^{2}}\KummerconfhyperU@{b-\tfrac{1}{2}}{a+\tfrac{1}{2}}{x^{2}}`	${\displaystyle{\displaystyle\Re a>0,\Re(b+s)>0}}$	int(cos(2xt)KummerM(a, b, - (t)^(2))/GAMMA(b), t = 0..infinity) = (sqrt(Pi))/(2GAMMA(a))(x)^(2a - 1)* exp(- (x)^(2))*KummerU(b -(1)/(2), a +(1)/(2), (x)^(2))	Integrate[Cos[2xt]Hypergeometric1F1Regularized[a, b, - (t)^(2)], {t, 0, Infinity}, GenerateConditions->None] == Divide[Sqrt[Pi],2Gamma[a]](x)^(2a - 1)* Exp[- (x)^(2)]*HypergeometricU[b -Divide[1,2], a +Divide[1,2], (x)^(2)]	Failure	Aborted	Failed [51 / 54] Result: Float(undefined)+Float(undefined)I Test Values: {a = 3/2, b = -3/2, x = 3/2} Result: Float(undefined)+Float(undefined)I Test Values: {a = 3/2, b = -3/2, x = 1/2} ... skip entries to safe data	Skipped - Because timed out
13.10.E13	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-t}t^{b-1-\frac{1}{2}\nu}{% \mathbf{M}}\left(a,b,t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=x^{-a+% \frac{1}{2}\nu}e^{-x}{\mathbf{M}}\left(\nu-b+1,\nu-a+1,x\right)}}$ `\int_{0}^{\infty}e^{-t}t^{b-1-\frac{1}{2}\nu}\OlverconfhyperM@{a}{b}{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = x^{-a+\frac{1}{2}\nu}e^{-x}\OlverconfhyperM@{\nu-b+1}{\nu-a+1}{x}`	${\displaystyle{\displaystyle x>0,2\Re a<\Re\nu+\tfrac{5}{2},\Re b>0,\Re(\nu+k+% 1)>0,\Re(b+s)>0,\Re((\nu-a+1)+s)>0}}$	int(exp(- t)(t)^(b - 1 -(1)/(2)nu)* KummerM(a, b, t)/GAMMA(b)BesselJ(nu, 2sqrt(xt)), t = 0..infinity) = (x)^(- a +(1)/(2)nu)* exp(- x)*KummerM(nu - b + 1, nu - a + 1, x)/GAMMA(nu - a + 1)	Integrate[Exp[- t](t)^(b - 1 -Divide[1,2]\[Nu])* Hypergeometric1F1Regularized[a, b, t]BesselJ[\[Nu], 2Sqrt[xt]], {t, 0, Infinity}, GenerateConditions->None] == (x)^(- a +Divide[1,2]\[Nu])* Exp[- x]*Hypergeometric1F1Regularized[\[Nu]- b + 1, \[Nu]- a + 1, x]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
13.10.E14	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}\nu}{\mathbf{% M}}\left(a,b,t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=\frac{x^{\frac{% 1}{2}\nu}e^{-x}}{\Gamma\left(b-a\right)}U\left(a,a-b+\nu+2,x\right)}}$ `\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}\nu}\OlverconfhyperM@{a}{b}{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = \frac{x^{\frac{1}{2}\nu}e^{-x}}{\EulerGamma@{b-a}}\KummerconfhyperU@{a}{a-b+\nu+2}{x}`	${\displaystyle{\displaystyle x>0,-1<\Re\nu,\Re\nu<2\Re\left(b-a\right)-\tfrac{% 1}{2},\Re(\nu+k+1)>0,\Re(b-a)>0,\Re(b+s)>0}}$	int(exp(- t)(t)^((1)/(2)nu)* KummerM(a, b, t)/GAMMA(b)BesselJ(nu, 2sqrt(xt)), t = 0..infinity) = ((x)^((1)/(2)nu)* exp(- x))/(GAMMA(b - a))*KummerU(a, a - b + nu + 2, x)	Integrate[Exp[- t](t)^(Divide[1,2]\[Nu])* Hypergeometric1F1Regularized[a, b, t]BesselJ[\[Nu], 2Sqrt[xt]], {t, 0, Infinity}, GenerateConditions->None] == Divide[(x)^(Divide[1,2]\[Nu])* Exp[- x],Gamma[b - a]]*HypergeometricU[a, a - b + \[Nu]+ 2, x]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
13.10.E15	${\displaystyle{\displaystyle\int_{0}^{\infty}t^{\frac{1}{2}\nu}U\left(a,b,t% \right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=\frac{\Gamma\left(\nu-b+2% \right)}{\Gamma\left(a\right)}x^{\frac{1}{2}\nu}U\left(\nu-b+2,\nu-a+2,x\right% )}}$ `\int_{0}^{\infty}t^{\frac{1}{2}\nu}\KummerconfhyperU@{a}{b}{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = \frac{\EulerGamma@{\nu-b+2}}{\EulerGamma@{a}}x^{\frac{1}{2}\nu}\KummerconfhyperU@{\nu-b+2}{\nu-a+2}{x}`	${\displaystyle{\displaystyle x>0,\max\left(\Re b-2<\Re\nu,-1\right)<\Re\nu,\Re% \nu<2\Re a+\tfrac{1}{2},\Re(\nu+k+1)>0,\Re(\nu-b+2)>0,\Re a>0}}$	int((t)^((1)/(2)nu) KummerU(a, b, t)BesselJ(nu, 2sqrt(xt)), t = 0..infinity) = (GAMMA(nu - b + 2))/(GAMMA(a))(x)^((1)/(2)nu) KummerU(nu - b + 2, nu - a + 2, x)	Integrate[(t)^(Divide[1,2]\[Nu]) HypergeometricU[a, b, t]BesselJ[\[Nu], 2Sqrt[xt]], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[\[Nu]- b + 2],Gamma[a]](x)^(Divide[1,2]\[Nu]) HypergeometricU[\[Nu]- b + 2, \[Nu]- a + 2, x]	Successful	Aborted	-	Skipped - Because timed out
13.10.E16	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}\nu}U\left(a,% b,t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=\Gamma\left(\nu-b+2\right)% x^{\frac{1}{2}\nu}e^{-x}{\mathbf{M}}\left(a,a-b+\nu+2,x\right)}}$ `\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}\nu}\KummerconfhyperU@{a}{b}{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = \EulerGamma@{\nu-b+2}x^{\frac{1}{2}\nu}e^{-x}\OlverconfhyperM@{a}{a-b+\nu+2}{x}`	${\displaystyle{\displaystyle x>0,\max\left(\Re b-2<\Re\nu,-1\right)<\Re\nu,\Re% (\nu+k+1)>0,\Re(\nu-b+2)>0,\Re((a-b+\nu+2)+s)>0}}$	int(exp(- t)(t)^((1)/(2)nu)* KummerU(a, b, t)BesselJ(nu, 2sqrt(xt)), t = 0..infinity) = GAMMA(nu - b + 2)(x)^((1)/(2)nu) exp(- x)*KummerM(a, a - b + nu + 2, x)/GAMMA(a - b + nu + 2)	Integrate[Exp[- t](t)^(Divide[1,2]\[Nu])* HypergeometricU[a, b, t]BesselJ[\[Nu], 2Sqrt[xt]], {t, 0, Infinity}, GenerateConditions->None] == Gamma[\[Nu]- b + 2](x)^(Divide[1,2]\[Nu]) Exp[- x]*Hypergeometric1F1Regularized[a, a - b + \[Nu]+ 2, x]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
13.11.E1	${\displaystyle{\displaystyle M\left(a,b,z\right)=\Gamma\left(a-\tfrac{1}{2}% \right)e^{\frac{1}{2}z}\left(\tfrac{1}{4}z\right)^{\frac{1}{2}-a}\\sum_{s=0}^% {\infty}\frac{{\left(2a-1\right)_{s}}{\left(2a-b\right)_{s}}}{{\left(b\right)_% {s}}s!}\\left(a-\tfrac{1}{2}+s\right)\I_{a-\frac{1}{2}+s}\left(\tfrac{1}{2}z% \right)}}$ `\KummerconfhyperM@{a}{b}{z} = \EulerGamma@{a-\tfrac{1}{2}}e^{\frac{1}{2}z}\left(\tfrac{1}{4}z\right)^{\frac{1}{2}-a}\\sum_{s=0}^{\infty}\frac{\Pochhammersym{2a-1}{s}\Pochhammersym{2a-b}{s}}{\Pochhammersym{b}{s}s!}\\left(a-\tfrac{1}{2}+s\right)\\modBesselI{a-\frac{1}{2}+s}@{\tfrac{1}{2}z}`	${\displaystyle{\displaystyle\Re(a-\tfrac{1}{2})>0,\Re((a-\frac{1}{2}+s)+k+1)>0}}$	KummerM(a, b, z) = GAMMA(a -(1)/(2))exp((1)/(2)z)((1)/(4)z)^((1)/(2)- a)* sum((pochhammer(2a - 1, s)pochhammer(2a - b, s))/(pochhammer(b, s)factorial(s))(a -(1)/(2)+ s) BesselI(a -(1)/(2)+ s, (1)/(2)*z), s = 0..infinity)	Hypergeometric1F1[a, b, z] == Gamma[a -Divide[1,2]]Exp[Divide[1,2]z](Divide[1,4]z)^(Divide[1,2]- a)* Sum[Divide[Pochhammer[2a - 1, s]Pochhammer[2a - b, s],Pochhammer[b, s](s)!](a -Divide[1,2]+ s) BesselI[a -Divide[1,2]+ s, Divide[1,2]*z], {s, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Manual Skip!	Failed [84 / 84] Result: Plus[Complex[-3.202632216430895, 12.150063432924489], Times[Complex[-5.9381784278055925, 1.66646925063829], NSum[Times[Plus[1.0, s], BesselI[Plus[1.0, s], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Factorial[s], -1], Power[Pochhammer[-1.5, s], -1], Pochhammer[2.0, s], Pochhammer[4.5, s]] Test Values: {s, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[a, 1.5], Rule[b, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[3.448639860241066, -0.8097281072366314], Times[Complex[0.28180823919021325, 3.102430445912792], NSum[Times[Plus[1.0, s], BesselI[Plus[1.0, s], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]], Power[Factorial[s], -1], Power[Pochhammer[-1.5, s], -1], Pochhammer[2.0, s], Pochhammer[4.5, s]] Test Values: {s, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[a, 1.5], Rule[b, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
13.12.E1	${\displaystyle{\displaystyle M\left(a,b,z\right)M\left(-a,-b,-z\right)+\frac{a% (a-b)z^{2}}{b^{2}(1-b^{2})}M\left(1+a,2+b,z\right)M\left(1-a,2-b,-z\right)=1}}$ `\KummerconfhyperM@{a}{b}{z}\KummerconfhyperM@{-a}{-b}{-z}+\frac{a(a-b)z^{2}}{b^{2}(1-b^{2})}\KummerconfhyperM@{1+a}{2+b}{z}\KummerconfhyperM@{1-a}{2-b}{-z} = 1`		KummerM(a, b, z)KummerM(- a, - b, - z)+(a(a - b)(z)^(2))/((b)^(2)(1 - (b)^(2)))KummerM(1 + a, 2 + b, z)KummerM(1 - a, 2 - b, - z) = 1	Hypergeometric1F1[a, b, z]Hypergeometric1F1[- a, - b, - z]+Divide[a(a - b)(z)^(2),(b)^(2)(1 - (b)^(2))]Hypergeometric1F1[1 + a, 2 + b, z]Hypergeometric1F1[1 - a, 2 - b, - z] == 1	Failure	Failure	Error	Failed [84 / 252] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data

@@ Line 1: / Line 1: @@
+<div style="width: 100%; height: 75vh; overflow: auto;">
+{| class="wikitable sortable" style="margin: 0;"
+|-
+! scope="col" style="position: sticky; top: 0;" | DLMF
+! scope="col" style="position: sticky; top: 0;" | Formula
+! scope="col" style="position: sticky; top: 0;" | Constraints
+! scope="col" style="position: sticky; top: 0;" | Maple
+! scope="col" style="position: sticky; top: 0;" | Mathematica
+! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
+! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
+! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
+! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
+|-
+| [https://dlmf.nist.gov/13.2.E1 13.2.E1] || [[Item:Q4291|<math>z\deriv[2]{w}{z}+(b-z)\deriv{w}{z}-aw = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z\deriv[2]{w}{z}+(b-z)\deriv{w}{z}-aw = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>z*diff(w, [z$(2)])+(b - z)*diff(w, z)- a*w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>z*D[w, {z, 2}]+(b - z)*D[w, z]- a*w == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.299038106+.7500000000*I
+Test Values: {a = -3/2, b = -3/2, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.299038106+.7500000000*I
+Test Values: {a = -3/2, b = -3/2, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.299038105676658, 0.7499999999999999]
+Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.299038105676658, 0.7499999999999999]
+Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.2.E2 13.2.E2] || [[Item:Q4292|<math>\KummerconfhyperM@{a}{b}{z} = \sum_{s=0}^{\infty}\frac{\Pochhammersym{a}{s}}{\Pochhammersym{b}{s}s!}z^{s}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperM@{a}{b}{z} = \sum_{s=0}^{\infty}\frac{\Pochhammersym{a}{s}}{\Pochhammersym{b}{s}s!}z^{s}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerM(a, b, z) = sum((pochhammer(a, s))/(pochhammer(b, s)*factorial(s))*(z)^(s), s = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric1F1[a, b, z] == Sum[Divide[Pochhammer[a, s],Pochhammer[b, s]*(s)!]*(z)^(s), {s, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 252]
+|-
+| [https://dlmf.nist.gov/13.2.E3 13.2.E3] || [[Item:Q4293|<math>\OlverconfhyperM@{a}{b}{z} = \sum_{s=0}^{\infty}\frac{\Pochhammersym{a}{s}}{\EulerGamma@{b+s}s!}z^{s}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\OlverconfhyperM@{a}{b}{z} = \sum_{s=0}^{\infty}\frac{\Pochhammersym{a}{s}}{\EulerGamma@{b+s}s!}z^{s}</syntaxhighlight> || <math>\realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>KummerM(a, b, z)/GAMMA(b) = sum((pochhammer(a, s))/(GAMMA(b + s)*factorial(s))*(z)^(s), s = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric1F1Regularized[a, b, z] == Sum[Divide[Pochhammer[a, s],Gamma[b + s]*(s)!]*(z)^(s), {s, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [35 / 252]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.2.E4 13.2.E4] || [[Item:Q4294|<math>\KummerconfhyperM@{a}{b}{z} = \EulerGamma@{b}\OlverconfhyperM@{a}{b}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperM@{a}{b}{z} = \EulerGamma@{b}\OlverconfhyperM@{a}{b}{z}</syntaxhighlight> || <math>\realpart@@{b} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>KummerM(a, b, z) = GAMMA(b)*KummerM(a, b, z)/GAMMA(b)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric1F1[a, b, z] == Gamma[b]*Hypergeometric1F1Regularized[a, b, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 126]
+|-
+| [https://dlmf.nist.gov/13.2.E5 13.2.E5] || [[Item:Q4295|<math>\lim_{b\to-n}\frac{\KummerconfhyperM@{a}{b}{z}}{\EulerGamma@{b}} = \OlverconfhyperM@{a}{-n}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\lim_{b\to-n}\frac{\KummerconfhyperM@{a}{b}{z}}{\EulerGamma@{b}} = \OlverconfhyperM@{a}{-n}{z}</syntaxhighlight> || <math>\realpart@@{b} > 0, \realpart@@{((-n)+s)} > 0</math> || <syntaxhighlight lang=mathematica>limit((KummerM(a, b, z))/(GAMMA(b)), b = - n) = KummerM(a, - n, z)/GAMMA(- n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Limit[Divide[Hypergeometric1F1[a, b, z],Gamma[b]], b -> - n, GenerateConditions->None] == Hypergeometric1F1Regularized[a, - n, z]</syntaxhighlight> || Failure || Successful || Successful [Tested: 0] || <div class="toccolours mw-collapsible mw-collapsed">Failed [112 / 126]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -1.5], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.2.E5 13.2.E5] || [[Item:Q4295|<math>\OlverconfhyperM@{a}{-n}{z} = \frac{\Pochhammersym{a}{n+1}}{(n+1)!}z^{n+1}\KummerconfhyperM@{a+n+1}{n+2}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\OlverconfhyperM@{a}{-n}{z} = \frac{\Pochhammersym{a}{n+1}}{(n+1)!}z^{n+1}\KummerconfhyperM@{a+n+1}{n+2}{z}</syntaxhighlight> || <math>\realpart@@{b} > 0, \realpart@@{((-n)+s)} > 0</math> || <syntaxhighlight lang=mathematica>KummerM(a, - n, z)/GAMMA(- n) = (pochhammer(a, n + 1))/(factorial(n + 1))*(z)^(n + 1)* KummerM(a + n + 1, n + 2, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric1F1Regularized[a, - n, z] == Divide[Pochhammer[a, n + 1],(n + 1)!]*(z)^(n + 1)* Hypergeometric1F1[a + n + 1, n + 2, z]</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 126]
+|-
+| [https://dlmf.nist.gov/13.2.E7 13.2.E7] || [[Item:Q4297|<math>\KummerconfhyperU@{-m}{b}{z} = (-1)^{m}\Pochhammersym{b}{m}\KummerconfhyperM@{-m}{b}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{-m}{b}{z} = (-1)^{m}\Pochhammersym{b}{m}\KummerconfhyperM@{-m}{b}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerU(- m, b, z) = (- 1)^(m)* pochhammer(b, m)*KummerM(- m, b, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[- m, b, z] == (- 1)^(m)* Pochhammer[b, m]*Hypergeometric1F1[- m, b, z]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 126]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[b, -2], Rule[m, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[b, -2], Rule[m, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.2.E7 13.2.E7] || [[Item:Q4297|<math>(-1)^{m}\Pochhammersym{b}{m}\KummerconfhyperM@{-m}{b}{z} = (-1)^{m}\sum_{s=0}^{m}\binom{m}{s}\Pochhammersym{b+s}{m-s}(-z)^{s}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(-1)^{m}\Pochhammersym{b}{m}\KummerconfhyperM@{-m}{b}{z} = (-1)^{m}\sum_{s=0}^{m}\binom{m}{s}\Pochhammersym{b+s}{m-s}(-z)^{s}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(- 1)^(m)* pochhammer(b, m)*KummerM(- m, b, z) = (- 1)^(m)* sum(binomial(m,s)*pochhammer(b + s, m - s)*(- z)^(s), s = 0..m)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(- 1)^(m)* Pochhammer[b, m]*Hypergeometric1F1[- m, b, z] == (- 1)^(m)* Sum[Binomial[m,s]*Pochhammer[b + s, m - s]*(- z)^(s), {s, 0, m}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 126]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[b, -2], Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[b, -2], Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.2.E8 13.2.E8] || [[Item:Q4298|<math>\KummerconfhyperU@{a}{a+n+1}{z} = \frac{(-1)^{n}\Pochhammersym{1-a-n}{n}}{z^{a+n}}\KummerconfhyperM@{-n}{1-a-n}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{a}{a+n+1}{z} = \frac{(-1)^{n}\Pochhammersym{1-a-n}{n}}{z^{a+n}}\KummerconfhyperM@{-n}{1-a-n}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerU(a, a + n + 1, z) = ((- 1)^(n)* pochhammer(1 - a - n, n))/((z)^(a + n))*KummerM(- n, 1 - a - n, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[a, a + n + 1, z] == Divide[(- 1)^(n)* Pochhammer[1 - a - n, n],(z)^(a + n)]*Hypergeometric1F1[- n, 1 - a - n, z]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 126]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -2], Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -2], Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.2.E8 13.2.E8] || [[Item:Q4298|<math>\frac{(-1)^{n}\Pochhammersym{1-a-n}{n}}{z^{a+n}}\KummerconfhyperM@{-n}{1-a-n}{z} = z^{-a}\sum_{s=0}^{n}\binom{n}{s}\Pochhammersym{a}{s}z^{-s}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{(-1)^{n}\Pochhammersym{1-a-n}{n}}{z^{a+n}}\KummerconfhyperM@{-n}{1-a-n}{z} = z^{-a}\sum_{s=0}^{n}\binom{n}{s}\Pochhammersym{a}{s}z^{-s}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>((- 1)^(n)* pochhammer(1 - a - n, n))/((z)^(a + n))*KummerM(- n, 1 - a - n, z) = (z)^(- a)* sum(binomial(n,s)*pochhammer(a, s)*(z)^(- s), s = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[(- 1)^(n)* Pochhammer[1 - a - n, n],(z)^(a + n)]*Hypergeometric1F1[- n, 1 - a - n, z] == (z)^(- a)* Sum[Binomial[n,s]*Pochhammer[a, s]*(z)^(- s), {s, 0, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 126]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -2], Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -2], Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.2.E9 13.2.E9] || [[Item:Q4299|<math>\KummerconfhyperU@{a}{n+1}{z} = \frac{(-1)^{n+1}}{n!\EulerGamma@{a-n}}\sum_{k=0}^{\infty}\frac{\Pochhammersym{a}{k}}{\Pochhammersym{n+1}{k}k!}z^{k}\left(\ln@@{z}+\digamma@{a+k}-\digamma@{1+k}-\digamma@{n+k+1}\right)+\frac{1}{\EulerGamma@{a}}\sum_{k=1}^{n}\frac{(k-1)!\Pochhammersym{1-a+k}{n-k}}{(n-k)!}z^{-k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{a}{n+1}{z} = \frac{(-1)^{n+1}}{n!\EulerGamma@{a-n}}\sum_{k=0}^{\infty}\frac{\Pochhammersym{a}{k}}{\Pochhammersym{n+1}{k}k!}z^{k}\left(\ln@@{z}+\digamma@{a+k}-\digamma@{1+k}-\digamma@{n+k+1}\right)+\frac{1}{\EulerGamma@{a}}\sum_{k=1}^{n}\frac{(k-1)!\Pochhammersym{1-a+k}{n-k}}{(n-k)!}z^{-k}</syntaxhighlight> || <math>\realpart@@{(a-n)} > 0, \realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>KummerU(a, n + 1, z) = ((- 1)^(n + 1))/(factorial(n)*GAMMA(a - n))*sum((pochhammer(a, k))/(pochhammer(n + 1, k)*factorial(k))*(z)^(k)*(ln(z)+ Psi(a + k)- Psi(1 + k)- Psi(n + k + 1)), k = 0..infinity)+(1)/(GAMMA(a))*sum((factorial(k - 1)*pochhammer(1 - a + k, n - k))/(factorial(n - k))*(z)^(- k), k = 1..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[a, n + 1, z] == Divide[(- 1)^(n + 1),(n)!*Gamma[a - n]]*Sum[Divide[Pochhammer[a, k],Pochhammer[n + 1, k]*(k)!]*(z)^(k)*(Log[z]+ PolyGamma[a + k]- PolyGamma[1 + k]- PolyGamma[n + k + 1]), {k, 0, Infinity}, GenerateConditions->None]+Divide[1,Gamma[a]]*Sum[Divide[(k - 1)!*Pochhammer[1 - a + k, n - k],(n - k)!]*(z)^(- k), {k, 1, n}, GenerateConditions->None]</syntaxhighlight> || Aborted || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 14]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(undefined)+Float(undefined)*I
+Test Values: {a = 2, z = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(undefined)+Float(undefined)*I
+Test Values: {a = 2, z = -1/2+1/2*I*3^(1/2), n = 1}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/13.2.E10 13.2.E10] || [[Item:Q4300|<math>\KummerconfhyperU@{-m}{n+1}{z} = (-1)^{m}\Pochhammersym{n+1}{m}\KummerconfhyperM@{-m}{n+1}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{-m}{n+1}{z} = (-1)^{m}\Pochhammersym{n+1}{m}\KummerconfhyperM@{-m}{n+1}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerU(- m, n + 1, z) = (- 1)^(m)* pochhammer(n + 1, m)*KummerM(- m, n + 1, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[- m, n + 1, z] == (- 1)^(m)* Pochhammer[n + 1, m]*Hypergeometric1F1[- m, n + 1, z]</syntaxhighlight> || Failure || Failure || Successful [Tested: 63] || Successful [Tested: 63]
+|-
+| [https://dlmf.nist.gov/13.2.E10 13.2.E10] || [[Item:Q4300|<math>(-1)^{m}\Pochhammersym{n+1}{m}\KummerconfhyperM@{-m}{n+1}{z} = (-1)^{m}\sum_{s=0}^{m}\binom{m}{s}\Pochhammersym{n+s+1}{m-s}(-z)^{s}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(-1)^{m}\Pochhammersym{n+1}{m}\KummerconfhyperM@{-m}{n+1}{z} = (-1)^{m}\sum_{s=0}^{m}\binom{m}{s}\Pochhammersym{n+s+1}{m-s}(-z)^{s}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(- 1)^(m)* pochhammer(n + 1, m)*KummerM(- m, n + 1, z) = (- 1)^(m)* sum(binomial(m,s)*pochhammer(n + s + 1, m - s)*(- z)^(s), s = 0..m)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(- 1)^(m)* Pochhammer[n + 1, m]*Hypergeometric1F1[- m, n + 1, z] == (- 1)^(m)* Sum[Binomial[m,s]*Pochhammer[n + s + 1, m - s]*(- z)^(s), {s, 0, m}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 63] || Successful [Tested: 63]
+|-
+| [https://dlmf.nist.gov/13.2.E11 13.2.E11] || [[Item:Q4301|<math>\KummerconfhyperU@{a}{-n}{z} = z^{n+1}\KummerconfhyperU@{a+n+1}{n+2}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{a}{-n}{z} = z^{n+1}\KummerconfhyperU@{a+n+1}{n+2}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerU(a, - n, z) = (z)^(n + 1)* KummerU(a + n + 1, n + 2, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[a, - n, z] == (z)^(n + 1)* HypergeometricU[a + n + 1, n + 2, z]</syntaxhighlight> || Failure || Successful || Successful [Tested: 126] || Successful [Tested: 126]
+|-
+| [https://dlmf.nist.gov/13.2.E12 13.2.E12] || [[Item:Q4302|<math>\KummerconfhyperU@{a}{b}{ze^{2\pi\iunit m}} = \frac{2\pi\iunit e^{-\pi\iunit bm}\sin@{\pi bm}}{\EulerGamma@{1+a-b}\sin@{\pi b}}\OlverconfhyperM@{a}{b}{z}+e^{-2\pi\iunit bm}\KummerconfhyperU@{a}{b}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{a}{b}{ze^{2\pi\iunit m}} = \frac{2\pi\iunit e^{-\pi\iunit bm}\sin@{\pi bm}}{\EulerGamma@{1+a-b}\sin@{\pi b}}\OlverconfhyperM@{a}{b}{z}+e^{-2\pi\iunit bm}\KummerconfhyperU@{a}{b}{z}</syntaxhighlight> || <math>\realpart@@{(1+a-b)} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>KummerU(a, b, z*exp(2*Pi*I*m)) = (2*Pi*I*exp(- Pi*I*b*m)*sin(Pi*b*m))/(GAMMA(1 + a - b)*sin(Pi*b))*KummerM(a, b, z)/GAMMA(b)+ exp(- 2*Pi*I*b*m)*KummerU(a, b, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[a, b, z*Exp[2*Pi*I*m]] == Divide[2*Pi*I*Exp[- Pi*I*b*m]*Sin[Pi*b*m],Gamma[1 + a - b]*Sin[Pi*b]]*Hypergeometric1F1Regularized[a, b, z]+ Exp[- 2*Pi*I*b*m]*HypergeometricU[a, b, z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [230 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.101548209-1.031304846*I
+Test Values: {a = -3/2, b = -3/2, z = 1/2*3^(1/2)+1/2*I, m = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.101548218-1.031304823*I
+Test Values: {a = -3/2, b = -3/2, z = 1/2*3^(1/2)+1/2*I, m = 3}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [230 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.10154820915393259, -1.0313048488210503]
+Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.1015482091539317, -1.03130484882105]
+Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[m, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.2.E33 13.2.E33] || [[Item:Q4325|<math>\Wronskian@{\OlverconfhyperM@{a}{b}{z},z^{1-b}\OlverconfhyperM@{a-b+1}{2-b}{z}} = \sin@{\pi b}z^{-b}e^{z}/\pi</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\OlverconfhyperM@{a}{b}{z},z^{1-b}\OlverconfhyperM@{a-b+1}{2-b}{z}} = \sin@{\pi b}z^{-b}e^{z}/\pi</syntaxhighlight> || <math>\realpart@@{(b+s)} > 0, \realpart@@{((2-b)+s)} > 0</math> || <syntaxhighlight lang=mathematica>(KummerM(a, b, z)/GAMMA(b))*diff((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b), z)-diff(KummerM(a, b, z)/GAMMA(b), z)*((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b)) = sin(Pi*b)*(z)^(- b)* exp(z)/Pi</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{Hypergeometric1F1Regularized[a, b, z], (z)^(1 - b)* Hypergeometric1F1Regularized[a - b + 1, 2 - b, z]}, z] == Sin[Pi*b]*(z)^(- b)* Exp[z]/Pi</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 252]
+|-
+| [https://dlmf.nist.gov/13.2.E34 13.2.E34] || [[Item:Q4326|<math>\Wronskian@{\OlverconfhyperM@{a}{b}{z},\KummerconfhyperU@{a}{b}{z}} = -\ifrac{z^{-b}e^{z}}{\EulerGamma@{a}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\OlverconfhyperM@{a}{b}{z},\KummerconfhyperU@{a}{b}{z}} = -\ifrac{z^{-b}e^{z}}{\EulerGamma@{a}}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>(KummerM(a, b, z)/GAMMA(b))*diff(KummerU(a, b, z), z)-diff(KummerM(a, b, z)/GAMMA(b), z)*(KummerU(a, b, z)) = -((z)^(- b)* exp(z))/(GAMMA(a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{Hypergeometric1F1Regularized[a, b, z], HypergeometricU[a, b, z]}, z] == -Divide[(z)^(- b)* Exp[z],Gamma[a]]</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 126]
+|-
+| [https://dlmf.nist.gov/13.2.E35 13.2.E35] || [[Item:Q4327|<math>\Wronskian@{\OlverconfhyperM@{a}{b}{z},e^{z}\KummerconfhyperU@{b-a}{b}{e^{+\pi\iunit}z}} = \ifrac{e^{- b\pi\iunit}z^{-b}e^{z}}{\EulerGamma@{b-a}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\OlverconfhyperM@{a}{b}{z},e^{z}\KummerconfhyperU@{b-a}{b}{e^{+\pi\iunit}z}} = \ifrac{e^{- b\pi\iunit}z^{-b}e^{z}}{\EulerGamma@{b-a}}</syntaxhighlight> || <math>\realpart@@{(b-a)} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>(KummerM(a, b, z)/GAMMA(b))*diff(exp(z)*KummerU(b - a, b, exp(+ Pi*I)*z), z)-diff(KummerM(a, b, z)/GAMMA(b), z)*(exp(z)*KummerU(b - a, b, exp(+ Pi*I)*z)) = (exp(- b*Pi*I)*(z)^(- b)* exp(z))/(GAMMA(b - a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{Hypergeometric1F1Regularized[a, b, z], Exp[z]*HypergeometricU[b - a, b, Exp[+ Pi*I]*z]}, z] == Divide[Exp[- b*Pi*I]*(z)^(- b)* Exp[z],Gamma[b - a]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [23 / 105]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.6693440963-2.281274239*I
+Test Values: {a = -3/2, b = 3/2, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.4620307839+.3929465556*I
+Test Values: {a = -3/2, b = 3/2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 105]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.6693440961046373, -2.2812742393329124]
+Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.46203078407110554, 0.39294655583435506]
+Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.2.E35 13.2.E35] || [[Item:Q4327|<math>\Wronskian@{\OlverconfhyperM@{a}{b}{z},e^{z}\KummerconfhyperU@{b-a}{b}{e^{-\pi\iunit}z}} = \ifrac{e^{+ b\pi\iunit}z^{-b}e^{z}}{\EulerGamma@{b-a}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\OlverconfhyperM@{a}{b}{z},e^{z}\KummerconfhyperU@{b-a}{b}{e^{-\pi\iunit}z}} = \ifrac{e^{+ b\pi\iunit}z^{-b}e^{z}}{\EulerGamma@{b-a}}</syntaxhighlight> || <math>\realpart@@{(b-a)} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>(KummerM(a, b, z)/GAMMA(b))*diff(exp(z)*KummerU(b - a, b, exp(- Pi*I)*z), z)-diff(KummerM(a, b, z)/GAMMA(b), z)*(exp(z)*KummerU(b - a, b, exp(- Pi*I)*z)) = (exp(+ b*Pi*I)*(z)^(- b)* exp(z))/(GAMMA(b - a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{Hypergeometric1F1Regularized[a, b, z], Exp[z]*HypergeometricU[b - a, b, Exp[- Pi*I]*z]}, z] == Divide[Exp[+ b*Pi*I]*(z)^(- b)* Exp[z],Gamma[b - a]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [53 / 105]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.068139482+1.255929884*I
+Test Values: {a = -3/2, b = 3/2, z = 1/2-1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .1184211651-.4036057902*I
+Test Values: {a = -3/2, b = 3/2, z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [50 / 105]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.0681394822800954, 1.2559298845291709]
+Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.11842116492450601, -0.40360579036441874]
+Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.2.E36 13.2.E36] || [[Item:Q4328|<math>\Wronskian@{z^{1-b}\OlverconfhyperM@{a-b+1}{2-b}{z},\KummerconfhyperU@{a}{b}{z}} = -\ifrac{z^{-b}e^{z}}{\EulerGamma@{a-b+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{z^{1-b}\OlverconfhyperM@{a-b+1}{2-b}{z},\KummerconfhyperU@{a}{b}{z}} = -\ifrac{z^{-b}e^{z}}{\EulerGamma@{a-b+1}}</syntaxhighlight> || <math>\realpart@@{(a-b+1)} > 0, \realpart@@{((2-b)+s)} > 0</math> || <syntaxhighlight lang=mathematica>((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b))*diff(KummerU(a, b, z), z)-diff((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b), z)*(KummerU(a, b, z)) = -((z)^(- b)* exp(z))/(GAMMA(a - b + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{(z)^(1 - b)* Hypergeometric1F1Regularized[a - b + 1, 2 - b, z], HypergeometricU[a, b, z]}, z] == -Divide[(z)^(- b)* Exp[z],Gamma[a - b + 1]]</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 161]
+|-
+| [https://dlmf.nist.gov/13.2.E37 13.2.E37] || [[Item:Q4329|<math>\Wronskian@{z^{1-b}\OlverconfhyperM@{a-b+1}{2-b}{z},e^{z}\KummerconfhyperU@{b-a}{b}{e^{+\pi\iunit}z}} = -\ifrac{z^{-b}e^{z}}{\EulerGamma@{1-a}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{z^{1-b}\OlverconfhyperM@{a-b+1}{2-b}{z},e^{z}\KummerconfhyperU@{b-a}{b}{e^{+\pi\iunit}z}} = -\ifrac{z^{-b}e^{z}}{\EulerGamma@{1-a}}</syntaxhighlight> || <math>\realpart@@{(1-a)} > 0, \realpart@@{((2-b)+s)} > 0</math> || <syntaxhighlight lang=mathematica>((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b))*diff(exp(z)*KummerU(b - a, b, exp(+ Pi*I)*z), z)-diff((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b), z)*(exp(z)*KummerU(b - a, b, exp(+ Pi*I)*z)) = -((z)^(- b)* exp(z))/(GAMMA(1 - a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{(z)^(1 - b)* Hypergeometric1F1Regularized[a - b + 1, 2 - b, z], Exp[z]*HypergeometricU[b - a, b, Exp[+ Pi*I]*z]}, z] == -Divide[(z)^(- b)* Exp[z],Gamma[1 - a]]</syntaxhighlight> || Failure || Aborted || Error || Successful [Tested: 168]
+|-
+| [https://dlmf.nist.gov/13.2.E37 13.2.E37] || [[Item:Q4329|<math>\Wronskian@{z^{1-b}\OlverconfhyperM@{a-b+1}{2-b}{z},e^{z}\KummerconfhyperU@{b-a}{b}{e^{-\pi\iunit}z}} = -\ifrac{z^{-b}e^{z}}{\EulerGamma@{1-a}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{z^{1-b}\OlverconfhyperM@{a-b+1}{2-b}{z},e^{z}\KummerconfhyperU@{b-a}{b}{e^{-\pi\iunit}z}} = -\ifrac{z^{-b}e^{z}}{\EulerGamma@{1-a}}</syntaxhighlight> || <math>\realpart@@{(1-a)} > 0, \realpart@@{((2-b)+s)} > 0</math> || <syntaxhighlight lang=mathematica>((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b))*diff(exp(z)*KummerU(b - a, b, exp(- Pi*I)*z), z)-diff((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b), z)*(exp(z)*KummerU(b - a, b, exp(- Pi*I)*z)) = -((z)^(- b)* exp(z))/(GAMMA(1 - a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{(z)^(1 - b)* Hypergeometric1F1Regularized[a - b + 1, 2 - b, z], Exp[z]*HypergeometricU[b - a, b, Exp[- Pi*I]*z]}, z] == -Divide[(z)^(- b)* Exp[z],Gamma[1 - a]]</syntaxhighlight> || Failure || Aborted || Error || Successful [Tested: 168]
+|-
+| [https://dlmf.nist.gov/13.2.E38 13.2.E38] || [[Item:Q4330|<math>\Wronskian@{\KummerconfhyperU@{a}{b}{z},e^{z}\KummerconfhyperU@{b-a}{b}{e^{+\pi\iunit}z}} = e^{+(a-b)\pi\iunit}z^{-b}e^{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\KummerconfhyperU@{a}{b}{z},e^{z}\KummerconfhyperU@{b-a}{b}{e^{+\pi\iunit}z}} = e^{+(a-b)\pi\iunit}z^{-b}e^{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(KummerU(a, b, z))*diff(exp(z)*KummerU(b - a, b, exp(+ Pi*I)*z), z)-diff(KummerU(a, b, z), z)*(exp(z)*KummerU(b - a, b, exp(+ Pi*I)*z)) = exp(+(a - b)*Pi*I)*(z)^(- b)* exp(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{HypergeometricU[a, b, z], Exp[z]*HypergeometricU[b - a, b, Exp[+ Pi*I]*z]}, z] == Exp[+(a - b)*Pi*I]*(z)^(- b)* Exp[z]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [38 / 252]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 4.753561418-.1121990572*I
+Test Values: {a = -3/2, b = -2, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.142634185-.4073142366*I
+Test Values: {a = -3/2, b = -2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [32 / 252]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[4.753561408836843, -0.1121990577209182]
+Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.1426341834354088, -0.40731423683768475]
+Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.2.E38 13.2.E38] || [[Item:Q4330|<math>\Wronskian@{\KummerconfhyperU@{a}{b}{z},e^{z}\KummerconfhyperU@{b-a}{b}{e^{-\pi\iunit}z}} = e^{-(a-b)\pi\iunit}z^{-b}e^{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\KummerconfhyperU@{a}{b}{z},e^{z}\KummerconfhyperU@{b-a}{b}{e^{-\pi\iunit}z}} = e^{-(a-b)\pi\iunit}z^{-b}e^{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(KummerU(a, b, z))*diff(exp(z)*KummerU(b - a, b, exp(- Pi*I)*z), z)-diff(KummerU(a, b, z), z)*(exp(z)*KummerU(b - a, b, exp(- Pi*I)*z)) = exp(-(a - b)*Pi*I)*(z)^(- b)* exp(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{HypergeometricU[a, b, z], Exp[z]*HypergeometricU[b - a, b, Exp[- Pi*I]*z]}, z] == Exp[-(a - b)*Pi*I]*(z)^(- b)* Exp[z]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [80 / 252]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .5941419621-3.243473855*I
+Test Values: {a = -3/2, b = -2, z = 1/2-1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.4376938533+.7184072077*I
+Test Values: {a = -3/2, b = -2, z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [80 / 252]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.5941419683502733, -3.243473853028733]
+Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.4376938536795689, 0.7184072074542298]
+Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.2.E39 13.2.E39] || [[Item:Q4331|<math>\KummerconfhyperM@{a}{b}{z} = e^{z}\KummerconfhyperM@{b-a}{b}{-z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperM@{a}{b}{z} = e^{z}\KummerconfhyperM@{b-a}{b}{-z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerM(a, b, z) = exp(z)*KummerM(b - a, b, - z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric1F1[a, b, z] == Exp[z]*Hypergeometric1F1[b - a, b, - z]</syntaxhighlight> || Failure || Successful || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 252]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.2.E40 13.2.E40] || [[Item:Q4332|<math>\KummerconfhyperU@{a}{b}{z} = z^{1-b}\KummerconfhyperU@{a-b+1}{2-b}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{a}{b}{z} = z^{1-b}\KummerconfhyperU@{a-b+1}{2-b}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerU(a, b, z) = (z)^(1 - b)* KummerU(a - b + 1, 2 - b, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[a, b, z] == (z)^(1 - b)* HypergeometricU[a - b + 1, 2 - b, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 252]
+|-
+| [https://dlmf.nist.gov/13.2.E41 13.2.E41] || [[Item:Q4333|<math>\frac{1}{\EulerGamma@{b}}\KummerconfhyperM@{a}{b}{z} = \frac{e^{- a\pi\iunit}}{\EulerGamma@{b-a}}\KummerconfhyperU@{a}{b}{z}+\frac{e^{+(b-a)\pi\iunit}}{\EulerGamma@{a}}e^{z}\KummerconfhyperU@{b-a}{b}{e^{+\pi\iunit}z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{\EulerGamma@{b}}\KummerconfhyperM@{a}{b}{z} = \frac{e^{- a\pi\iunit}}{\EulerGamma@{b-a}}\KummerconfhyperU@{a}{b}{z}+\frac{e^{+(b-a)\pi\iunit}}{\EulerGamma@{a}}e^{z}\KummerconfhyperU@{b-a}{b}{e^{+\pi\iunit}z}</syntaxhighlight> || <math>\realpart@@{b} > 0, \realpart@@{(b-a)} > 0, \realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(GAMMA(b))*KummerM(a, b, z) = (exp(- a*Pi*I))/(GAMMA(b - a))*KummerU(a, b, z)+(exp(+(b - a)*Pi*I))/(GAMMA(a))*exp(z)*KummerU(b - a, b, exp(+ Pi*I)*z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,Gamma[b]]*Hypergeometric1F1[a, b, z] == Divide[Exp[- a*Pi*I],Gamma[b - a]]*HypergeometricU[a, b, z]+Divide[Exp[+(b - a)*Pi*I],Gamma[a]]*Exp[z]*HypergeometricU[b - a, b, Exp[+ Pi*I]*z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 3.583210384+1.512741910*I
+Test Values: {a = 3/2, b = 2, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.096602540+.7868998856*I
+Test Values: {a = 3/2, b = 2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[3.583210382577498, 1.512741908514331]
+Test Values: {Rule[a, 1.5], Rule[b, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.096602539454242, 0.7868998849931845]
+Test Values: {Rule[a, 1.5], Rule[b, 2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.2.E41 13.2.E41] || [[Item:Q4333|<math>\frac{1}{\EulerGamma@{b}}\KummerconfhyperM@{a}{b}{z} = \frac{e^{+ a\pi\iunit}}{\EulerGamma@{b-a}}\KummerconfhyperU@{a}{b}{z}+\frac{e^{-(b-a)\pi\iunit}}{\EulerGamma@{a}}e^{z}\KummerconfhyperU@{b-a}{b}{e^{-\pi\iunit}z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{\EulerGamma@{b}}\KummerconfhyperM@{a}{b}{z} = \frac{e^{+ a\pi\iunit}}{\EulerGamma@{b-a}}\KummerconfhyperU@{a}{b}{z}+\frac{e^{-(b-a)\pi\iunit}}{\EulerGamma@{a}}e^{z}\KummerconfhyperU@{b-a}{b}{e^{-\pi\iunit}z}</syntaxhighlight> || <math>\realpart@@{b} > 0, \realpart@@{(b-a)} > 0, \realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(GAMMA(b))*KummerM(a, b, z) = (exp(+ a*Pi*I))/(GAMMA(b - a))*KummerU(a, b, z)+(exp(-(b - a)*Pi*I))/(GAMMA(a))*exp(z)*KummerU(b - a, b, exp(- Pi*I)*z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,Gamma[b]]*Hypergeometric1F1[a, b, z] == Divide[Exp[+ a*Pi*I],Gamma[b - a]]*HypergeometricU[a, b, z]+Divide[Exp[-(b - a)*Pi*I],Gamma[a]]*Exp[z]*HypergeometricU[b - a, b, Exp[- Pi*I]*z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [15 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 2.239690726-1.798422043*I
+Test Values: {a = 3/2, b = 2, z = 1/2-1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .9984283068-.3592011980*I
+Test Values: {a = 3/2, b = 2, z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [15 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[2.239690726834086, -1.7984220417127512]
+Test Values: {Rule[a, 1.5], Rule[b, 2], Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.9984283065924617, -0.35920119796837185]
+Test Values: {Rule[a, 1.5], Rule[b, 2], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.2.E42 13.2.E42] || [[Item:Q4334|<math>\KummerconfhyperU@{a}{b}{z} = \frac{\EulerGamma@{1-b}}{\EulerGamma@{a-b+1}}\KummerconfhyperM@{a}{b}{z}+\frac{\EulerGamma@{b-1}}{\EulerGamma@{a}}z^{1-b}\KummerconfhyperM@{a-b+1}{2-b}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{a}{b}{z} = \frac{\EulerGamma@{1-b}}{\EulerGamma@{a-b+1}}\KummerconfhyperM@{a}{b}{z}+\frac{\EulerGamma@{b-1}}{\EulerGamma@{a}}z^{1-b}\KummerconfhyperM@{a-b+1}{2-b}{z}</syntaxhighlight> || <math>\realpart@@{(1-b)} > 0, \realpart@@{(a-b+1)} > 0, \realpart@@{(b-1)} > 0, \realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>KummerU(a, b, z) = (GAMMA(1 - b))/(GAMMA(a - b + 1))*KummerM(a, b, z)+(GAMMA(b - 1))/(GAMMA(a))*(z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[a, b, z] == Divide[Gamma[1 - b],Gamma[a - b + 1]]*Hypergeometric1F1[a, b, z]+Divide[Gamma[b - 1],Gamma[a]]*(z)^(1 - b)* Hypergeometric1F1[a - b + 1, 2 - b, z]</syntaxhighlight> || Successful || Successful || - || -
+|-
+| [https://dlmf.nist.gov/13.3.E1 13.3.E1] || [[Item:Q4335|<math>(b-a)\KummerconfhyperM@{a-1}{b}{z}+(2a-b+z)\KummerconfhyperM@{a}{b}{z}-a\KummerconfhyperM@{a+1}{b}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(b-a)\KummerconfhyperM@{a-1}{b}{z}+(2a-b+z)\KummerconfhyperM@{a}{b}{z}-a\KummerconfhyperM@{a+1}{b}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(b - a)*KummerM(a - 1, b, z)+(2*a - b + z)*KummerM(a, b, z)- a*KummerM(a + 1, b, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>(b - a)*Hypergeometric1F1[a - 1, b, z]+(2*a - b + z)*Hypergeometric1F1[a, b, z]- a*Hypergeometric1F1[a + 1, b, z] == 0</syntaxhighlight> || Successful || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 252]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.3.E2 13.3.E2] || [[Item:Q4336|<math>b(b-1)\KummerconfhyperM@{a}{b-1}{z}+b(1-b-z)\KummerconfhyperM@{a}{b}{z}+z(b-a)\KummerconfhyperM@{a}{b+1}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>b(b-1)\KummerconfhyperM@{a}{b-1}{z}+b(1-b-z)\KummerconfhyperM@{a}{b}{z}+z(b-a)\KummerconfhyperM@{a}{b+1}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>b*(b - 1)*KummerM(a, b - 1, z)+ b*(1 - b - z)*KummerM(a, b, z)+ z*(b - a)*KummerM(a, b + 1, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>b*(b - 1)*Hypergeometric1F1[a, b - 1, z]+ b*(1 - b - z)*Hypergeometric1F1[a, b, z]+ z*(b - a)*Hypergeometric1F1[a, b + 1, z] == 0</syntaxhighlight> || Successful || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 252]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.3.E3 13.3.E3] || [[Item:Q4337|<math>(a-b+1)\KummerconfhyperM@{a}{b}{z}-a\KummerconfhyperM@{a+1}{b}{z}+(b-1)\KummerconfhyperM@{a}{b-1}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(a-b+1)\KummerconfhyperM@{a}{b}{z}-a\KummerconfhyperM@{a+1}{b}{z}+(b-1)\KummerconfhyperM@{a}{b-1}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(a - b + 1)*KummerM(a, b, z)- a*KummerM(a + 1, b, z)+(b - 1)*KummerM(a, b - 1, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>(a - b + 1)*Hypergeometric1F1[a, b, z]- a*Hypergeometric1F1[a + 1, b, z]+(b - 1)*Hypergeometric1F1[a, b - 1, z] == 0</syntaxhighlight> || Successful || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [35 / 252]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.3.E4 13.3.E4] || [[Item:Q4338|<math>b\KummerconfhyperM@{a}{b}{z}-b\KummerconfhyperM@{a-1}{b}{z}-z\KummerconfhyperM@{a}{b+1}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>b\KummerconfhyperM@{a}{b}{z}-b\KummerconfhyperM@{a-1}{b}{z}-z\KummerconfhyperM@{a}{b+1}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>b*KummerM(a, b, z)- b*KummerM(a - 1, b, z)- z*KummerM(a, b + 1, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>b*Hypergeometric1F1[a, b, z]- b*Hypergeometric1F1[a - 1, b, z]- z*Hypergeometric1F1[a, b + 1, z] == 0</syntaxhighlight> || Successful || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 252]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.3.E5 13.3.E5] || [[Item:Q4339|<math>b(a+z)\KummerconfhyperM@{a}{b}{z}+z(a-b)\KummerconfhyperM@{a}{b+1}{z}-ab\KummerconfhyperM@{a+1}{b}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>b(a+z)\KummerconfhyperM@{a}{b}{z}+z(a-b)\KummerconfhyperM@{a}{b+1}{z}-ab\KummerconfhyperM@{a+1}{b}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>b*(a + z)*KummerM(a, b, z)+ z*(a - b)*KummerM(a, b + 1, z)- a*b*KummerM(a + 1, b, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>b*(a + z)*Hypergeometric1F1[a, b, z]+ z*(a - b)*Hypergeometric1F1[a, b + 1, z]- a*b*Hypergeometric1F1[a + 1, b, z] == 0</syntaxhighlight> || Successful || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 252]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.3.E6 13.3.E6] || [[Item:Q4340|<math>(a-1+z)\KummerconfhyperM@{a}{b}{z}+(b-a)\KummerconfhyperM@{a-1}{b}{z}+(1-b)\KummerconfhyperM@{a}{b-1}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(a-1+z)\KummerconfhyperM@{a}{b}{z}+(b-a)\KummerconfhyperM@{a-1}{b}{z}+(1-b)\KummerconfhyperM@{a}{b-1}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(a - 1 + z)*KummerM(a, b, z)+(b - a)*KummerM(a - 1, b, z)+(1 - b)*KummerM(a, b - 1, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>(a - 1 + z)*Hypergeometric1F1[a, b, z]+(b - a)*Hypergeometric1F1[a - 1, b, z]+(1 - b)*Hypergeometric1F1[a, b - 1, z] == 0</syntaxhighlight> || Successful || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 252]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.3.E7 13.3.E7] || [[Item:Q4341|<math>\KummerconfhyperU@{a-1}{b}{z}+(b-2a-z)\KummerconfhyperU@{a}{b}{z}+a(a-b+1)\KummerconfhyperU@{a+1}{b}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{a-1}{b}{z}+(b-2a-z)\KummerconfhyperU@{a}{b}{z}+a(a-b+1)\KummerconfhyperU@{a+1}{b}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerU(a - 1, b, z)+(b - 2*a - z)*KummerU(a, b, z)+ a*(a - b + 1)*KummerU(a + 1, b, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[a - 1, b, z]+(b - 2*a - z)*HypergeometricU[a, b, z]+ a*(a - b + 1)*HypergeometricU[a + 1, b, z] == 0</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 252]
+|-
+| [https://dlmf.nist.gov/13.3.E8 13.3.E8] || [[Item:Q4342|<math>(b-a-1)\KummerconfhyperU@{a}{b-1}{z}+(1-b-z)\KummerconfhyperU@{a}{b}{z}+z\KummerconfhyperU@{a}{b+1}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(b-a-1)\KummerconfhyperU@{a}{b-1}{z}+(1-b-z)\KummerconfhyperU@{a}{b}{z}+z\KummerconfhyperU@{a}{b+1}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(b - a - 1)*KummerU(a, b - 1, z)+(1 - b - z)*KummerU(a, b, z)+ z*KummerU(a, b + 1, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>(b - a - 1)*HypergeometricU[a, b - 1, z]+(1 - b - z)*HypergeometricU[a, b, z]+ z*HypergeometricU[a, b + 1, z] == 0</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 252]
+|-
+| [https://dlmf.nist.gov/13.3.E9 13.3.E9] || [[Item:Q4343|<math>\KummerconfhyperU@{a}{b}{z}-a\KummerconfhyperU@{a+1}{b}{z}-\KummerconfhyperU@{a}{b-1}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{a}{b}{z}-a\KummerconfhyperU@{a+1}{b}{z}-\KummerconfhyperU@{a}{b-1}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerU(a, b, z)- a*KummerU(a + 1, b, z)- KummerU(a, b - 1, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[a, b, z]- a*HypergeometricU[a + 1, b, z]- HypergeometricU[a, b - 1, z] == 0</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 252]
+|-
+| [https://dlmf.nist.gov/13.3.E10 13.3.E10] || [[Item:Q4344|<math>(b-a)\KummerconfhyperU@{a}{b}{z}+\KummerconfhyperU@{a-1}{b}{z}-z\KummerconfhyperU@{a}{b+1}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(b-a)\KummerconfhyperU@{a}{b}{z}+\KummerconfhyperU@{a-1}{b}{z}-z\KummerconfhyperU@{a}{b+1}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(b - a)*KummerU(a, b, z)+ KummerU(a - 1, b, z)- z*KummerU(a, b + 1, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>(b - a)*HypergeometricU[a, b, z]+ HypergeometricU[a - 1, b, z]- z*HypergeometricU[a, b + 1, z] == 0</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 252]
+|-
+| [https://dlmf.nist.gov/13.3.E11 13.3.E11] || [[Item:Q4345|<math>(a+z)\KummerconfhyperU@{a}{b}{z}-z\KummerconfhyperU@{a}{b+1}{z}+a(b-a-1)\KummerconfhyperU@{a+1}{b}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(a+z)\KummerconfhyperU@{a}{b}{z}-z\KummerconfhyperU@{a}{b+1}{z}+a(b-a-1)\KummerconfhyperU@{a+1}{b}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(a + z)*KummerU(a, b, z)- z*KummerU(a, b + 1, z)+ a*(b - a - 1)*KummerU(a + 1, b, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>(a + z)*HypergeometricU[a, b, z]- z*HypergeometricU[a, b + 1, z]+ a*(b - a - 1)*HypergeometricU[a + 1, b, z] == 0</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 252]
+|-
+| [https://dlmf.nist.gov/13.3.E12 13.3.E12] || [[Item:Q4346|<math>(a-1+z)\KummerconfhyperU@{a}{b}{z}-\KummerconfhyperU@{a-1}{b}{z}+(a-b+1)\KummerconfhyperU@{a}{b-1}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(a-1+z)\KummerconfhyperU@{a}{b}{z}-\KummerconfhyperU@{a-1}{b}{z}+(a-b+1)\KummerconfhyperU@{a}{b-1}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(a - 1 + z)*KummerU(a, b, z)- KummerU(a - 1, b, z)+(a - b + 1)*KummerU(a, b - 1, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>(a - 1 + z)*HypergeometricU[a, b, z]- HypergeometricU[a - 1, b, z]+(a - b + 1)*HypergeometricU[a, b - 1, z] == 0</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 252]
+|-
+| [https://dlmf.nist.gov/13.3.E13 13.3.E13] || [[Item:Q4347|<math>(a+1)z\KummerconfhyperM@{a+2}{b+2}{z}+(b+1)(b-z)\KummerconfhyperM@{a+1}{b+1}{z}-b(b+1)\KummerconfhyperM@{a}{b}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(a+1)z\KummerconfhyperM@{a+2}{b+2}{z}+(b+1)(b-z)\KummerconfhyperM@{a+1}{b+1}{z}-b(b+1)\KummerconfhyperM@{a}{b}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(a + 1)*z*KummerM(a + 2, b + 2, z)+(b + 1)*(b - z)*KummerM(a + 1, b + 1, z)- b*(b + 1)*KummerM(a, b, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>(a + 1)*z*Hypergeometric1F1[a + 2, b + 2, z]+(b + 1)*(b - z)*Hypergeometric1F1[a + 1, b + 1, z]- b*(b + 1)*Hypergeometric1F1[a, b, z] == 0</syntaxhighlight> || Successful || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [35 / 252]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.3.E14 13.3.E14] || [[Item:Q4348|<math>(a+1)z\KummerconfhyperU@{a+2}{b+2}{z}+(z-b)\KummerconfhyperU@{a+1}{b+1}{z}-\KummerconfhyperU@{a}{b}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(a+1)z\KummerconfhyperU@{a+2}{b+2}{z}+(z-b)\KummerconfhyperU@{a+1}{b+1}{z}-\KummerconfhyperU@{a}{b}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(a + 1)*z*KummerU(a + 2, b + 2, z)+(z - b)*KummerU(a + 1, b + 1, z)- KummerU(a, b, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>(a + 1)*z*HypergeometricU[a + 2, b + 2, z]+(z - b)*HypergeometricU[a + 1, b + 1, z]- HypergeometricU[a, b, z] == 0</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 252]
+|-
+| [https://dlmf.nist.gov/13.3.E15 13.3.E15] || [[Item:Q4349|<math>\deriv{}{z}\KummerconfhyperM@{a}{b}{z} = \frac{a}{b}\KummerconfhyperM@{a+1}{b+1}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv{}{z}\KummerconfhyperM@{a}{b}{z} = \frac{a}{b}\KummerconfhyperM@{a+1}{b+1}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(KummerM(a, b, z), z) = (a)/(b)*KummerM(a + 1, b + 1, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Hypergeometric1F1[a, b, z], z] == Divide[a,b]*Hypergeometric1F1[a + 1, b + 1, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 252]
+|-
+| [https://dlmf.nist.gov/13.3.E16 13.3.E16] || [[Item:Q4350|<math>\deriv[n]{}{z}\KummerconfhyperM@{a}{b}{z} = \frac{\Pochhammersym{a}{n}}{\Pochhammersym{b}{n}}\KummerconfhyperM@{a+n}{b+n}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[n]{}{z}\KummerconfhyperM@{a}{b}{z} = \frac{\Pochhammersym{a}{n}}{\Pochhammersym{b}{n}}\KummerconfhyperM@{a+n}{b+n}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(KummerM(a, b, z), [z$(n)]) = (pochhammer(a, n))/(pochhammer(b, n))*KummerM(a + n, b + n, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Hypergeometric1F1[a, b, z], {z, n}] == Divide[Pochhammer[a, n],Pochhammer[b, n]]*Hypergeometric1F1[a + n, b + n, z]</syntaxhighlight> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.3.E17 13.3.E17] || [[Item:Q4351|<math>\left(z\deriv{}{z}z\right)^{n}\left(z^{a-1}\KummerconfhyperM@{a}{b}{z}\right) = \Pochhammersym{a}{n}z^{a+n-1}\KummerconfhyperM@{a+n}{b}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(z\deriv{}{z}z\right)^{n}\left(z^{a-1}\KummerconfhyperM@{a}{b}{z}\right) = \Pochhammersym{a}{n}z^{a+n-1}\KummerconfhyperM@{a+n}{b}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(z*diff(z, z))^(n)*((z)^(a - 1)* KummerM(a, b, z)) = pochhammer(a, n)*(z)^(a + n - 1)* KummerM(a + n, b, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(z*D[z, z])^(n)*((z)^(a - 1)* Hypergeometric1F1[a, b, z]) == Pochhammer[a, n]*(z)^(a + n - 1)* Hypergeometric1F1[a + n, b, z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 3.392872106-2.234328368*I
+Test Values: {a = -3/2, b = -3/2, z = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.628540387-.5000628115*I
+Test Values: {a = -3/2, b = -3/2, z = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[3.392872106018638, -2.234328368828302]
+Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.628540387739978, -0.5000628109822313]
+Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.3.E18 13.3.E18] || [[Item:Q4352|<math>\deriv[n]{}{z}\left(z^{b-1}\KummerconfhyperM@{a}{b}{z}\right) = \Pochhammersym{b-n}{n}z^{b-n-1}\KummerconfhyperM@{a}{b-n}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[n]{}{z}\left(z^{b-1}\KummerconfhyperM@{a}{b}{z}\right) = \Pochhammersym{b-n}{n}z^{b-n-1}\KummerconfhyperM@{a}{b-n}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff((z)^(b - 1)* KummerM(a, b, z), [z$(n)]) = pochhammer(b - n, n)*(z)^(b - n - 1)* KummerM(a, b - n, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[(z)^(b - 1)* Hypergeometric1F1[a, b, z], {z, n}] == Pochhammer[b - n, n]*(z)^(b - n - 1)* Hypergeometric1F1[a, b - n, z]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.23854907479223686, -4.055477620017901], Times[Complex[-0.2588190451025206, -0.9659258262890682], DifferenceRoot[Function[{, }
+Test Values: {Equal[Plus[Times[Plus[, -1.5], Plus[-1, Times[-1, ], 1], Plus[Times[-1, ], 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], []], Times[-1, Plus[-1, Times[-1, ], 1], Plus[Power[, 2], Power[, 3], Times[2, , -1.5], Times[2, Power[, 2], -1.5], Power[-1.5, 2], Times[, Power[-1.5, 2]], Times[-1, , 1], Times[-1, Power[, 2], 1], Times[-1, -1.5, 1], Times[-1, , -1.5, 1], Times[-1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-2, Power[, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, , 1, Pow<syntaxhighlight lang=mathematica>Result: Plus[Complex[7.020632087540109, 10.129888243360973], Times[Complex[-1.4142135623730947, -1.414213562373095], DifferenceRoot[Function[{, }
+Test Values: {Equal[Plus[Times[Plus[, -1.5], Plus[-1, Times[-1, ], 2], Plus[Times[-1, ], 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], []], Times[-1, Plus[-1, Times[-1, ], 2], Plus[Power[, 2], Power[, 3], Times[2, , -1.5], Times[2, Power[, 2], -1.5], Power[-1.5, 2], Times[, Power[-1.5, 2]], Times[-1, , 2], Times[-1, Power[, 2], 2], Times[-1, -1.5, 2], Times[-1, , -1.5, 2], Times[-1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-2, Power[, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, , 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1.5, 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[1, ]]], Times[Plus[1, ], Plus[, -1.5, Times[-1, 2]], Plus[-2, Times[-4, ], Times[-2, Power[, 2]], Times[-3, -1.5], Times[-2, , -1.5], Times[2, 2], Times[2, , 2], Times[-1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Times[, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[2, ]]], Times[Plus[1, ], Plus[2, ], Plus[, -1.5, Times[-1, 2]], Plus[1, , -1.5, Times[-1, 2]], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Times[Binomial[Plus[-1, -1.5], 2], Hypergeometric1F1[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[2], Times[Binomial[Plus[-1, -1.5], 2], Plus[Hypergeometric1F1[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1.5, 2, Power[Plus[Power[-1.5, 2], Times[-1, -1.5, 2]], -1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Hypergeometric1F1[Plus[1, -1.5], Plus[1, -1.5], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][3.0]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.3.E19 13.3.E19] || [[Item:Q4353|<math>\left(z\deriv{}{z}z\right)^{n}\left(z^{b-a-1}e^{-z}\KummerconfhyperM@{a}{b}{z}\right) = \Pochhammersym{b-a}{n}z^{b-a+n-1}e^{-z}\KummerconfhyperM@{a-n}{b}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(z\deriv{}{z}z\right)^{n}\left(z^{b-a-1}e^{-z}\KummerconfhyperM@{a}{b}{z}\right) = \Pochhammersym{b-a}{n}z^{b-a+n-1}e^{-z}\KummerconfhyperM@{a-n}{b}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(z*diff(z, z))^(n)*((z)^(b - a - 1)* exp(- z)*KummerM(a, b, z)) = pochhammer(b - a, n)*(z)^(b - a + n - 1)* exp(- z)*KummerM(a - n, b, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(z*D[z, z])^(n)*((z)^(b - a - 1)* Exp[- z]*Hypergeometric1F1[a, b, z]) == Pochhammer[b - a, n]*(z)^(b - a + n - 1)* Exp[- z]*Hypergeometric1F1[a - n, b, z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [298 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.000000000-.649969050e-10*I
+Test Values: {a = -3/2, b = -3/2, z = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .8660254040+.4999999999*I
+Test Values: {a = -3/2, b = -3/2, z = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [298 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.0, -5.551115123125783*^-17]
+Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.8660254037844388, 0.49999999999999983]
+Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.3.E20 13.3.E20] || [[Item:Q4354|<math>\deriv[n]{}{z}\left(e^{-z}\KummerconfhyperM@{a}{b}{z}\right) = (-1)^{n}\frac{\Pochhammersym{b-a}{n}}{\Pochhammersym{b}{n}}e^{-z}\KummerconfhyperM@{a}{b+n}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[n]{}{z}\left(e^{-z}\KummerconfhyperM@{a}{b}{z}\right) = (-1)^{n}\frac{\Pochhammersym{b-a}{n}}{\Pochhammersym{b}{n}}e^{-z}\KummerconfhyperM@{a}{b+n}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(exp(- z)*KummerM(a, b, z), [z$(n)]) = (- 1)^(n)*(pochhammer(b - a, n))/(pochhammer(b, n))*exp(- z)*KummerM(a, b + n, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Exp[- z]*Hypergeometric1F1[a, b, z], {z, n}] == (- 1)^(n)*Divide[Pochhammer[b - a, n],Pochhammer[b, n]]*Exp[- z]*Hypergeometric1F1[a, b + n, z]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.0, 0.0], Times[Complex[-0.36912880004696536, 0.20165598253870784], DifferenceRoot[Function[{, }
+Test Values: {Equal[Plus[Times[Plus[, -1.5], Plus[-1, Times[-1, ], 1], Plus[Times[-1, ], 1], []], Times[Plus[-1, Times[-1, ], 1], Plus[, Times[2, Power[, 2]], Times[, -1.5], -1.5, Times[, -1.5], Times[-1, , 1], Times[-1, -1.5, 1], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[1, ]]], Times[-1, Plus[1, ], Plus[Times[-1, ], Times[-1, Power[, 2]], Times[-1, -1.5], Times[-1, , -1.5], Times[, 1], Times[-1.5, 1], Times[3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, 1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[2, ]]], Times[Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Tim<syntaxhighlight lang=mathematica>Result: Plus[Complex[0.0, 0.0], Times[Complex[0.7382576000939307, -0.4033119650774157], DifferenceRoot[Function[{, }
+Test Values: {Equal[Plus[Times[Plus[, -1.5], Plus[-1, Times[-1, ], 2], Plus[Times[-1, ], 2], []], Times[Plus[-1, Times[-1, ], 2], Plus[, Times[2, Power[, 2]], Times[, -1.5], -1.5, Times[, -1.5], Times[-1, , 2], Times[-1, -1.5, 2], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[1, ]]], Times[-1, Plus[1, ], Plus[Times[-1, ], Times[-1, Power[, 2]], Times[-1, -1.5], Times[-1, , -1.5], Times[, 2], Times[-1.5, 2], Times[3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[2, ]]], Times[Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Times[Power[Factorial[2], -1], Hypergeometric1F1[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[2], Times[Power[-1.5, -1], Power[Factorial[2], -1], Plus[Times[-1.5, Hypergeometric1F1[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, 2, Hypergeometric1F1[Plus[1, -1.5], Plus[1, -1.5], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][3.0]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.3.E21 13.3.E21] || [[Item:Q4355|<math>\deriv[n]{}{z}\left(z^{b-1}e^{-z}\KummerconfhyperM@{a}{b}{z}\right) = \Pochhammersym{b-n}{n}z^{b-n-1}e^{-z}\KummerconfhyperM@{a-n}{b-n}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[n]{}{z}\left(z^{b-1}e^{-z}\KummerconfhyperM@{a}{b}{z}\right) = \Pochhammersym{b-n}{n}z^{b-n-1}e^{-z}\KummerconfhyperM@{a-n}{b-n}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff((z)^(b - 1)* exp(- z)*KummerM(a, b, z), [z$(n)]) = pochhammer(b - n, n)*(z)^(b - n - 1)* exp(- z)*KummerM(a - n, b - n, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[(z)^(b - 1)* Exp[- z]*Hypergeometric1F1[a, b, z], {z, n}] == Pochhammer[b - n, n]*(z)^(b - n - 1)* Exp[- z]*Hypergeometric1F1[a - n, b - n, z]</syntaxhighlight> || Failure || Aborted || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.6470476127563014, -2.4148145657226703], DifferenceRoot[Function[{, }
+Test Values: {Equal[Plus[Times[Plus[1, , Times[-1, -1.5]], []], Times[-1, Plus[1, ], Plus[-2, Times[-1, ], -1.5, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[1, ]]], Times[Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[-1, -1.5]], Hypergeometric1F1[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[Power[-1.5, -1], Power[E, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[-2, -1.5]], Plus[Times[-1, -1.5, Hypergeometric1F1[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[Power[-1.5, 2], Hypergeometric1F1[-1.5, -1.5, Pow<syntaxhighlight lang=mathematica>Result: Plus[Complex[6.187184335382289, 6.187184335382291], Times[2.0, DifferenceRoot[Function[{, }
+Test Values: {Equal[Plus[Times[Plus[1, , Times[-1, -1.5]], []], Times[-1, Plus[1, ], Plus[-2, Times[-1, ], -1.5, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[1, ]]], Times[Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[-1, -1.5]], Hypergeometric1F1[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[Power[-1.5, -1], Power[E, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[-2, -1.5]], Plus[Times[-1, -1.5, Hypergeometric1F1[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[Power[-1.5, 2], Hypergeometric1F1[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Hypergeometric1F1[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Hypergeometric1F1[Plus[1, -1.5], Plus[1, -1.5], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][2.0]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.3.E22 13.3.E22] || [[Item:Q4356|<math>\deriv{}{z}\KummerconfhyperU@{a}{b}{z} = -a\KummerconfhyperU@{a+1}{b+1}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv{}{z}\KummerconfhyperU@{a}{b}{z} = -a\KummerconfhyperU@{a+1}{b+1}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(KummerU(a, b, z), z) = - a*KummerU(a + 1, b + 1, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[HypergeometricU[a, b, z], z] == - a*HypergeometricU[a + 1, b + 1, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 252]
+|-
+| [https://dlmf.nist.gov/13.3.E23 13.3.E23] || [[Item:Q4357|<math>\deriv[n]{}{z}\KummerconfhyperU@{a}{b}{z} = (-1)^{n}\Pochhammersym{a}{n}\KummerconfhyperU@{a+n}{b+n}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[n]{}{z}\KummerconfhyperU@{a}{b}{z} = (-1)^{n}\Pochhammersym{a}{n}\KummerconfhyperU@{a+n}{b+n}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(KummerU(a, b, z), [z$(n)]) = (- 1)^(n)* pochhammer(a, n)*KummerU(a + n, b + n, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[HypergeometricU[a, b, z], {z, n}] == (- 1)^(n)* Pochhammer[a, n]*HypergeometricU[a + n, b + n, z]</syntaxhighlight> || Failure || Successful || Error || Successful [Tested: 300]
+|-
+| [https://dlmf.nist.gov/13.3.E24 13.3.E24] || [[Item:Q4358|<math>\left(z\deriv{}{z}z\right)^{n}\left(z^{a-1}\KummerconfhyperU@{a}{b}{z}\right) = \Pochhammersym{a}{n}\Pochhammersym{a-b+1}{n}z^{a+n-1}\KummerconfhyperU@{a+n}{b}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(z\deriv{}{z}z\right)^{n}\left(z^{a-1}\KummerconfhyperU@{a}{b}{z}\right) = \Pochhammersym{a}{n}\Pochhammersym{a-b+1}{n}z^{a+n-1}\KummerconfhyperU@{a+n}{b}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(z*diff(z, z))^(n)*((z)^(a - 1)* KummerU(a, b, z)) = pochhammer(a, n)*pochhammer(a - b + 1, n)*(z)^(a + n - 1)* KummerU(a + n, b, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(z*D[z, z])^(n)*((z)^(a - 1)* HypergeometricU[a, b, z]) == Pochhammer[a, n]*Pochhammer[a - b + 1, n]*(z)^(a + n - 1)* HypergeometricU[a + n, b, z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [295 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 4.557501915-2.807038782*I
+Test Values: {a = -3/2, b = -3/2, z = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 2.124956377+.5363245788*I
+Test Values: {a = -3/2, b = -3/2, z = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [295 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[4.557501914022213, -2.807038783226017]
+Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[2.124956376243804, 0.5363245787128816]
+Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.3.E25 13.3.E25] || [[Item:Q4359|<math>\deriv[n]{}{z}\left(z^{b-1}\KummerconfhyperU@{a}{b}{z}\right) = (-1)^{n}\Pochhammersym{a-b+1}{n}z^{b-n-1}\KummerconfhyperU@{a}{b-n}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[n]{}{z}\left(z^{b-1}\KummerconfhyperU@{a}{b}{z}\right) = (-1)^{n}\Pochhammersym{a-b+1}{n}z^{b-n-1}\KummerconfhyperU@{a}{b-n}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff((z)^(b - 1)* KummerU(a, b, z), [z$(n)]) = (- 1)^(n)* pochhammer(a - b + 1, n)*(z)^(b - n - 1)* KummerU(a, b - n, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[(z)^(b - 1)* HypergeometricU[a, b, z], {z, n}] == (- 1)^(n)* Pochhammer[a - b + 1, n]*(z)^(b - n - 1)* HypergeometricU[a, b - n, z]</syntaxhighlight> || Failure || Aborted || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.1522159386707833, -5.3504318269524465], Times[Complex[-0.2588190451025206, -0.9659258262890682], DifferenceRoot[Function[{, }
+Test Values: {Equal[Plus[Times[Plus[, -1.5], Plus[-1, Times[-1, ], 1], Plus[Times[-1, ], 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], []], Times[-1, Plus[-1, Times[-1, ], 1], Plus[Power[, 2], Power[, 3], Times[2, , -1.5], Times[2, Power[, 2], -1.5], Power[-1.5, 2], Times[, Power[-1.5, 2]], Times[-1, , 1], Times[-1, Power[, 2], 1], Times[-1, -1.5, 1], Times[-1, , -1.5, 1], Times[-1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-2, Power[, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, , 1, Pow<syntaxhighlight lang=mathematica>Result: Plus[Complex[9.411642901699432, 13.489513219804685], Times[Complex[-1.4142135623730947, -1.414213562373095], DifferenceRoot[Function[{, }
+Test Values: {Equal[Plus[Times[Plus[, -1.5], Plus[-1, Times[-1, ], 2], Plus[Times[-1, ], 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], []], Times[-1, Plus[-1, Times[-1, ], 2], Plus[Power[, 2], Power[, 3], Times[2, , -1.5], Times[2, Power[, 2], -1.5], Power[-1.5, 2], Times[, Power[-1.5, 2]], Times[-1, , 2], Times[-1, Power[, 2], 2], Times[-1, -1.5, 2], Times[-1, , -1.5, 2], Times[-1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-2, Power[, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, , 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1.5, 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[1, ]]], Times[Plus[1, ], Plus[, -1.5, Times[-1, 2]], Plus[-2, Times[-4, ], Times[-2, Power[, 2]], Times[-3, -1.5], Times[-2, , -1.5], Times[2, 2], Times[2, , 2], Times[-1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Times[, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[2, ]]], Times[Plus[1, ], Plus[2, ], Plus[, -1.5, Times[-1, 2]], Plus[1, , -1.5, Times[-1, 2]], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Times[Binomial[Plus[-1, -1.5], 2], HypergeometricU[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[2], Times[Binomial[Plus[-1, -1.5], 2], Plus[HypergeometricU[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, -1.5, Power[Plus[-1.5, Times[-1, 2]], -1], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], HypergeometricU[Plus[1, -1.5], Plus[1, -1.5], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][3.0]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.3.E26 13.3.E26] || [[Item:Q4360|<math>\left(z\deriv{}{z}z\right)^{n}\left(z^{b-a-1}e^{-z}\KummerconfhyperU@{a}{b}{z}\right) = (-1)^{n}z^{b-a+n-1}e^{-z}\KummerconfhyperU@{a-n}{b}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(z\deriv{}{z}z\right)^{n}\left(z^{b-a-1}e^{-z}\KummerconfhyperU@{a}{b}{z}\right) = (-1)^{n}z^{b-a+n-1}e^{-z}\KummerconfhyperU@{a-n}{b}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(z*diff(z, z))^(n)*((z)^(b - a - 1)* exp(- z)*KummerU(a, b, z)) = (- 1)^(n)* (z)^(b - a + n - 1)* exp(- z)*KummerU(a - n, b, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(z*D[z, z])^(n)*((z)^(b - a - 1)* Exp[- z]*HypergeometricU[a, b, z]) == (- 1)^(n)* (z)^(b - a + n - 1)* Exp[- z]*HypergeometricU[a - n, b, z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [298 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.496936093+.1242553737*I
+Test Values: {a = -3/2, b = -3/2, z = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.600796058+1.474329192*I
+Test Values: {a = -3/2, b = -3/2, z = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [298 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.4969360926980415, 0.12425537363460365]
+Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.6007960572551263, 1.4743291911897365]
+Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.3.E27 13.3.E27] || [[Item:Q4361|<math>\deriv[n]{}{z}\left(e^{-z}\KummerconfhyperU@{a}{b}{z}\right) = (-1)^{n}e^{-z}\KummerconfhyperU@{a}{b+n}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[n]{}{z}\left(e^{-z}\KummerconfhyperU@{a}{b}{z}\right) = (-1)^{n}e^{-z}\KummerconfhyperU@{a}{b+n}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(exp(- z)*KummerU(a, b, z), [z$(n)]) = (- 1)^(n)* exp(- z)*KummerU(a, b + n, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Exp[- z]*HypergeometricU[a, b, z], {z, n}] == (- 1)^(n)* Exp[- z]*HypergeometricU[a, b + n, z]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.40360579036441874, 0.11842116492450602], Times[Complex[-0.36912880004696536, 0.20165598253870784], DifferenceRoot[Function[{, }
+Test Values: {Equal[Plus[Times[Plus[, -1.5], Plus[-1, Times[-1, ], 1], Plus[Times[-1, ], 1], []], Times[Plus[-1, Times[-1, ], 1], Plus[, Times[2, Power[, 2]], Times[, -1.5], -1.5, Times[, -1.5], Times[-1, , 1], Times[-1, -1.5, 1], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[1, ]]], Times[-1, Plus[1, ], Plus[Times[-1, ], Times[-1, Power[, 2]], Times[-1, -1.5], Times[-1, , -1.5], Times[, 1], Times[-1.5, 1], Times[3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, 1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[2, ]]], Times[Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[3, ]]]], 0],<syntaxhighlight lang=mathematica>Result: Plus[Complex[0.20950938468408564, -0.2672919019422666], Times[Complex[0.7382576000939307, -0.4033119650774157], DifferenceRoot[Function[{, }
+Test Values: {Equal[Plus[Times[Plus[, -1.5], Plus[-1, Times[-1, ], 2], Plus[Times[-1, ], 2], []], Times[Plus[-1, Times[-1, ], 2], Plus[, Times[2, Power[, 2]], Times[, -1.5], -1.5, Times[, -1.5], Times[-1, , 2], Times[-1, -1.5, 2], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[1, ]]], Times[-1, Plus[1, ], Plus[Times[-1, ], Times[-1, Power[, 2]], Times[-1, -1.5], Times[-1, , -1.5], Times[, 2], Times[-1.5, 2], Times[3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[2, ]]], Times[Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Times[Power[Factorial[2], -1], HypergeometricU[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[2], Times[Power[Factorial[2], -1], Plus[HypergeometricU[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1.5, 2, HypergeometricU[Plus[1, -1.5], Plus[1, -1.5], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][3.0]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.3.E28 13.3.E28] || [[Item:Q4362|<math>\deriv[n]{}{z}\left(z^{b-1}e^{-z}\KummerconfhyperU@{a}{b}{z}\right) = (-1)^{n}z^{b-n-1}e^{-z}\KummerconfhyperU@{a-n}{b-n}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[n]{}{z}\left(z^{b-1}e^{-z}\KummerconfhyperU@{a}{b}{z}\right) = (-1)^{n}z^{b-n-1}e^{-z}\KummerconfhyperU@{a-n}{b-n}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff((z)^(b - 1)* exp(- z)*KummerU(a, b, z), [z$(n)]) = (- 1)^(n)* (z)^(b - n - 1)* exp(- z)*KummerU(a - n, b - n, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[(z)^(b - 1)* Exp[- z]*HypergeometricU[a, b, z], {z, n}] == (- 1)^(n)* (z)^(b - n - 1)* Exp[- z]*HypergeometricU[a - n, b - n, z]</syntaxhighlight> || Failure || Aborted || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.9968056293665363, -3.1564168178949528], DifferenceRoot[Function[{, }
+Test Values: {Equal[Plus[Times[Plus[1, , Times[-1, -1.5]], []], Times[-1, Plus[1, ], Plus[-2, Times[-1, ], -1.5, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[1, ]]], Times[Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[-1, -1.5]], HypergeometricU[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[-1, Power[E, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[-2, -1.5]], Plus[HypergeometricU[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, -1.5, HypergeometricU[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi<syntaxhighlight lang=mathematica>Result: Plus[Complex[8.32628899631003, 8.182173774638818], Times[2.0, DifferenceRoot[Function[{, }
+Test Values: {Equal[Plus[Times[Plus[1, , Times[-1, -1.5]], []], Times[-1, Plus[1, ], Plus[-2, Times[-1, ], -1.5, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[1, ]]], Times[Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[-1, -1.5]], HypergeometricU[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[-1, Power[E, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[-2, -1.5]], Plus[HypergeometricU[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, -1.5, HypergeometricU[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], HypergeometricU[-1.5, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], HypergeometricU[Plus[1, -1.5], Plus[1, -1.5], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][2.0]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.3.E29 13.3.E29] || [[Item:Q4363|<math>\left(z\deriv{}{z}z\right)^{n} = z^{n}\deriv[n]{}{z}z^{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(z\deriv{}{z}z\right)^{n} = z^{n}\deriv[n]{}{z}z^{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(z*diff(z, z))^(n) = (z)^(n)* diff((z)^(n), [z$(n)])</syntaxhighlight> || <syntaxhighlight lang=mathematica>(z*D[z, z])^(n) == (z)^(n)* D[(z)^(n), {z, n}]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.1616869430e-8-5.000000005*I
+Test Values: {z = 1/2*3^(1/2)+1/2*I, n = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -5.000000005+.1616869430e-8*I
+Test Values: {z = -1/2+1/2*I*3^(1/2), n = 3}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.0, -5.0]
+Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -5.0
+Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.4.E1 13.4.E1] || [[Item:Q4364|<math>\OlverconfhyperM@{a}{b}{z} = \frac{1}{\EulerGamma@{a}\EulerGamma@{b-a}}\int_{0}^{1}e^{zt}t^{a-1}(1-t)^{b-a-1}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\OlverconfhyperM@{a}{b}{z} = \frac{1}{\EulerGamma@{a}\EulerGamma@{b-a}}\int_{0}^{1}e^{zt}t^{a-1}(1-t)^{b-a-1}\diff{t}</syntaxhighlight> || <math>\realpart@@{b} > \realpart@@{a}, \realpart@@{a} > 0, \realpart@@{(b-a)} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>KummerM(a, b, z)/GAMMA(b) = (1)/(GAMMA(a)*GAMMA(b - a))*int(exp(z*t)*(t)^(a - 1)*(1 - t)^(b - a - 1), t = 0..1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric1F1Regularized[a, b, z] == Divide[1,Gamma[a]*Gamma[b - a]]*Integrate[Exp[z*t]*(t)^(a - 1)*(1 - t)^(b - a - 1), {t, 0, 1}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 21]
+|-
+| [https://dlmf.nist.gov/13.4.E2 13.4.E2] || [[Item:Q4365|<math>\OlverconfhyperM@{a}{b}{z} = \frac{1}{\EulerGamma@{b-c}}\int_{0}^{1}\OlverconfhyperM@{a}{c}{zt}t^{c-1}(1-t)^{b-c-1}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\OlverconfhyperM@{a}{b}{z} = \frac{1}{\EulerGamma@{b-c}}\int_{0}^{1}\OlverconfhyperM@{a}{c}{zt}t^{c-1}(1-t)^{b-c-1}\diff{t}</syntaxhighlight> || <math>\realpart@@{b} > \realpart@@{c}, \realpart@@{c} > 0, \realpart@@{(b-c)} > 0, \realpart@@{(b+s)} > 0, \realpart@@{(c+s)} > 0</math> || <syntaxhighlight lang=mathematica>KummerM(a, b, z)/GAMMA(b) = (1)/(GAMMA(b - c))*int(KummerM(a, c, z*t)/GAMMA(c)*(t)^(c - 1)*(1 - t)^(b - c - 1), t = 0..1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric1F1Regularized[a, b, z] == Divide[1,Gamma[b - c]]*Integrate[Hypergeometric1F1Regularized[a, c, z*t]*(t)^(c - 1)*(1 - t)^(b - c - 1), {t, 0, 1}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 126]
+|-
+| [https://dlmf.nist.gov/13.4.E3 13.4.E3] || [[Item:Q4366|<math>\OlverconfhyperM@{a}{b}{-z} = \frac{z^{\frac{1}{2}-\frac{1}{2}b}}{\EulerGamma@{a}}\int_{0}^{\infty}e^{-t}t^{a-\frac{1}{2}b-\frac{1}{2}}\BesselJ{b-1}@{2\sqrt{zt}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\OlverconfhyperM@{a}{b}{-z} = \frac{z^{\frac{1}{2}-\frac{1}{2}b}}{\EulerGamma@{a}}\int_{0}^{\infty}e^{-t}t^{a-\frac{1}{2}b-\frac{1}{2}}\BesselJ{b-1}@{2\sqrt{zt}}\diff{t}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{((b-1)+k+1)} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>KummerM(a, b, - z)/GAMMA(b) = ((z)^((1)/(2)-(1)/(2)*b))/(GAMMA(a))*int(exp(- t)*(t)^(a -(1)/(2)*b -(1)/(2))* BesselJ(b - 1, 2*sqrt(z*t)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric1F1Regularized[a, b, - z] == Divide[(z)^(Divide[1,2]-Divide[1,2]*b),Gamma[a]]*Integrate[Exp[- t]*(t)^(a -Divide[1,2]*b -Divide[1,2])* BesselJ[b - 1, 2*Sqrt[z*t]], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Error || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/13.4.E4 13.4.E4] || [[Item:Q4367|<math>\KummerconfhyperU@{a}{b}{z} = \frac{1}{\EulerGamma@{a}}\int_{0}^{\infty}e^{-zt}t^{a-1}(1+t)^{b-a-1}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{a}{b}{z} = \frac{1}{\EulerGamma@{a}}\int_{0}^{\infty}e^{-zt}t^{a-1}(1+t)^{b-a-1}\diff{t}</syntaxhighlight> || <math>\realpart@@{a} > 0, |\phase{z}| < \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>KummerU(a, b, z) = (1)/(GAMMA(a))*int(exp(- z*t)*(t)^(a - 1)*(1 + t)^(b - a - 1), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[a, b, z] == Divide[1,Gamma[a]]*Integrate[Exp[- z*t]*(t)^(a - 1)*(1 + t)^(b - a - 1), {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 90]
+|-
+| [https://dlmf.nist.gov/13.4.E5 13.4.E5] || [[Item:Q4368|<math>\KummerconfhyperU@{a}{b}{z} = \frac{z^{1-a}}{\EulerGamma@{a}\EulerGamma@{1+a-b}}\int_{0}^{\infty}\frac{\KummerconfhyperU@{b-a}{b}{t}e^{-t}t^{a-1}}{t+z}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{a}{b}{z} = \frac{z^{1-a}}{\EulerGamma@{a}\EulerGamma@{1+a-b}}\int_{0}^{\infty}\frac{\KummerconfhyperU@{b-a}{b}{t}e^{-t}t^{a-1}}{t+z}\diff{t}</syntaxhighlight> || <math>|\phase{z}| < \pi, \realpart@@{a} > \max\left(\realpart@@{b-1}, \realpart@@{a} > 0, \realpart@@{(1+a-b)} > 0</math> || <syntaxhighlight lang=mathematica>KummerU(a, b, z) = ((z)^(1 - a))/(GAMMA(a)*GAMMA(1 + a - b))*int((KummerU(b - a, b, t)*exp(- t)*(t)^(a - 1))/(t + z), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[a, b, z] == Divide[(z)^(1 - a),Gamma[a]*Gamma[1 + a - b]]*Integrate[Divide[HypergeometricU[b - a, b, t]*Exp[- t]*(t)^(a - 1),t + z], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/13.4.E6 13.4.E6] || [[Item:Q4369|<math>\KummerconfhyperU@{a}{b}{z} = \frac{(-1)^{n}z^{1-b-n}}{\EulerGamma@{1+a-b}}\int_{0}^{\infty}\frac{\OlverconfhyperM@{b-a}{b}{t}e^{-t}t^{b+n-1}}{t+z}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{a}{b}{z} = \frac{(-1)^{n}z^{1-b-n}}{\EulerGamma@{1+a-b}}\int_{0}^{\infty}\frac{\OlverconfhyperM@{b-a}{b}{t}e^{-t}t^{b+n-1}}{t+z}\diff{t}</syntaxhighlight> || <math>\abs{\phase@@{z}} < \pi, -\realpart@@{b} < n, n < 1+\realpart@{a-b}, \realpart@@{(1+a-b)} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>KummerU(a, b, z) = ((- 1)^(n)* (z)^(1 - b - n))/(GAMMA(1 + a - b))*int((KummerM(b - a, b, t)/GAMMA(b)*exp(- t)*(t)^(b + n - 1))/(t + z), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[a, b, z] == Divide[(- 1)^(n)* (z)^(1 - b - n),Gamma[1 + a - b]]*Integrate[Divide[Hypergeometric1F1Regularized[b - a, b, t]*Exp[- t]*(t)^(b + n - 1),t + z], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/13.4.E7 13.4.E7] || [[Item:Q4370|<math>\KummerconfhyperU@{a}{b}{z} = \frac{2z^{\frac{1}{2}-\frac{1}{2}b}}{\EulerGamma@{a}\EulerGamma@{a-b+1}}\*\int_{0}^{\infty}e^{-t}t^{a-\frac{1}{2}b-\frac{1}{2}}\modBesselK{b-1}@{2\sqrt{zt}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{a}{b}{z} = \frac{2z^{\frac{1}{2}-\frac{1}{2}b}}{\EulerGamma@{a}\EulerGamma@{a-b+1}}\*\int_{0}^{\infty}e^{-t}t^{a-\frac{1}{2}b-\frac{1}{2}}\modBesselK{b-1}@{2\sqrt{zt}}\diff{t}</syntaxhighlight> || <math>\realpart@@{a} > \max\left(\realpart@@{b-1}, \realpart@@{a} > 0, \realpart@@{(a-b+1)} > 0</math> || <syntaxhighlight lang=mathematica>KummerU(a, b, z) = (2*(z)^((1)/(2)-(1)/(2)*b))/(GAMMA(a)*GAMMA(a - b + 1))* int(exp(- t)*(t)^(a -(1)/(2)*b -(1)/(2))* BesselK(b - 1, 2*sqrt(z*t)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[a, b, z] == Divide[2*(z)^(Divide[1,2]-Divide[1,2]*b),Gamma[a]*Gamma[a - b + 1]]* Integrate[Exp[- t]*(t)^(a -Divide[1,2]*b -Divide[1,2])* BesselK[b - 1, 2*Sqrt[z*t]], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Aborted || - || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/13.4.E8 13.4.E8] || [[Item:Q4371|<math>\KummerconfhyperU@{a}{b}{z} = z^{c-a}\*\int_{0}^{\infty}e^{-zt}t^{c-1}\genhyperOlverF{2}{1}@{a,a-b+1}{c}{-t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{a}{b}{z} = z^{c-a}\*\int_{0}^{\infty}e^{-zt}t^{c-1}\genhyperOlverF{2}{1}@{a,a-b+1}{c}{-t}\diff{t}</syntaxhighlight> || <math>|\phase{z}| < \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>KummerU(a, b, z) = (z)^(c - a)* int(exp(- z*t)*(t)^(c - 1)* hypergeom([a , a - b + 1], [c], - t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[a, b, z] == (z)^(c - a)* Integrate[Exp[- z*t]*(t)^(c - 1)* HypergeometricPFQRegularized[{a , a - b + 1}, {c}, - t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [294 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(undefined)+Float(undefined)*I
+Test Values: {a = -3/2, b = -3/2, c = -3/2, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(undefined)+Float(undefined)*I
+Test Values: {a = -3/2, b = -3/2, c = -3/2, z = 1/2-1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/13.4.E9 13.4.E9] || [[Item:Q4372|<math>\OlverconfhyperM@{a}{b}{z} = \frac{\EulerGamma@{1+a-b}}{2\pi\iunit\EulerGamma@{a}}\int_{0}^{(1+)}e^{zt}t^{a-1}{(t-1)^{b-a-1}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\OlverconfhyperM@{a}{b}{z} = \frac{\EulerGamma@{1+a-b}}{2\pi\iunit\EulerGamma@{a}}\int_{0}^{(1+)}e^{zt}t^{a-1}{(t-1)^{b-a-1}}\diff{t}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{(1+a-b)} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>KummerM(a, b, z)/GAMMA(b) = (GAMMA(1 + a - b))/(2*Pi*I*GAMMA(a))*int(exp(z*t)*(t)^(a - 1)*(t - 1)^(b - a - 1), t = 0..(1 +))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric1F1Regularized[a, b, z] == Divide[Gamma[1 + a - b],2*Pi*I*Gamma[a]]*Integrate[Exp[z*t]*(t)^(a - 1)*(t - 1)^(b - a - 1), {t, 0, (1 +)}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || Error
+|-
+| [https://dlmf.nist.gov/13.4.E10 13.4.E10] || [[Item:Q4373|<math>\OlverconfhyperM@{a}{b}{z} = e^{-a\pi\iunit}\frac{\EulerGamma@{1-a}}{2\pi\iunit\EulerGamma@{b-a}}\int_{1}^{(0+)}e^{zt}t^{a-1}{(1-t)^{b-a-1}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\OlverconfhyperM@{a}{b}{z} = e^{-a\pi\iunit}\frac{\EulerGamma@{1-a}}{2\pi\iunit\EulerGamma@{b-a}}\int_{1}^{(0+)}e^{zt}t^{a-1}{(1-t)^{b-a-1}}\diff{t}</syntaxhighlight> || <math>\realpart@{b-a} > 0, \realpart@@{(1-a)} > 0, \realpart@@{(b-a)} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>KummerM(a, b, z)/GAMMA(b) = exp(- a*Pi*I)*(GAMMA(1 - a))/(2*Pi*I*GAMMA(b - a))*int(exp(z*t)*(t)^(a - 1)*(1 - t)^(b - a - 1), t = 1..(0 +))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric1F1Regularized[a, b, z] == Exp[- a*Pi*I]*Divide[Gamma[1 - a],2*Pi*I*Gamma[b - a]]*Integrate[Exp[z*t]*(t)^(a - 1)*(1 - t)^(b - a - 1), {t, 1, (0 +)}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || Error
+|-
+| [https://dlmf.nist.gov/13.4.E11 13.4.E11] || [[Item:Q4374|<math>\OlverconfhyperM@{a}{b}{z} = e^{-b\pi\iunit}\EulerGamma@{1-a}\EulerGamma@{1+a-b}\*\frac{1}{4\pi^{2}}\int_{\alpha}^{(0+,1+,0-,1-)}e^{zt}t^{a-1}{(1-t)^{b-a-1}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\OlverconfhyperM@{a}{b}{z} = e^{-b\pi\iunit}\EulerGamma@{1-a}\EulerGamma@{1+a-b}\*\frac{1}{4\pi^{2}}\int_{\alpha}^{(0+,1+,0-,1-)}e^{zt}t^{a-1}{(1-t)^{b-a-1}}\diff{t}</syntaxhighlight> || <math>\realpart@@{(1-a)} > 0, \realpart@@{(1+a-b)} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>KummerM(a, b, z)/GAMMA(b) = exp(- b*Pi*I)*GAMMA(1 - a)*GAMMA(1 + a - b)*(1)/(4*(Pi)^(2))*int(exp(z*t)*(t)^(a - 1)*(1 - t)^(b - a - 1), t = alpha..(0 + , 1 + , 0 - , 1 -))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric1F1Regularized[a, b, z] == Exp[- b*Pi*I]*Gamma[1 - a]*Gamma[1 + a - b]*Divide[1,4*(Pi)^(2)]*Integrate[Exp[z*t]*(t)^(a - 1)*(1 - t)^(b - a - 1), {t, \[Alpha], (0 + , 1 + , 0 - , 1 -)}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || Error
+|-
+| [https://dlmf.nist.gov/13.4.E12 13.4.E12] || [[Item:Q4375|<math>\OlverconfhyperM@{a}{c}{z} = \frac{\EulerGamma@{b}}{2\pi\iunit}z^{1-b}\int_{-\infty}^{(0+,1+)}e^{zt}t^{-b}\genhyperOlverF{2}{1}@{a,b}{c}{\ifrac{1}{t}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\OlverconfhyperM@{a}{c}{z} = \frac{\EulerGamma@{b}}{2\pi\iunit}z^{1-b}\int_{-\infty}^{(0+,1+)}e^{zt}t^{-b}\genhyperOlverF{2}{1}@{a,b}{c}{\ifrac{1}{t}}\diff{t}</syntaxhighlight> || <math>\abs{\phase@@{z}} < \frac{1}{2}\pi, \realpart@@{b} > 0, \realpart@@{(c+s)} > 0</math> || <syntaxhighlight lang=mathematica>KummerM(a, c, z)/GAMMA(c) = (GAMMA(b))/(2*Pi*I)*(z)^(1 - b)* int(exp(z*t)*(t)^(- b)* hypergeom([a , b], [c], (1)/(t)), t = - infinity..(0 + , 1 +))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric1F1Regularized[a, c, z] == Divide[Gamma[b],2*Pi*I]*(z)^(1 - b)* Integrate[Exp[z*t]*(t)^(- b)* HypergeometricPFQRegularized[{a , b}, {c}, Divide[1,t]], {t, - Infinity, (0 + , 1 +)}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || Error
+|-
+| [https://dlmf.nist.gov/13.4.E13 13.4.E13] || [[Item:Q4376|<math>\OlverconfhyperM@{a}{b}{z} = \frac{z^{1-b}}{2\pi\iunit}\int_{-\infty}^{(0+,1+)}e^{zt}t^{-b}\!\left(1-\frac{1}{t}\right)^{-a}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\OlverconfhyperM@{a}{b}{z} = \frac{z^{1-b}}{2\pi\iunit}\int_{-\infty}^{(0+,1+)}e^{zt}t^{-b}\!\left(1-\frac{1}{t}\right)^{-a}\diff{t}</syntaxhighlight> || <math>|\phase{z}| < \frac{1}{2}\pi, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>KummerM(a, b, z)/GAMMA(b) = ((z)^(1 - b))/(2*Pi*I)*int(exp(z*t)*(t)^(- b)*(1 -(1)/(t))^(- a), t = - infinity..(0 + , 1 +))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric1F1Regularized[a, b, z] == Divide[(z)^(1 - b),2*Pi*I]*Integrate[Exp[z*t]*(t)^(- b)*(1 -Divide[1,t])^(- a), {t, - Infinity, (0 + , 1 +)}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || Error
+|-
+| [https://dlmf.nist.gov/13.4.E14 13.4.E14] || [[Item:Q4377|<math>\KummerconfhyperU@{a}{b}{z} = e^{-a\pi\iunit}\frac{\EulerGamma@{1-a}}{2\pi\iunit}\int_{\infty}^{(0+)}e^{-zt}t^{a-1}{(1+t)^{b-a-1}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{a}{b}{z} = e^{-a\pi\iunit}\frac{\EulerGamma@{1-a}}{2\pi\iunit}\int_{\infty}^{(0+)}e^{-zt}t^{a-1}{(1+t)^{b-a-1}}\diff{t}</syntaxhighlight> || <math>\abs{\phase@@{z}} < \frac{1}{2}\pi, \realpart@@{(1-a)} > 0</math> || <syntaxhighlight lang=mathematica>KummerU(a, b, z) = exp(- a*Pi*I)*(GAMMA(1 - a))/(2*Pi*I)*int(exp(- z*t)*(t)^(a - 1)*(1 + t)^(b - a - 1), t = infinity..(0 +))</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[a, b, z] == Exp[- a*Pi*I]*Divide[Gamma[1 - a],2*Pi*I]*Integrate[Exp[- z*t]*(t)^(a - 1)*(1 + t)^(b - a - 1), {t, Infinity, (0 +)}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || Error
+|-
+| [https://dlmf.nist.gov/13.4.E15 13.4.E15] || [[Item:Q4378|<math>\frac{\KummerconfhyperU@{a}{b}{z}}{\EulerGamma@{c}\EulerGamma@{c-b+1}} = \frac{z^{1-c}}{2\pi\iunit}\int_{-\infty}^{(0+)}e^{zt}t^{-c}\genhyperOlverF{2}{1}@{a,c}{a+c-b+1}{1-\frac{1}{t}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\KummerconfhyperU@{a}{b}{z}}{\EulerGamma@{c}\EulerGamma@{c-b+1}} = \frac{z^{1-c}}{2\pi\iunit}\int_{-\infty}^{(0+)}e^{zt}t^{-c}\genhyperOlverF{2}{1}@{a,c}{a+c-b+1}{1-\frac{1}{t}}\diff{t}</syntaxhighlight> || <math>\abs{\phase@@{z}} < \frac{1}{2}\pi, \realpart@@{c} > 0, \realpart@@{(c-b+1)} > 0</math> || <syntaxhighlight lang=mathematica>(KummerU(a, b, z))/(GAMMA(c)*GAMMA(c - b + 1)) = ((z)^(1 - c))/(2*Pi*I)*int(exp(z*t)*(t)^(- c)* hypergeom([a , c], [a + c - b + 1], 1 -(1)/(t)), t = - infinity..(0 +))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[HypergeometricU[a, b, z],Gamma[c]*Gamma[c - b + 1]] == Divide[(z)^(1 - c),2*Pi*I]*Integrate[Exp[z*t]*(t)^(- c)* HypergeometricPFQRegularized[{a , c}, {a + c - b + 1}, 1 -Divide[1,t]], {t, - Infinity, (0 +)}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || Error
+|-
+| [https://dlmf.nist.gov/13.4.E16 13.4.E16] || [[Item:Q4379|<math>\OlverconfhyperM@{a}{b}{-z} = \frac{1}{2\pi\iunit\EulerGamma@{a}}\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{a+t}\EulerGamma@{-t}}{\EulerGamma@{b+t}}z^{t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\OlverconfhyperM@{a}{b}{-z} = \frac{1}{2\pi\iunit\EulerGamma@{a}}\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{a+t}\EulerGamma@{-t}}{\EulerGamma@{b+t}}z^{t}\diff{t}</syntaxhighlight> || <math>|\phase{z}| < \tfrac{1}{2}\pi, \realpart@@{a} > 0, \realpart@@{(a+t)} > 0, \realpart@@{(-t)} > 0, \realpart@@{(b+t)} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>KummerM(a, b, - z)/GAMMA(b) = (1)/(2*Pi*I*GAMMA(a))*int((GAMMA(a + t)*GAMMA(- t))/(GAMMA(b + t))*(z)^(t), t = - I*infinity..I*infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric1F1Regularized[a, b, - z] == Divide[1,2*Pi*I*Gamma[a]]*Integrate[Divide[Gamma[a + t]*Gamma[- t],Gamma[b + t]]*(z)^(t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/13.4.E17 13.4.E17] || [[Item:Q4380|<math>\KummerconfhyperU@{a}{b}{z} = \frac{z^{-a}}{2\pi\iunit}\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{a+t}\EulerGamma@{1+a-b+t}\EulerGamma@{-t}}{\EulerGamma@{a}\EulerGamma@{1+a-b}}z^{-t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{a}{b}{z} = \frac{z^{-a}}{2\pi\iunit}\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{a+t}\EulerGamma@{1+a-b+t}\EulerGamma@{-t}}{\EulerGamma@{a}\EulerGamma@{1+a-b}}z^{-t}\diff{t}</syntaxhighlight> || <math>|\phase{z}| < \tfrac{3}{2}\pi, \realpart@@{(a+t)} > 0, \realpart@@{(1+a-b+t)} > 0, \realpart@@{(-t)} > 0, \realpart@@{a} > 0, \realpart@@{(1+a-b)} > 0</math> || <syntaxhighlight lang=mathematica>KummerU(a, b, z) = ((z)^(- a))/(2*Pi*I)*int((GAMMA(a + t)*GAMMA(1 + a - b + t)*GAMMA(- t))/(GAMMA(a)*GAMMA(1 + a - b))*(z)^(- t), t = - I*infinity..I*infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[a, b, z] == Divide[(z)^(- a),2*Pi*I]*Integrate[Divide[Gamma[a + t]*Gamma[1 + a - b + t]*Gamma[- t],Gamma[a]*Gamma[1 + a - b]]*(z)^(- t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/13.4.E18 13.4.E18] || [[Item:Q4381|<math>\KummerconfhyperU@{a}{b}{z} = \frac{z^{1-b}e^{z}}{2\pi\iunit}\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{b-1+t}\EulerGamma@{t}}{\EulerGamma@{a+t}}z^{-t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{a}{b}{z} = \frac{z^{1-b}e^{z}}{2\pi\iunit}\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{b-1+t}\EulerGamma@{t}}{\EulerGamma@{a+t}}z^{-t}\diff{t}</syntaxhighlight> || <math>|\phase{z}| < \tfrac{1}{2}\pi, \realpart@@{(b-1+t)} > 0, \realpart@@{t} > 0, \realpart@@{(a+t)} > 0</math> || <syntaxhighlight lang=mathematica>KummerU(a, b, z) = ((z)^(1 - b)* exp(z))/(2*Pi*I)*int((GAMMA(b - 1 + t)*GAMMA(t))/(GAMMA(a + t))*(z)^(- t), t = - I*infinity..I*infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[a, b, z] == Divide[(z)^(1 - b)* Exp[z],2*Pi*I]*Integrate[Divide[Gamma[b - 1 + t]*Gamma[t],Gamma[a + t]]*(z)^(- t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/13.6.E1 13.6.E1] || [[Item:Q4388|<math>\KummerconfhyperM@{a}{a}{z} = e^{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperM@{a}{a}{z} = e^{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerM(a, a, z) = exp(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric1F1[a, a, z] == Exp[z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 42]
+|-
+| [https://dlmf.nist.gov/13.6.E2 13.6.E2] || [[Item:Q4389|<math>\KummerconfhyperM@{1}{2}{2z} = \frac{e^{z}}{z}\sinh@@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperM@{1}{2}{2z} = \frac{e^{z}}{z}\sinh@@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerM(1, 2, 2*z) = (exp(z))/(z)*sinh(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric1F1[1, 2, 2*z] == Divide[Exp[z],z]*Sinh[z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
+|-
+| [https://dlmf.nist.gov/13.6.E3 13.6.E3] || [[Item:Q4390|<math>\KummerconfhyperM@{0}{b}{z} = \KummerconfhyperU@{0}{b}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperM@{0}{b}{z} = \KummerconfhyperU@{0}{b}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerM(0, b, z) = KummerU(0, b, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric1F1[0, b, z] == HypergeometricU[0, b, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 42]
+|-
+| [https://dlmf.nist.gov/13.6.E3 13.6.E3] || [[Item:Q4390|<math>\KummerconfhyperU@{0}{b}{z} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{0}{b}{z} = 1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerU(0, b, z) = 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[0, b, z] == 1</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 42]
+|-
+| [https://dlmf.nist.gov/13.6.E4 13.6.E4] || [[Item:Q4391|<math>\KummerconfhyperU@{a}{a+1}{z} = z^{-a}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{a}{a+1}{z} = z^{-a}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerU(a, a + 1, z) = (z)^(- a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[a, a + 1, z] == (z)^(- a)</syntaxhighlight> || Failure || Successful || Successful [Tested: 42] || Successful [Tested: 42]
+|-
+| [https://dlmf.nist.gov/13.6.E5 13.6.E5] || [[Item:Q4392|<math>\KummerconfhyperM@{a}{a+1}{-z} = e^{-z}\KummerconfhyperM@{1}{a+1}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperM@{a}{a+1}{-z} = e^{-z}\KummerconfhyperM@{1}{a+1}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerM(a, a + 1, - z) = exp(- z)*KummerM(1, a + 1, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric1F1[a, a + 1, - z] == Exp[- z]*Hypergeometric1F1[1, a + 1, z]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.6.E5 13.6.E5] || [[Item:Q4392|<math>e^{-z}\KummerconfhyperM@{1}{a+1}{z} = az^{-a}\incgamma@{a}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{-z}\KummerconfhyperM@{1}{a+1}{z} = az^{-a}\incgamma@{a}{z}</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>exp(- z)*KummerM(1, a + 1, z) = a*(z)^(- a)* GAMMA(a)-GAMMA(a, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[- z]*Hypergeometric1F1[1, a + 1, z] == a*(z)^(- a)* Gamma[a, 0, z]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .1786149082+.5798847761*I
+Test Values: {a = 3/2, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 4.103691021-1.156198608*I
+Test Values: {a = 3/2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 21]
+|-
+| [https://dlmf.nist.gov/13.6.E6 13.6.E6] || [[Item:Q4393|<math>\KummerconfhyperU@{a}{a}{z} = z^{1-a}\KummerconfhyperU@{1}{2-a}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{a}{a}{z} = z^{1-a}\KummerconfhyperU@{1}{2-a}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerU(a, a, z) = (z)^(1 - a)* KummerU(1, 2 - a, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[a, a, z] == (z)^(1 - a)* HypergeometricU[1, 2 - a, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 42]
+|-
+| [https://dlmf.nist.gov/13.6.E6 13.6.E6] || [[Item:Q4393|<math>z^{1-a}\KummerconfhyperU@{1}{2-a}{z} = z^{1-a}e^{z}\genexpintE{a}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z^{1-a}\KummerconfhyperU@{1}{2-a}{z} = z^{1-a}e^{z}\genexpintE{a}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(z)^(1 - a)* KummerU(1, 2 - a, z) = (z)^(1 - a)* exp(z)*Ei(a, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(z)^(1 - a)* HypergeometricU[1, 2 - a, z] == (z)^(1 - a)* Exp[z]*ExpIntegralE[a, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 42]
+|-
+| [https://dlmf.nist.gov/13.6.E6 13.6.E6] || [[Item:Q4393|<math>z^{1-a}e^{z}\genexpintE{a}@{z} = e^{z}\incGamma@{1-a}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z^{1-a}e^{z}\genexpintE{a}@{z} = e^{z}\incGamma@{1-a}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(z)^(1 - a)* exp(z)*Ei(a, z) = exp(z)*GAMMA(1 - a, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(z)^(1 - a)* Exp[z]*ExpIntegralE[a, z] == Exp[z]*Gamma[1 - a, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 42]
+|-
+| [https://dlmf.nist.gov/13.6.E7 13.6.E7] || [[Item:Q4394|<math>\KummerconfhyperM@{\tfrac{1}{2}}{\tfrac{3}{2}}{-z^{2}} = \frac{\sqrt{\pi}}{2z}\erf@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperM@{\tfrac{1}{2}}{\tfrac{3}{2}}{-z^{2}} = \frac{\sqrt{\pi}}{2z}\erf@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerM((1)/(2), (3)/(2), - (z)^(2)) = (sqrt(Pi))/(2*z)*erf(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric1F1[Divide[1,2], Divide[3,2], - (z)^(2)] == Divide[Sqrt[Pi],2*z]*Erf[z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
+|-
+| [https://dlmf.nist.gov/13.6.E8 13.6.E8] || [[Item:Q4395|<math>\KummerconfhyperU@{\tfrac{1}{2}}{\tfrac{1}{2}}{z^{2}} = \sqrt{\pi}e^{z^{2}}\erfc@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{\tfrac{1}{2}}{\tfrac{1}{2}}{z^{2}} = \sqrt{\pi}e^{z^{2}}\erfc@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerU((1)/(2), (1)/(2), (z)^(2)) = sqrt(Pi)*exp((z)^(2))*erfc(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[Divide[1,2], Divide[1,2], (z)^(2)] == Sqrt[Pi]*Exp[(z)^(2)]*Erfc[z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .418096912e-1+2.795226389*I
+Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -2.288685714-4.974950146*I
+Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.041809690497868646, 2.7952263885381483]
+Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-2.28868571442365, -4.974950145988551]
+Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br></div></div>
+|-
+| [https://dlmf.nist.gov/13.6.E9 13.6.E9] || [[Item:Q4396|<math>\KummerconfhyperM@{\nu+\tfrac{1}{2}}{2\nu+1}{2z} = \EulerGamma@{1+\nu}e^{z}\left(\ifrac{z}{2}\right)^{-\nu}\modBesselI{\nu}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperM@{\nu+\tfrac{1}{2}}{2\nu+1}{2z} = \EulerGamma@{1+\nu}e^{z}\left(\ifrac{z}{2}\right)^{-\nu}\modBesselI{\nu}@{z}</syntaxhighlight> || <math>\realpart@@{(1+\nu)} > 0, \realpart@@{(\nu+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>KummerM(nu +(1)/(2), 2*nu + 1, 2*z) = GAMMA(1 + nu)*exp(z)*((z)/(2))^(- nu)* BesselI(nu, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric1F1[\[Nu]+Divide[1,2], 2*\[Nu]+ 1, 2*z] == Gamma[1 + \[Nu]]*Exp[z]*(Divide[z,2])^(- \[Nu])* BesselI[\[Nu], z]</syntaxhighlight> || Successful || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 56]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.026957443693084, -2.3780953180269115]
+Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -0.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.5295327248436391, -0.1815534052901876]
+Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, -0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.6.E10 13.6.E10] || [[Item:Q4397|<math>\KummerconfhyperU@{\nu+\tfrac{1}{2}}{2\nu+1}{2z} = \frac{1}{\sqrt{\pi}}e^{z}\left(2z\right)^{-\nu}\modBesselK{\nu}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{\nu+\tfrac{1}{2}}{2\nu+1}{2z} = \frac{1}{\sqrt{\pi}}e^{z}\left(2z\right)^{-\nu}\modBesselK{\nu}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerU(nu +(1)/(2), 2*nu + 1, 2*z) = (1)/(sqrt(Pi))*exp(z)*(2*z)^(- nu)* BesselK(nu, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[\[Nu]+Divide[1,2], 2*\[Nu]+ 1, 2*z] == Divide[1,Sqrt[Pi]]*Exp[z]*(2*z)^(- \[Nu])* BesselK[\[Nu], z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 70]
+|-
+| [https://dlmf.nist.gov/13.6.E11 13.6.E11] || [[Item:Q4398|<math>\KummerconfhyperU@{\tfrac{5}{6}}{\tfrac{5}{3}}{\tfrac{4}{3}z^{3/2}} = \sqrt{\pi}\frac{3^{5/6}\exp@{\tfrac{2}{3}z^{3/2}}}{2^{2/3}z}\AiryAi@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{\tfrac{5}{6}}{\tfrac{5}{3}}{\tfrac{4}{3}z^{3/2}} = \sqrt{\pi}\frac{3^{5/6}\exp@{\tfrac{2}{3}z^{3/2}}}{2^{2/3}z}\AiryAi@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerU((5)/(6), (5)/(3), (4)/(3)*(z)^(3/2)) = sqrt(Pi)*((3)^(5/6)* exp((2)/(3)*(z)^(3/2)))/((2)^(2/3)* z)*AiryAi(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[Divide[5,6], Divide[5,3], Divide[4,3]*(z)^(3/2)] == Sqrt[Pi]*Divide[(3)^(5/6)* Exp[Divide[2,3]*(z)^(3/2)],(2)^(2/3)* z]*AiryAi[z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .7957982359-.7292249892*I
+Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.7957982355202466, -0.7292249896477329]
+Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br></div></div>
+|-
+| [https://dlmf.nist.gov/13.6.E12 13.6.E12] || [[Item:Q4399|<math>\KummerconfhyperU@{\tfrac{1}{2}a+\tfrac{1}{4}}{\tfrac{1}{2}}{\tfrac{1}{2}z^{2}} = 2^{\frac{1}{2}a+\frac{1}{4}}e^{\frac{1}{4}z^{2}}\paraU@{a}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{\tfrac{1}{2}a+\tfrac{1}{4}}{\tfrac{1}{2}}{\tfrac{1}{2}z^{2}} = 2^{\frac{1}{2}a+\frac{1}{4}}e^{\frac{1}{4}z^{2}}\paraU@{a}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerU((1)/(2)*a +(1)/(4), (1)/(2), (1)/(2)*(z)^(2)) = (2)^((1)/(2)*a +(1)/(4))* exp((1)/(4)*(z)^(2))*CylinderU(a, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[Divide[1,2]*a +Divide[1,4], Divide[1,2], Divide[1,2]*(z)^(2)] == (2)^(Divide[1,2]*a +Divide[1,4])* Exp[Divide[1,4]*(z)^(2)]*ParabolicCylinderD[- 1/2 -(a), z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .7071067808-1.224744871*I
+Test Values: {a = -3/2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.224744871+.7071067810*I
+Test Values: {a = -3/2, z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.7071067811865475, -1.224744871391589]
+Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.224744871391589, 0.7071067811865475]
+Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.6.E13 13.6.E13] || [[Item:Q4400|<math>\KummerconfhyperU@{\tfrac{1}{2}a+\tfrac{3}{4}}{\tfrac{3}{2}}{\tfrac{1}{2}z^{2}} = 2^{\frac{1}{2}a+\frac{3}{4}}\frac{e^{\frac{1}{4}z^{2}}}{z}\paraU@{a}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{\tfrac{1}{2}a+\tfrac{3}{4}}{\tfrac{3}{2}}{\tfrac{1}{2}z^{2}} = 2^{\frac{1}{2}a+\frac{3}{4}}\frac{e^{\frac{1}{4}z^{2}}}{z}\paraU@{a}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerU((1)/(2)*a +(3)/(4), (3)/(2), (1)/(2)*(z)^(2)) = (2)^((1)/(2)*a +(3)/(4))*(exp((1)/(4)*(z)^(2)))/(z)*CylinderU(a, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[Divide[1,2]*a +Divide[3,4], Divide[3,2], Divide[1,2]*(z)^(2)] == (2)^(Divide[1,2]*a +Divide[3,4])*Divide[Exp[Divide[1,4]*(z)^(2)],z]*ParabolicCylinderD[- 1/2 -(a), z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 3.981039608+.280376847*I
+Test Values: {a = 3/2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 9.425210776+2.041008108*I
+Test Values: {a = 3/2, z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[3.9810396073031904, 0.2803768494018799]
+Test Values: {Rule[a, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[9.42521077642933, 2.0410081046172346]
+Test Values: {Rule[a, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.6.E14 13.6.E14] || [[Item:Q4401|<math>\KummerconfhyperM@{\tfrac{1}{2}a+\tfrac{1}{4}}{\tfrac{1}{2}}{\tfrac{1}{2}z^{2}} = \frac{2^{\frac{1}{2}a-\frac{3}{4}}\EulerGamma@{\tfrac{1}{2}a+\tfrac{3}{4}}e^{\frac{1}{4}z^{2}}}{\sqrt{\pi}}\*\left(\paraU@{a}{z}+\paraU@{a}{-z}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperM@{\tfrac{1}{2}a+\tfrac{1}{4}}{\tfrac{1}{2}}{\tfrac{1}{2}z^{2}} = \frac{2^{\frac{1}{2}a-\frac{3}{4}}\EulerGamma@{\tfrac{1}{2}a+\tfrac{3}{4}}e^{\frac{1}{4}z^{2}}}{\sqrt{\pi}}\*\left(\paraU@{a}{z}+\paraU@{a}{-z}\right)</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{2}a+\tfrac{3}{4})} > 0</math> || <syntaxhighlight lang=mathematica>KummerM((1)/(2)*a +(1)/(4), (1)/(2), (1)/(2)*(z)^(2)) = ((2)^((1)/(2)*a -(3)/(4))* GAMMA((1)/(2)*a +(3)/(4))*exp((1)/(4)*(z)^(2)))/(sqrt(Pi))*(CylinderU(a, z)+ CylinderU(a, - z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric1F1[Divide[1,2]*a +Divide[1,4], Divide[1,2], Divide[1,2]*(z)^(2)] == Divide[(2)^(Divide[1,2]*a -Divide[3,4])* Gamma[Divide[1,2]*a +Divide[3,4]]*Exp[Divide[1,4]*(z)^(2)],Sqrt[Pi]]*(ParabolicCylinderD[- 1/2 -(a), z]+ ParabolicCylinderD[- 1/2 -(a), - z])</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 28]
+|-
+| [https://dlmf.nist.gov/13.6.E15 13.6.E15] || [[Item:Q4402|<math>\KummerconfhyperM@{\tfrac{1}{2}a+\tfrac{3}{4}}{\tfrac{3}{2}}{\tfrac{1}{2}z^{2}} = \frac{2^{\frac{1}{2}a-\frac{5}{4}}\EulerGamma@{\tfrac{1}{2}a+\tfrac{1}{4}}e^{\frac{1}{4}z^{2}}}{z\sqrt{\pi}}\*\left(\paraU@{a}{-z}-\paraU@{a}{z}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperM@{\tfrac{1}{2}a+\tfrac{3}{4}}{\tfrac{3}{2}}{\tfrac{1}{2}z^{2}} = \frac{2^{\frac{1}{2}a-\frac{5}{4}}\EulerGamma@{\tfrac{1}{2}a+\tfrac{1}{4}}e^{\frac{1}{4}z^{2}}}{z\sqrt{\pi}}\*\left(\paraU@{a}{-z}-\paraU@{a}{z}\right)</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{2}a+\tfrac{1}{4})} > 0</math> || <syntaxhighlight lang=mathematica>KummerM((1)/(2)*a +(3)/(4), (3)/(2), (1)/(2)*(z)^(2)) = ((2)^((1)/(2)*a -(5)/(4))* GAMMA((1)/(2)*a +(1)/(4))*exp((1)/(4)*(z)^(2)))/(z*sqrt(Pi))*(CylinderU(a, - z)- CylinderU(a, z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric1F1[Divide[1,2]*a +Divide[3,4], Divide[3,2], Divide[1,2]*(z)^(2)] == Divide[(2)^(Divide[1,2]*a -Divide[5,4])* Gamma[Divide[1,2]*a +Divide[1,4]]*Exp[Divide[1,4]*(z)^(2)],z*Sqrt[Pi]]*(ParabolicCylinderD[- 1/2 -(a), - z]- ParabolicCylinderD[- 1/2 -(a), z])</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 21]
+|-
+| [https://dlmf.nist.gov/13.6.E16 13.6.E16] || [[Item:Q4403|<math>\KummerconfhyperM@{-n}{\tfrac{1}{2}}{z^{2}} = (-1)^{n}\frac{n!}{(2n)!}\HermitepolyH{2n}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperM@{-n}{\tfrac{1}{2}}{z^{2}} = (-1)^{n}\frac{n!}{(2n)!}\HermitepolyH{2n}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerM(- n, (1)/(2), (z)^(2)) = (- 1)^(n)*(factorial(n))/(factorial(2*n))*HermiteH(2*n, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric1F1[- n, Divide[1,2], (z)^(2)] == (- 1)^(n)*Divide[(n)!,(2*n)!]*HermiteH[2*n, z]</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
+|-
+| [https://dlmf.nist.gov/13.6.E17 13.6.E17] || [[Item:Q4404|<math>\KummerconfhyperM@{-n}{\tfrac{3}{2}}{z^{2}} = (-1)^{n}\frac{n!}{(2n+1)!2z}\HermitepolyH{2n+1}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperM@{-n}{\tfrac{3}{2}}{z^{2}} = (-1)^{n}\frac{n!}{(2n+1)!2z}\HermitepolyH{2n+1}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerM(- n, (3)/(2), (z)^(2)) = (- 1)^(n)*(factorial(n))/(factorial(2*n + 1)*2*z)*HermiteH(2*n + 1, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric1F1[- n, Divide[3,2], (z)^(2)] == (- 1)^(n)*Divide[(n)!,(2*n + 1)!*2*z]*HermiteH[2*n + 1, z]</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
+|-
+| [https://dlmf.nist.gov/13.6.E18 13.6.E18] || [[Item:Q4405|<math>\KummerconfhyperU@{\tfrac{1}{2}-\tfrac{1}{2}n}{\tfrac{3}{2}}{z^{2}} = 2^{-n}z^{-1}\HermitepolyH{n}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{\tfrac{1}{2}-\tfrac{1}{2}n}{\tfrac{3}{2}}{z^{2}} = 2^{-n}z^{-1}\HermitepolyH{n}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerU((1)/(2)-(1)/(2)*n, (3)/(2), (z)^(2)) = (2)^(- n)* (z)^(- 1)* HermiteH(n, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[Divide[1,2]-Divide[1,2]*n, Divide[3,2], (z)^(2)] == (2)^(- n)* (z)^(- 1)* HermiteH[n, z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .5000000003-2.598076212*I
+Test Values: {z = -1/2+1/2*I*3^(1/2), n = 2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .8660254044+1.500000000*I
+Test Values: {z = -1/2*3^(1/2)-1/2*I, n = 2}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.5000000000000006, -2.5980762113533156]
+Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.8660254037844388, 1.5]
+Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br></div></div>
+|-
+| [https://dlmf.nist.gov/13.6.E19 13.6.E19] || [[Item:Q4406|<math>\KummerconfhyperU@{-n}{\alpha+1}{z} = (-1)^{n}\Pochhammersym{\alpha+1}{n}\KummerconfhyperM@{-n}{\alpha+1}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{-n}{\alpha+1}{z} = (-1)^{n}\Pochhammersym{\alpha+1}{n}\KummerconfhyperM@{-n}{\alpha+1}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerU(- n, alpha + 1, z) = (- 1)^(n)* pochhammer(alpha + 1, n)*KummerM(- n, alpha + 1, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[- n, \[Alpha]+ 1, z] == (- 1)^(n)* Pochhammer[\[Alpha]+ 1, n]*Hypergeometric1F1[- n, \[Alpha]+ 1, z]</syntaxhighlight> || Failure || Failure || Successful [Tested: 63] || Successful [Tested: 63]
+|-
+| [https://dlmf.nist.gov/13.6.E19 13.6.E19] || [[Item:Q4406|<math>(-1)^{n}\Pochhammersym{\alpha+1}{n}\KummerconfhyperM@{-n}{\alpha+1}{z} = (-1)^{n}n!\LaguerrepolyL[\alpha]{n}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(-1)^{n}\Pochhammersym{\alpha+1}{n}\KummerconfhyperM@{-n}{\alpha+1}{z} = (-1)^{n}n!\LaguerrepolyL[\alpha]{n}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(- 1)^(n)* pochhammer(alpha + 1, n)*KummerM(- n, alpha + 1, z) = (- 1)^(n)* factorial(n)*LaguerreL(n, alpha, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(- 1)^(n)* Pochhammer[\[Alpha]+ 1, n]*Hypergeometric1F1[- n, \[Alpha]+ 1, z] == (- 1)^(n)* (n)!*LaguerreL[n, \[Alpha], z]</syntaxhighlight> || Missing Macro Error || Successful || Skip - symbolical successful subtest || Successful [Tested: 63]
+|-
+| [https://dlmf.nist.gov/13.6.E20 13.6.E20] || [[Item:Q4407|<math>\KummerconfhyperU@{-n}{z-n+1}{a} = \Pochhammersym{-z}{n}\KummerconfhyperM@{-n}{z-n+1}{a}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{-n}{z-n+1}{a} = \Pochhammersym{-z}{n}\KummerconfhyperM@{-n}{z-n+1}{a}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerU(- n, z - n + 1, a) = pochhammer(- z, n)*KummerM(- n, z - n + 1, a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[- n, z - n + 1, a] == Pochhammer[- z, n]*Hypergeometric1F1[- n, z - n + 1, a]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 126]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -1.5], Rule[n, 3], Rule[z, 2]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, 1.5], Rule[n, 3], Rule[z, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.6.E20 13.6.E20] || [[Item:Q4407|<math>\Pochhammersym{-z}{n}\KummerconfhyperM@{-n}{z-n+1}{a} = a^{n}\CharlierpolyC{n}@{z}{a}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Pochhammersym{-z}{n}\KummerconfhyperM@{-n}{z-n+1}{a} = a^{n}\CharlierpolyC{n}@{z}{a}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Pochhammer[- z, n]*Hypergeometric1F1[- n, z - n + 1, a] == (a)^(n)* HypergeometricPFQ[{-(n), -(z)}, {}, -Divide[1,a]]</syntaxhighlight> || Missing Macro Error || Missing Macro Error || Skip - symbolical successful subtest || Skip - symbolical successful subtest
+|-
+| [https://dlmf.nist.gov/13.6.E21 13.6.E21] || [[Item:Q4408|<math>\KummerconfhyperU@{a}{b}{z} = z^{-a}\genhyperF{2}{0}@{a,a-b+1}{-}{-z^{-1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{a}{b}{z} = z^{-a}\genhyperF{2}{0}@{a,a-b+1}{-}{-z^{-1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerU(a, b, z) = (z)^(- a)* hypergeom([a , a - b + 1], [-], - (z)^(- 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[a, b, z] == (z)^(- a)* HypergeometricPFQ[{a , a - b + 1}, {-}, - (z)^(- 1)]</syntaxhighlight> || Error || Failure || - || Error
+|-
+| [https://dlmf.nist.gov/13.7.E4 13.7.E4] || [[Item:Q4412|<math>\KummerconfhyperU@{a}{b}{z} = z^{-a}\sum_{s=0}^{n-1}\frac{\Pochhammersym{a}{s}\Pochhammersym{a-b+1}{s}}{s!}(-z)^{-s}+\varepsilon_{n}(z)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{a}{b}{z} = z^{-a}\sum_{s=0}^{n-1}\frac{\Pochhammersym{a}{s}\Pochhammersym{a-b+1}{s}}{s!}(-z)^{-s}+\varepsilon_{n}(z)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerU(a, b, z) = (z)^(- a)* sum((pochhammer(a, s)*pochhammer(a - b + 1, s))/(factorial(s))*(- z)^(- s), s = 0..n - 1)+ varepsilon[n](z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[a, b, z] == (z)^(- a)* Sum[Divide[Pochhammer[a, s]*Pochhammer[a - b + 1, s],(s)!]*(- z)^(- s), {s, 0, n - 1}, GenerateConditions->None]+ Subscript[\[CurlyEpsilon], n][z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.515657870-.5735934827*I
+Test Values: {a = -3/2, b = -3/2, z = 1/2*3^(1/2)+1/2*I, varepsilon[n] = 1/2*3^(1/2)+1/2*I, n = 1, varepsilon = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.515657870-.5735934827*I
+Test Values: {a = -3/2, b = -3/2, z = 1/2*3^(1/2)+1/2*I, varepsilon[n] = 1/2*3^(1/2)+1/2*I, n = 1, varepsilon = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.515657869456145, -0.5735934817267648]
+Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ε, 1], Rule[Subscript[ε, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.515657869456145, -0.5735934817267648]
+Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ε, 2], Rule[Subscript[ε, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.7.E10 13.7.E10] || [[Item:Q4422|<math>\KummerconfhyperU@{a}{b}{z} = z^{-a}\sum_{s=0}^{n-1}\frac{\Pochhammersym{a}{s}\Pochhammersym{a-b+1}{s}}{s!}(-z)^{-s}+R_{n}(a,b,z)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperU@{a}{b}{z} = z^{-a}\sum_{s=0}^{n-1}\frac{\Pochhammersym{a}{s}\Pochhammersym{a-b+1}{s}}{s!}(-z)^{-s}+R_{n}(a,b,z)</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{(a-b+1)} > 0</math> || <syntaxhighlight lang=mathematica>KummerU(a, b, z) = (z)^(- a)* sum((pochhammer(a, s)*pochhammer(a - b + 1, s))/(factorial(s))*(- z)^(- s)+(((- 1)^(n)* 2*Pi*(z)^(a - b))/(GAMMA(a)*GAMMA(a - b + 1))*(sum((pochhammer(1 - a, s)*pochhammer(b - a, s))/(factorial(s))*(- z)^(- s)* G[n + 2*a - b - s](z), s = 0..m - 1)+ pochhammer(1 - a, m)*pochhammer(b - a, m)*R[m , n](a , b , z))), s = 0..n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricU[a, b, z] == (z)^(- a)* Sum[Divide[Pochhammer[a, s]*Pochhammer[a - b + 1, s],(s)!]*(- z)^(- s)+(Divide[(- 1)^(n)* 2*Pi*(z)^(a - b),Gamma[a]*Gamma[a - b + 1]]*(Sum[Divide[Pochhammer[1 - a, s]*Pochhammer[b - a, s],(s)!]*(- z)^(- s)* Subscript[G, n + 2*a - b - s][z], {s, 0, m - 1}, GenerateConditions->None]+ Pochhammer[1 - a, m]*Pochhammer[b - a, m]*Subscript[R, m , n][a , b , z])), {s, 0, n - 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .1211969897-.2855680854e-1*I+(-.7071067811+.7071067809*I)*(2.023326709-.5908179514*I+(.8862269255-1.534990063*I)*(1.500000000, -1.500000000, .8660254040+.5000000000*I))
+Test Values: {a = 3/2, b = -3/2, z = 1/2*3^(1/2)+1/2*I, G[n+2*a-b-s] = 1/2*3^(1/2)+1/2*I, R[m,n] = 1/2*3^(1/2)+1/2*I, m = 1, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .1211969897-.2855680854e-1*I+(-.7071067811+.7071067809*I)*(-6.242805838+4.181635900*I+(-1.772453851+3.069980127*I)*(1.500000000, -1.500000000, .8660254040+.5000000000*I))
+Test Values: {a = 3/2, b = -3/2, z = 1/2*3^(1/2)+1/2*I, G[n+2*a-b-s] = 1/2*3^(1/2)+1/2*I, R[m,n] = 1/2*3^(1/2)+1/2*I, m = 1, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || Error
+|-
+| [https://dlmf.nist.gov/13.8.E3 13.8.E3] || [[Item:Q4428|<math>\left(e^{t}-1\right)^{a-1}\exp@{t+z(1-e^{-t})} = \sum_{s=0}^{\infty}q_{s}(z,a)t^{s+a-1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(e^{t}-1\right)^{a-1}\exp@{t+z(1-e^{-t})} = \sum_{s=0}^{\infty}q_{s}(z,a)t^{s+a-1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(exp(t)- 1)^(a - 1)* exp(t + z*(1 - exp(- t))) = sum(q[s](z , a)* (t)^(s + a - 1), s = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(Exp[t]- 1)^(a - 1)* Exp[t + z*(1 - Exp[- t])] == Sum[Subscript[q, s][z , a]* (t)^(s + a - 1), {s, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || Error
+|-
+| [https://dlmf.nist.gov/13.8#Ex1 13.8#Ex1] || [[Item:Q4440|<math>p_{k}(z) = \sum_{s=0}^{k}\binom{k}{s}\Pochhammersym{1-b+s}{k-s}z^{s}c_{k+s}(z)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>p_{k}(z) = \sum_{s=0}^{k}\binom{k}{s}\Pochhammersym{1-b+s}{k-s}z^{s}c_{k+s}(z)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>p[k](z) = sum(binomial(k,s)*pochhammer(1 - b + s, k - s)*(z)^(s)* c[k + s](z), s = 0..k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[p, k][z] == Sum[Binomial[k,s]*Pochhammer[1 - b + s, k - s]*(z)^(s)* Subscript[c, k + s][z], {s, 0, k}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.7500000009-2.299038107*I
+Test Values: {b = -3/2, z = 1/2*3^(1/2)+1/2*I, c[k+s] = 1/2*3^(1/2)+1/2*I, p[k] = 1/2*3^(1/2)+1/2*I, k = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -3.375000005-14.57772229*I
+Test Values: {b = -3/2, z = 1/2*3^(1/2)+1/2*I, c[k+s] = 1/2*3^(1/2)+1/2*I, p[k] = 1/2*3^(1/2)+1/2*I, k = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/13.8#Ex2 13.8#Ex2] || [[Item:Q4441|<math>q_{k}(z) = \sum_{s=0}^{k}\binom{k}{s}\Pochhammersym{2-b+s}{k-s}z^{s}c_{k+s+1}(z)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>q_{k}(z) = \sum_{s=0}^{k}\binom{k}{s}\Pochhammersym{2-b+s}{k-s}z^{s}c_{k+s+1}(z)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>q[k](z) = sum(binomial(k,s)*pochhammer(2 - b + s, k - s)*(z)^(s)* c[k + s + 1](z), s = 0..k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[q, k][z] == Sum[Binomial[k,s]*Pochhammer[2 - b + s, k - s]*(z)^(s)* Subscript[c, k + s + 1][z], {s, 0, k}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.250000001-3.165063511*I
+Test Values: {b = -3/2, z = 1/2*3^(1/2)+1/2*I, c[k+s+1] = 1/2*3^(1/2)+1/2*I, q[k] = 1/2*3^(1/2)+1/2*I, k = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -6.875000009-22.63990012*I
+Test Values: {b = -3/2, z = 1/2*3^(1/2)+1/2*I, c[k+s+1] = 1/2*3^(1/2)+1/2*I, q[k] = 1/2*3^(1/2)+1/2*I, k = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/13.8.E16 13.8.E16] || [[Item:Q4442|<math>(k+1)c_{k+1}(z)+\sum_{s=0}^{k}\left(\frac{b\BernoullinumberB{s+1}}{(s+1)!}+\frac{z(s+1)\BernoullinumberB{s+2}}{(s+2)!}\right)c_{k-s}(z) = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(k+1)c_{k+1}(z)+\sum_{s=0}^{k}\left(\frac{b\BernoullinumberB{s+1}}{(s+1)!}+\frac{z(s+1)\BernoullinumberB{s+2}}{(s+2)!}\right)c_{k-s}(z) = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(k + 1)*c[k + 1](z)+ sum(((b*bernoulli(s + 1))/(factorial(s + 1))+(z*(s + 1)*bernoulli(s + 2))/(factorial(s + 2)))*c[k - s](z), s = 0..k) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>(k + 1)*Subscript[c, k + 1][z]+ Sum[(Divide[b*BernoulliB[s + 1],(s + 1)!]+Divide[z*(s + 1)*BernoulliB[s + 2],(s + 2)!])*Subscript[c, k - s][z], {s, 0, k}, GenerateConditions->None] == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 2.313541668+4.086338379*I
+Test Values: {b = -3/2, z = 1/2*3^(1/2)+1/2*I, c[1+k] = 1/2*3^(1/2)+1/2*I, c[k-s] = 1/2*3^(1/2)+1/2*I, k = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.377763239+3.777643283*I
+Test Values: {b = -3/2, z = 1/2*3^(1/2)+1/2*I, c[1+k] = 1/2*3^(1/2)+1/2*I, c[k-s] = -1/2+1/2*I*3^(1/2), k = 3}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/13.8#Ex3 13.8#Ex3] || [[Item:Q4443|<math>\pderiv{f}{t} = \left(b\left(\frac{1}{t}-\frac{1}{e^{t}-1}\right)-z\left(\frac{1}{t^{2}}-\frac{e^{t}}{\left(e^{t}-1\right)^{2}}\right)\right)f</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\pderiv{f}{t} = \left(b\left(\frac{1}{t}-\frac{1}{e^{t}-1}\right)-z\left(\frac{1}{t^{2}}-\frac{e^{t}}{\left(e^{t}-1\right)^{2}}\right)\right)f</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(f, t) = (b*((1)/(t)-(1)/(exp(t)- 1))- z*((1)/((t)^(2))-(exp(t))/((exp(t)- 1)^(2))))*f</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[f, t] == (b*(Divide[1,t]-Divide[1,Exp[t]- 1])- z*(Divide[1,(t)^(2)]-Divide[Exp[t],(Exp[t]- 1)^(2)]))*f</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .8434854075+.5301342049*I
+Test Values: {b = -3/2, f = 1/2*3^(1/2)+1/2*I, t = -3/2, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .7413969054+.5027796732*I
+Test Values: {b = -3/2, f = 1/2*3^(1/2)+1/2*I, t = -3/2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.8434854065788572, 0.5301342044541701]
+Test Values: {Rule[b, -1.5], Rule[f, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.7413969045334019, 0.5027796727745873]
+Test Values: {Rule[b, -1.5], Rule[f, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.9.E1 13.9.E1] || [[Item:Q4444|<math>p(a,b) = \ceiling{-a}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>p(a,b) = \ceiling{-a}</syntaxhighlight> || <math>a < 0, b \geq 0</math> || <syntaxhighlight lang=mathematica>p(a , b) = ceil(- a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>p[a , b] == Ceiling[- a]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [90 / 90]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: (.8660254040+.5000000000*I)*(-1.500000000, 1.500000000)-2.
+Test Values: {a = -3/2, b = 3/2, p = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: (-.5000000000+.8660254040*I)*(-1.500000000, 1.500000000)-2.
+Test Values: {a = -3/2, b = 3/2, p = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Error
+|- style="background: #dfe6e9;"
+| [https://dlmf.nist.gov/13.9.E2 13.9.E2] || [[Item:Q4445|<math>p(a,b) = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>p(a,b) = 0</syntaxhighlight> || <math>a \geq 0, b \geq 0</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">p(a , b) = 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">p[a , b] == 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
+|- style="background: #dfe6e9;"
+| [https://dlmf.nist.gov/13.9.E3 13.9.E3] || [[Item:Q4446|<math>p(a,b) = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>p(a,b) = 1</syntaxhighlight> || <math>a \geq 0, -1 < b, b < 0</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">p(a , b) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">p[a , b] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
+|-
+| [https://dlmf.nist.gov/13.9.E4 13.9.E4] || [[Item:Q4447|<math>p(a,b) = \floor{-\tfrac{1}{2}b}-\floor{-\tfrac{1}{2}(b+1)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>p(a,b) = \floor{-\tfrac{1}{2}b}-\floor{-\tfrac{1}{2}(b+1)}</syntaxhighlight> || <math>a \geq 0, b \leq -1</math> || <syntaxhighlight lang=mathematica>p(a , b) = floor(-(1)/(2)*b)- floor(-(1)/(2)*(b + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>p[a , b] == Floor[-Divide[1,2]*b]- Floor[-Divide[1,2]*(b + 1)]</syntaxhighlight> || Failure || Failure || Error || Error
+|-
+| [https://dlmf.nist.gov/13.9.E5 13.9.E5] || [[Item:Q4448|<math>p(a,b) = \ceiling{-a}-\ceiling{-b}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>p(a,b) = \ceiling{-a}-\ceiling{-b}</syntaxhighlight> || <math>\ceiling{-a} \geq \ceiling{-b}, a < 0, b < 0</math> || <syntaxhighlight lang=mathematica>p(a , b) = ceil(- a)- ceil(- b)</syntaxhighlight> || <syntaxhighlight lang=mathematica>p[a , b] == Ceiling[- a]- Ceiling[- b]</syntaxhighlight> || Failure || Failure || Error || Error
+|-
+| [https://dlmf.nist.gov/13.9.E6 13.9.E6] || [[Item:Q4449|<math>p(a,b) = \floor{\tfrac{1}{2}\left(\ceiling{-b}-\ceiling{-a}+1\right)}-\floor{\tfrac{1}{2}\left(\ceiling{-b}-\ceiling{-a}\right)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>p(a,b) = \floor{\tfrac{1}{2}\left(\ceiling{-b}-\ceiling{-a}+1\right)}-\floor{\tfrac{1}{2}\left(\ceiling{-b}-\ceiling{-a}\right)}</syntaxhighlight> || <math>\ceiling{-b} > \ceiling{-a}, \ceiling{-a} > 0</math> || <syntaxhighlight lang=mathematica>p(a , b) = floor((1)/(2)*(ceil(- b)- ceil(- a)+ 1))- floor((1)/(2)*(ceil(- b)- ceil(- a)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>p[a , b] == Floor[Divide[1,2]*(Ceiling[- b]- Ceiling[- a]+ 1)]- Floor[Divide[1,2]*(Ceiling[- b]- Ceiling[- a])]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 20]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: (.8660254040+.5000000000*I)*(-.5000000000, -1.500000000)-1.
+Test Values: {a = -1/2, b = -3/2, p = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: (-.5000000000+.8660254040*I)*(-.5000000000, -1.500000000)-1.
+Test Values: {a = -1/2, b = -3/2, p = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Error
+|-
+| [https://dlmf.nist.gov/13.9.E11 13.9.E11] || [[Item:Q4454|<math>T(a,b) = \floor{-a}+1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>T(a,b) = \floor{-a}+1</syntaxhighlight> || <math>a < 0, \EulerGamma@{a}\EulerGamma@{a-b+1} > 0</math> || <syntaxhighlight lang=mathematica>T(a , b) = floor(- a)+ 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>T[a , b] == Floor[- a]+ 1</syntaxhighlight> || Failure || Failure || Error || Error
+|-
+| [https://dlmf.nist.gov/13.9.E12 13.9.E12] || [[Item:Q4455|<math>T(a,b) = \floor{-a}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>T(a,b) = \floor{-a}</syntaxhighlight> || <math>a < 0, \EulerGamma@{a}\EulerGamma@{a-b+1} < 0</math> || <syntaxhighlight lang=mathematica>T(a , b) = floor(- a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>T[a , b] == Floor[- a]</syntaxhighlight> || Failure || Failure || Error || Error
+|- style="background: #dfe6e9;"
+| [https://dlmf.nist.gov/13.9.E13 13.9.E13] || [[Item:Q4456|<math>T(a,b) = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>T(a,b) = 0</syntaxhighlight> || <math>a > 0</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">T(a , b) = 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">T[a , b] == 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
+|-
+| [https://dlmf.nist.gov/13.9.E14 13.9.E14] || [[Item:Q4457|<math>P(a,b) = \ceiling{b-a-1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>P(a,b) = \ceiling{b-a-1}</syntaxhighlight> || <math>a+1 < b</math> || <syntaxhighlight lang=mathematica>P(a , b) = ceil(b - a - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>P[a , b] == Ceiling[b - a - 1]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [100 / 100]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: (.8660254040+.5000000000*I)*(-1.500000000, 1.500000000)-2.
+Test Values: {P = 1/2*3^(1/2)+1/2*I, a = -3/2, b = 3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: (.8660254040+.5000000000*I)*(-1.500000000, .5000000000)-1.
+Test Values: {P = 1/2*3^(1/2)+1/2*I, a = -3/2, b = 1/2}</syntaxhighlight><br>... skip entries to safe data</div></div> || Error
+|- style="background: #dfe6e9;"
+| [https://dlmf.nist.gov/13.9.E15 13.9.E15] || [[Item:Q4458|<math>P(a,b) = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>P(a,b) = 0</syntaxhighlight> || <math>a+1 \geq b</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">P(a , b) = 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">P[a , b] == 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
+|-
+| [https://dlmf.nist.gov/13.10.E1 13.10.E1] || [[Item:Q4460|<math>\int\OlverconfhyperM@{a}{b}{z}\diff{z} = \frac{1}{a-1}\OlverconfhyperM@{a-1}{b-1}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\OlverconfhyperM@{a}{b}{z}\diff{z} = \frac{1}{a-1}\OlverconfhyperM@{a-1}{b-1}{z}</syntaxhighlight> || <math>\realpart@@{(b+s)} > 0, \realpart@@{((b-1)+s)} > 0</math> || <syntaxhighlight lang=mathematica>int(KummerM(a, b, z)/GAMMA(b), z) = (1)/(a - 1)*KummerM(a - 1, b - 1, z)/GAMMA(b - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Hypergeometric1F1Regularized[a, b, z], z, GenerateConditions->None] == Divide[1,a - 1]*Hypergeometric1F1Regularized[a - 1, b - 1, z]</syntaxhighlight> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [252 / 252]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.4231421876608173, 0.0]
+Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.42314218766081735, 0.0]
+Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.10.E2 13.10.E2] || [[Item:Q4461|<math>\int\KummerconfhyperU@{a}{b}{z}\diff{z} = -\frac{1}{a-1}\KummerconfhyperU@{a-1}{b-1}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\KummerconfhyperU@{a}{b}{z}\diff{z} = -\frac{1}{a-1}\KummerconfhyperU@{a-1}{b-1}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(KummerU(a, b, z), z) = -(1)/(a - 1)*KummerU(a - 1, b - 1, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[HypergeometricU[a, b, z], z, GenerateConditions->None] == -Divide[1,a - 1]*HypergeometricU[a - 1, b - 1, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 252]
+|-
+| [https://dlmf.nist.gov/13.10.E3 13.10.E3] || [[Item:Q4462|<math>\int_{0}^{\infty}e^{-zt}t^{b-1}\OlverconfhyperM@{a}{c}{kt}\diff{t} = \EulerGamma@{b}z^{-b}\genhyperOlverF{2}{1}@{a,b}{c}{\ifrac{k}{z}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-zt}t^{b-1}\OlverconfhyperM@{a}{c}{kt}\diff{t} = \EulerGamma@{b}z^{-b}\genhyperOlverF{2}{1}@{a,b}{c}{\ifrac{k}{z}}</syntaxhighlight> || <math>\realpart@@{b} > 0, \realpart@@{z} > \max\left(\realpart@@{k}, \realpart@@{(c+s)} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(- z*t)*(t)^(b - 1)* KummerM(a, c, k*t)/GAMMA(c), t = 0..infinity) = GAMMA(b)*(z)^(- b)* hypergeom([a , b], [c], (k)/(z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[- z*t]*(t)^(b - 1)* Hypergeometric1F1Regularized[a, c, k*t], {t, 0, Infinity}, GenerateConditions->None] == Gamma[b]*(z)^(- b)* HypergeometricPFQRegularized[{a , b}, {c}, Divide[k,z]]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(undefined)+Float(undefined)*I
+Test Values: {a = -3/2, b = -3/2, c = -3/2, z = 1/2*3^(1/2)+1/2*I, k = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(undefined)+Float(undefined)*I
+Test Values: {a = -3/2, b = -3/2, c = -3/2, z = 1/2*3^(1/2)+1/2*I, k = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/13.10.E4 13.10.E4] || [[Item:Q4463|<math>\int_{0}^{\infty}e^{-zt}t^{b-1}\OlverconfhyperM@{a}{b}{t}\diff{t} = z^{-b}\left(1-\frac{1}{z}\right)^{-a}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-zt}t^{b-1}\OlverconfhyperM@{a}{b}{t}\diff{t} = z^{-b}\left(1-\frac{1}{z}\right)^{-a}</syntaxhighlight> || <math>\realpart@@{b} > 0, \realpart@@{z} > 1, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(- z*t)*(t)^(b - 1)* KummerM(a, b, t)/GAMMA(b), t = 0..infinity) = (z)^(- b)*(1 -(1)/(z))^(- a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[- z*t]*(t)^(b - 1)* Hypergeometric1F1Regularized[a, b, t], {t, 0, Infinity}, GenerateConditions->None] == (z)^(- b)*(1 -Divide[1,z])^(- a)</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [24 / 36]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.2095131204
+Test Values: {a = -3/2, b = 3/2, z = 3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.2500000000
+Test Values: {a = -3/2, b = 3/2, z = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/13.10.E5 13.10.E5] || [[Item:Q4464|<math>\int_{0}^{\infty}e^{-t}t^{b-1}\OlverconfhyperM@{a}{c}{t}\diff{t} = \frac{\EulerGamma@{b}\EulerGamma@{c-a-b}}{\EulerGamma@{c-a}\EulerGamma@{c-b}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-t}t^{b-1}\OlverconfhyperM@{a}{c}{t}\diff{t} = \frac{\EulerGamma@{b}\EulerGamma@{c-a-b}}{\EulerGamma@{c-a}\EulerGamma@{c-b}}</syntaxhighlight> || <math>\realpart@{c-a} > \realpart@@{b}, \realpart@@{b} > 0, \realpart@@{(c-a-b)} > 0, \realpart@@{(c-a)} > 0, \realpart@@{(c-b)} > 0, \realpart@@{(c+s)} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(- t)*(t)^(b - 1)* KummerM(a, c, t)/GAMMA(c), t = 0..infinity) = (GAMMA(b)*GAMMA(c - a - b))/(GAMMA(c - a)*GAMMA(c - b))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[- t]*(t)^(b - 1)* Hypergeometric1F1Regularized[a, c, t], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[b]*Gamma[c - a - b],Gamma[c - a]*Gamma[c - b]]</syntaxhighlight> || Successful || Aborted || - || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/13.10.E6 13.10.E6] || [[Item:Q4465|<math>\int_{0}^{\infty}e^{-zt-t^{2}}t^{2b-2}\OlverconfhyperM@{a}{b}{t^{2}}\diff{t} = \tfrac{1}{2}\pi^{-\frac{1}{2}}\EulerGamma@{b-\tfrac{1}{2}}\KummerconfhyperU@{b-\tfrac{1}{2}}{a+\tfrac{1}{2}}{\tfrac{1}{4}z^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-zt-t^{2}}t^{2b-2}\OlverconfhyperM@{a}{b}{t^{2}}\diff{t} = \tfrac{1}{2}\pi^{-\frac{1}{2}}\EulerGamma@{b-\tfrac{1}{2}}\KummerconfhyperU@{b-\tfrac{1}{2}}{a+\tfrac{1}{2}}{\tfrac{1}{4}z^{2}}</syntaxhighlight> || <math>\realpart@@{b} > \tfrac{1}{2}, \realpart@@{z} > 0, \realpart@@{(b-\tfrac{1}{2})} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(- z*t - (t)^(2))*(t)^(2*b - 2)* KummerM(a, b, (t)^(2))/GAMMA(b), t = 0..infinity) = (1)/(2)*(Pi)^(-(1)/(2))* GAMMA(b -(1)/(2))*KummerU(b -(1)/(2), a +(1)/(2), (1)/(4)*(z)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[- z*t - (t)^(2)]*(t)^(2*b - 2)* Hypergeometric1F1Regularized[a, b, (t)^(2)], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,2]*(Pi)^(-Divide[1,2])* Gamma[b -Divide[1,2]]*HypergeometricU[b -Divide[1,2], a +Divide[1,2], Divide[1,4]*(z)^(2)]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/13.10.E7 13.10.E7] || [[Item:Q4466|<math>\int_{0}^{\infty}e^{-zt}t^{b-1}\KummerconfhyperU@{a}{c}{t}\diff{t} = \EulerGamma@{b}\EulerGamma@{b-c+1}\*z^{-b}\genhyperOlverF{2}{1}@{a,b}{a+b-c+1}{1-\frac{1}{z}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-zt}t^{b-1}\KummerconfhyperU@{a}{c}{t}\diff{t} = \EulerGamma@{b}\EulerGamma@{b-c+1}\*z^{-b}\genhyperOlverF{2}{1}@{a,b}{a+b-c+1}{1-\frac{1}{z}}</syntaxhighlight> || <math>\realpart@@{b} > \max\left(\realpart@@{c-1}, \realpart@@{z} > 0, \realpart@@{b} > 0, \realpart@@{(b-c+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(- z*t)*(t)^(b - 1)* KummerU(a, c, t), t = 0..infinity) = GAMMA(b)*GAMMA(b - c + 1)* (z)^(- b)* hypergeom([a , b], [a + b - c + 1], 1 -(1)/(z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[- z*t]*(t)^(b - 1)* HypergeometricU[a, c, t], {t, 0, Infinity}, GenerateConditions->None] == Gamma[b]*Gamma[b - c + 1]* (z)^(- b)* HypergeometricPFQRegularized[{a , b}, {a + b - c + 1}, 1 -Divide[1,z]]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/13.10.E8 13.10.E8] || [[Item:Q4467|<math>\frac{1}{2\pi\iunit}\int_{-\infty}^{(0+)}e^{tz}t^{-a}\OlverconfhyperM@{a}{b}{\ifrac{y}{t}}\diff{t} = \frac{1}{\EulerGamma@{a}}z^{\frac{1}{2}(2a-b-1)}y^{\frac{1}{2}(1-b)}\modBesselI{b-1}@{2\sqrt{zy}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{2\pi\iunit}\int_{-\infty}^{(0+)}e^{tz}t^{-a}\OlverconfhyperM@{a}{b}{\ifrac{y}{t}}\diff{t} = \frac{1}{\EulerGamma@{a}}z^{\frac{1}{2}(2a-b-1)}y^{\frac{1}{2}(1-b)}\modBesselI{b-1}@{2\sqrt{zy}}</syntaxhighlight> || <math>\realpart@@{z} > 0, \realpart@@{a} > 0, \realpart@@{(b+s)} > 0, \realpart@@{((b-1)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(2*Pi*I)*int(exp(t*(x + y*I))*(t)^(- a)* KummerM(a, b, (y)/(t))/GAMMA(b), t = - infinity..(0 +)) = (1)/(GAMMA(a))*(x + y*I)^((1)/(2)*(2*a - b - 1))* (y)^((1)/(2)*(1 - b))* BesselI(b - 1, 2*sqrt((x + y*I)*y))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,2*Pi*I]*Integrate[Exp[t*(x + y*I)]*(t)^(- a)* Hypergeometric1F1Regularized[a, b, Divide[y,t]], {t, - Infinity, (0 +)}, GenerateConditions->None] == Divide[1,Gamma[a]]*(x + y*I)^(Divide[1,2]*(2*a - b - 1))* (y)^(Divide[1,2]*(1 - b))* BesselI[b - 1, 2*Sqrt[(x + y*I)*y]]</syntaxhighlight> || Error || Failure || - || Error
+|-
+| [https://dlmf.nist.gov/13.10.E9 13.10.E9] || [[Item:Q4468|<math>\frac{1}{2\pi\iunit}\int_{-\infty}^{(0+)}e^{tz}t^{-a}\KummerconfhyperU@{a}{b}{\ifrac{y}{t}}\diff{t} = \frac{2z^{\frac{1}{2}(2a-b-1)}y^{\frac{1}{2}(1-b)}}{\EulerGamma@{a}\EulerGamma@{a-b+1}}\modBesselK{b-1}@{2\sqrt{zy}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{2\pi\iunit}\int_{-\infty}^{(0+)}e^{tz}t^{-a}\KummerconfhyperU@{a}{b}{\ifrac{y}{t}}\diff{t} = \frac{2z^{\frac{1}{2}(2a-b-1)}y^{\frac{1}{2}(1-b)}}{\EulerGamma@{a}\EulerGamma@{a-b+1}}\modBesselK{b-1}@{2\sqrt{zy}}</syntaxhighlight> || <math>\realpart@@{z} > 0, \realpart@@{a} > 0, \realpart@@{(a-b+1)} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(2*Pi*I)*int(exp(t*(x + y*I))*(t)^(- a)* KummerU(a, b, (y)/(t)), t = - infinity..(0 +)) = (2*(x + y*I)^((1)/(2)*(2*a - b - 1))* (y)^((1)/(2)*(1 - b)))/(GAMMA(a)*GAMMA(a - b + 1))*BesselK(b - 1, 2*sqrt((x + y*I)*y))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,2*Pi*I]*Integrate[Exp[t*(x + y*I)]*(t)^(- a)* HypergeometricU[a, b, Divide[y,t]], {t, - Infinity, (0 +)}, GenerateConditions->None] == Divide[2*(x + y*I)^(Divide[1,2]*(2*a - b - 1))* (y)^(Divide[1,2]*(1 - b)),Gamma[a]*Gamma[a - b + 1]]*BesselK[b - 1, 2*Sqrt[(x + y*I)*y]]</syntaxhighlight> || Error || Failure || - || Error
+|-
+| [https://dlmf.nist.gov/13.10.E10 13.10.E10] || [[Item:Q4469|<math>\int_{0}^{\infty}t^{\lambda-1}\OlverconfhyperM@{a}{b}{-t}\diff{t} = \frac{\EulerGamma@{\lambda}\EulerGamma@{a-\lambda}}{\EulerGamma@{a}\EulerGamma@{b-\lambda}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}t^{\lambda-1}\OlverconfhyperM@{a}{b}{-t}\diff{t} = \frac{\EulerGamma@{\lambda}\EulerGamma@{a-\lambda}}{\EulerGamma@{a}\EulerGamma@{b-\lambda}}</syntaxhighlight> || <math>0 < \realpart@@{\lambda}, \realpart@@{\lambda} < \realpart@@{a}, \realpart@@{(\lambda)} > 0, \realpart@@{(a-\lambda)} > 0, \realpart@@{a} > 0, \realpart@@{(b-\lambda)} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>int((t)^(lambda - 1)* KummerM(a, b, - t)/GAMMA(b), t = 0..infinity) = (GAMMA(lambda)*GAMMA(a - lambda))/(GAMMA(a)*GAMMA(b - lambda))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(t)^(\[Lambda]- 1)* Hypergeometric1F1Regularized[a, b, - t], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[\[Lambda]]*Gamma[a - \[Lambda]],Gamma[a]*Gamma[b - \[Lambda]]]</syntaxhighlight> || Successful || Aborted || - || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/13.10.E11 13.10.E11] || [[Item:Q4470|<math>\int_{0}^{\infty}t^{\lambda-1}\KummerconfhyperU@{a}{b}{t}\diff{t} = \frac{\EulerGamma@{\lambda}\EulerGamma@{a-\lambda}\EulerGamma@{\lambda-b+1}}{\EulerGamma@{a}\EulerGamma@{a-b+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}t^{\lambda-1}\KummerconfhyperU@{a}{b}{t}\diff{t} = \frac{\EulerGamma@{\lambda}\EulerGamma@{a-\lambda}\EulerGamma@{\lambda-b+1}}{\EulerGamma@{a}\EulerGamma@{a-b+1}}</syntaxhighlight> || <math>\max\left(\realpart@@{b-1} < \realpart@@{\lambda}, 0\right) < \realpart@@{\lambda}, \realpart@@{\lambda} < \realpart@@{a}, \realpart@@{(\lambda)} > 0, \realpart@@{(a-\lambda)} > 0, \realpart@@{(\lambda-b+1)} > 0, \realpart@@{a} > 0, \realpart@@{(a-b+1)} > 0</math> || <syntaxhighlight lang=mathematica>int((t)^(lambda - 1)* KummerU(a, b, t), t = 0..infinity) = (GAMMA(lambda)*GAMMA(a - lambda)*GAMMA(lambda - b + 1))/(GAMMA(a)*GAMMA(a - b + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(t)^(\[Lambda]- 1)* HypergeometricU[a, b, t], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[\[Lambda]]*Gamma[a - \[Lambda]]*Gamma[\[Lambda]- b + 1],Gamma[a]*Gamma[a - b + 1]]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 300]
+|-
+| [https://dlmf.nist.gov/13.10.E12 13.10.E12] || [[Item:Q4471|<math>\int_{0}^{\infty}\cos@{2xt}\OlverconfhyperM@{a}{b}{-t^{2}}\diff{t} = \frac{\sqrt{\pi}}{2\EulerGamma@{a}}x^{2a-1}e^{-x^{2}}\KummerconfhyperU@{b-\tfrac{1}{2}}{a+\tfrac{1}{2}}{x^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}\cos@{2xt}\OlverconfhyperM@{a}{b}{-t^{2}}\diff{t} = \frac{\sqrt{\pi}}{2\EulerGamma@{a}}x^{2a-1}e^{-x^{2}}\KummerconfhyperU@{b-\tfrac{1}{2}}{a+\tfrac{1}{2}}{x^{2}}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>int(cos(2*x*t)*KummerM(a, b, - (t)^(2))/GAMMA(b), t = 0..infinity) = (sqrt(Pi))/(2*GAMMA(a))*(x)^(2*a - 1)* exp(- (x)^(2))*KummerU(b -(1)/(2), a +(1)/(2), (x)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Cos[2*x*t]*Hypergeometric1F1Regularized[a, b, - (t)^(2)], {t, 0, Infinity}, GenerateConditions->None] == Divide[Sqrt[Pi],2*Gamma[a]]*(x)^(2*a - 1)* Exp[- (x)^(2)]*HypergeometricU[b -Divide[1,2], a +Divide[1,2], (x)^(2)]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [51 / 54]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(undefined)+Float(undefined)*I
+Test Values: {a = 3/2, b = -3/2, x = 3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(undefined)+Float(undefined)*I
+Test Values: {a = 3/2, b = -3/2, x = 1/2}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/13.10.E13 13.10.E13] || [[Item:Q4472|<math>\int_{0}^{\infty}e^{-t}t^{b-1-\frac{1}{2}\nu}\OlverconfhyperM@{a}{b}{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = x^{-a+\frac{1}{2}\nu}e^{-x}\OlverconfhyperM@{\nu-b+1}{\nu-a+1}{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-t}t^{b-1-\frac{1}{2}\nu}\OlverconfhyperM@{a}{b}{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = x^{-a+\frac{1}{2}\nu}e^{-x}\OlverconfhyperM@{\nu-b+1}{\nu-a+1}{x}</syntaxhighlight> || <math>x > 0, 2\realpart@@{a} < \realpart@@{\nu}+\tfrac{5}{2}, \realpart@@{b} > 0, \realpart@@{(\nu+k+1)} > 0, \realpart@@{(b+s)} > 0, \realpart@@{((\nu-a+1)+s)} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(- t)*(t)^(b - 1 -(1)/(2)*nu)* KummerM(a, b, t)/GAMMA(b)*BesselJ(nu, 2*sqrt(x*t)), t = 0..infinity) = (x)^(- a +(1)/(2)*nu)* exp(- x)*KummerM(nu - b + 1, nu - a + 1, x)/GAMMA(nu - a + 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[- t]*(t)^(b - 1 -Divide[1,2]*\[Nu])* Hypergeometric1F1Regularized[a, b, t]*BesselJ[\[Nu], 2*Sqrt[x*t]], {t, 0, Infinity}, GenerateConditions->None] == (x)^(- a +Divide[1,2]*\[Nu])* Exp[- x]*Hypergeometric1F1Regularized[\[Nu]- b + 1, \[Nu]- a + 1, x]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/13.10.E14 13.10.E14] || [[Item:Q4473|<math>\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}\nu}\OlverconfhyperM@{a}{b}{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = \frac{x^{\frac{1}{2}\nu}e^{-x}}{\EulerGamma@{b-a}}\KummerconfhyperU@{a}{a-b+\nu+2}{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}\nu}\OlverconfhyperM@{a}{b}{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = \frac{x^{\frac{1}{2}\nu}e^{-x}}{\EulerGamma@{b-a}}\KummerconfhyperU@{a}{a-b+\nu+2}{x}</syntaxhighlight> || <math>x > 0, -1 < \realpart@@{\nu}, \realpart@@{\nu} < 2\realpart@{b-a}-\tfrac{1}{2}, \realpart@@{(\nu+k+1)} > 0, \realpart@@{(b-a)} > 0, \realpart@@{(b+s)} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(- t)*(t)^((1)/(2)*nu)* KummerM(a, b, t)/GAMMA(b)*BesselJ(nu, 2*sqrt(x*t)), t = 0..infinity) = ((x)^((1)/(2)*nu)* exp(- x))/(GAMMA(b - a))*KummerU(a, a - b + nu + 2, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[- t]*(t)^(Divide[1,2]*\[Nu])* Hypergeometric1F1Regularized[a, b, t]*BesselJ[\[Nu], 2*Sqrt[x*t]], {t, 0, Infinity}, GenerateConditions->None] == Divide[(x)^(Divide[1,2]*\[Nu])* Exp[- x],Gamma[b - a]]*HypergeometricU[a, a - b + \[Nu]+ 2, x]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/13.10.E15 13.10.E15] || [[Item:Q4474|<math>\int_{0}^{\infty}t^{\frac{1}{2}\nu}\KummerconfhyperU@{a}{b}{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = \frac{\EulerGamma@{\nu-b+2}}{\EulerGamma@{a}}x^{\frac{1}{2}\nu}\KummerconfhyperU@{\nu-b+2}{\nu-a+2}{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}t^{\frac{1}{2}\nu}\KummerconfhyperU@{a}{b}{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = \frac{\EulerGamma@{\nu-b+2}}{\EulerGamma@{a}}x^{\frac{1}{2}\nu}\KummerconfhyperU@{\nu-b+2}{\nu-a+2}{x}</syntaxhighlight> || <math>x > 0, \max\left(\realpart@@{b-2} < \realpart@@{\nu}, -1\right) < \realpart@@{\nu}, \realpart@@{\nu} < 2\realpart@@{a}+\tfrac{1}{2}, \realpart@@{(\nu+k+1)} > 0, \realpart@@{(\nu-b+2)} > 0, \realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>int((t)^((1)/(2)*nu)* KummerU(a, b, t)*BesselJ(nu, 2*sqrt(x*t)), t = 0..infinity) = (GAMMA(nu - b + 2))/(GAMMA(a))*(x)^((1)/(2)*nu)* KummerU(nu - b + 2, nu - a + 2, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(t)^(Divide[1,2]*\[Nu])* HypergeometricU[a, b, t]*BesselJ[\[Nu], 2*Sqrt[x*t]], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[\[Nu]- b + 2],Gamma[a]]*(x)^(Divide[1,2]*\[Nu])* HypergeometricU[\[Nu]- b + 2, \[Nu]- a + 2, x]</syntaxhighlight> || Successful || Aborted || - || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/13.10.E16 13.10.E16] || [[Item:Q4475|<math>\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}\nu}\KummerconfhyperU@{a}{b}{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = \EulerGamma@{\nu-b+2}x^{\frac{1}{2}\nu}e^{-x}\OlverconfhyperM@{a}{a-b+\nu+2}{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}\nu}\KummerconfhyperU@{a}{b}{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = \EulerGamma@{\nu-b+2}x^{\frac{1}{2}\nu}e^{-x}\OlverconfhyperM@{a}{a-b+\nu+2}{x}</syntaxhighlight> || <math>x > 0, \max\left(\realpart@@{b-2} < \realpart@@{\nu}, -1\right) < \realpart@@{\nu}, \realpart@@{(\nu+k+1)} > 0, \realpart@@{(\nu-b+2)} > 0, \realpart@@{((a-b+\nu+2)+s)} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(- t)*(t)^((1)/(2)*nu)* KummerU(a, b, t)*BesselJ(nu, 2*sqrt(x*t)), t = 0..infinity) = GAMMA(nu - b + 2)*(x)^((1)/(2)*nu)* exp(- x)*KummerM(a, a - b + nu + 2, x)/GAMMA(a - b + nu + 2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[- t]*(t)^(Divide[1,2]*\[Nu])* HypergeometricU[a, b, t]*BesselJ[\[Nu], 2*Sqrt[x*t]], {t, 0, Infinity}, GenerateConditions->None] == Gamma[\[Nu]- b + 2]*(x)^(Divide[1,2]*\[Nu])* Exp[- x]*Hypergeometric1F1Regularized[a, a - b + \[Nu]+ 2, x]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/13.11.E1 13.11.E1] || [[Item:Q4476|<math>\KummerconfhyperM@{a}{b}{z} = \EulerGamma@{a-\tfrac{1}{2}}e^{\frac{1}{2}z}\left(\tfrac{1}{4}z\right)^{\frac{1}{2}-a}\*\sum_{s=0}^{\infty}\frac{\Pochhammersym{2a-1}{s}\Pochhammersym{2a-b}{s}}{\Pochhammersym{b}{s}s!}\*\left(a-\tfrac{1}{2}+s\right)\*\modBesselI{a-\frac{1}{2}+s}@{\tfrac{1}{2}z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperM@{a}{b}{z} = \EulerGamma@{a-\tfrac{1}{2}}e^{\frac{1}{2}z}\left(\tfrac{1}{4}z\right)^{\frac{1}{2}-a}\*\sum_{s=0}^{\infty}\frac{\Pochhammersym{2a-1}{s}\Pochhammersym{2a-b}{s}}{\Pochhammersym{b}{s}s!}\*\left(a-\tfrac{1}{2}+s\right)\*\modBesselI{a-\frac{1}{2}+s}@{\tfrac{1}{2}z}</syntaxhighlight> || <math>\realpart@@{(a-\tfrac{1}{2})} > 0, \realpart@@{((a-\frac{1}{2}+s)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>KummerM(a, b, z) = GAMMA(a -(1)/(2))*exp((1)/(2)*z)*((1)/(4)*z)^((1)/(2)- a)* sum((pochhammer(2*a - 1, s)*pochhammer(2*a - b, s))/(pochhammer(b, s)*factorial(s))*(a -(1)/(2)+ s)* BesselI(a -(1)/(2)+ s, (1)/(2)*z), s = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric1F1[a, b, z] == Gamma[a -Divide[1,2]]*Exp[Divide[1,2]*z]*(Divide[1,4]*z)^(Divide[1,2]- a)* Sum[Divide[Pochhammer[2*a - 1, s]*Pochhammer[2*a - b, s],Pochhammer[b, s]*(s)!]*(a -Divide[1,2]+ s)* BesselI[a -Divide[1,2]+ s, Divide[1,2]*z], {s, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Manual Skip! || <div class="toccolours mw-collapsible mw-collapsed">Failed [84 / 84]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-3.202632216430895, 12.150063432924489], Times[Complex[-5.9381784278055925, 1.66646925063829], NSum[Times[Plus[1.0, s], BesselI[Plus[1.0, s], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Factorial[s], -1], Power[Pochhammer[-1.5, s], -1], Pochhammer[2.0, s], Pochhammer[4.5, s]]
+Test Values: {s, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[a, 1.5], Rule[b, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[3.448639860241066, -0.8097281072366314], Times[Complex[0.28180823919021325, 3.102430445912792], NSum[Times[Plus[1.0, s], BesselI[Plus[1.0, s], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]], Power[Factorial[s], -1], Power[Pochhammer[-1.5, s], -1], Pochhammer[2.0, s], Pochhammer[4.5, s]]
+Test Values: {s, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[a, 1.5], Rule[b, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/13.12.E1 13.12.E1] || [[Item:Q4477|<math>\KummerconfhyperM@{a}{b}{z}\KummerconfhyperM@{-a}{-b}{-z}+\frac{a(a-b)z^{2}}{b^{2}(1-b^{2})}\KummerconfhyperM@{1+a}{2+b}{z}\KummerconfhyperM@{1-a}{2-b}{-z} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KummerconfhyperM@{a}{b}{z}\KummerconfhyperM@{-a}{-b}{-z}+\frac{a(a-b)z^{2}}{b^{2}(1-b^{2})}\KummerconfhyperM@{1+a}{2+b}{z}\KummerconfhyperM@{1-a}{2-b}{-z} = 1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KummerM(a, b, z)*KummerM(- a, - b, - z)+(a*(a - b)*(z)^(2))/((b)^(2)*(1 - (b)^(2)))*KummerM(1 + a, 2 + b, z)*KummerM(1 - a, 2 - b, - z) = 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric1F1[a, b, z]*Hypergeometric1F1[- a, - b, - z]+Divide[a*(a - b)*(z)^(2),(b)^(2)*(1 - (b)^(2))]*Hypergeometric1F1[1 + a, 2 + b, z]*Hypergeometric1F1[1 - a, 2 - b, - z] == 1</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [84 / 252]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|}
+</div>

Results of Confluent Hypergeometric Functions I: Difference between revisions

Latest revision as of 12:59, 22 May 2021

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