Hypergeometric Function - 15.5 Derivatives and Contiguous Functions

From testwiki
Revision as of 11:39, 28 June 2021 by Admin (talk | contribs) (Admin moved page Main Page to Verifying DLMF with Maple and Mathematica)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
15.5.E1 d d z ⁡ F ⁡ ( a , b ; c ; z ) = a ⁢ b c ⁢ F ⁡ ( a + 1 , b + 1 ; c + 1 ; z ) derivative 𝑧 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 𝑎 𝑏 𝑐 Gauss-hypergeometric-F 𝑎 1 𝑏 1 𝑐 1 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}F\left(a,b;c;z\right% )=\frac{ab}{c}F\left(a+1,b+1;c+1;z\right)}}
\deriv{}{z}\hyperF@{a}{b}{c}{z} = \frac{ab}{c}\hyperF@{a+1}{b+1}{c+1}{z}		 

diff(hypergeom([a, b], [c], z), z) = (a*b)/(c)*hypergeom([a + 1, b + 1], [c + 1], z)

D[Hypergeometric2F1[a, b, c, z], z] == Divide[a*b,c]*Hypergeometric2F1[a + 1, b + 1, c + 1, z]
Successful Successful - Successful [Tested: 300]
15.5.E2 d n d z n ⁡ F ⁡ ( a , b ; c ; z ) = ( a ) n ⁢ ( b ) n ( c ) n ⁢ F ⁡ ( a + n , b + n ; c + n ; z ) derivative 𝑧 𝑛 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 Pochhammer 𝑎 𝑛 Pochhammer 𝑏 𝑛 Pochhammer 𝑐 𝑛 Gauss-hypergeometric-F 𝑎 𝑛 𝑏 𝑛 𝑐 𝑛 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}F\left(a% ,b;c;z\right)=\frac{{\left(a\right)_{n}}{\left(b\right)_{n}}}{{\left(c\right)_% {n}}}\*F\left(a+n,b+n;c+n;z\right)}}
\deriv[n]{}{z}\hyperF@{a}{b}{c}{z} = \frac{\Pochhammersym{a}{n}\Pochhammersym{b}{n}}{\Pochhammersym{c}{n}}\*\hyperF@{a+n}{b+n}{c+n}{z}		 

diff(hypergeom([a, b], [c], z), [z$(n)]) = (pochhammer(a, n)*pochhammer(b, n))/(pochhammer(c, n))* hypergeom([a + n, b + n], [c + n], z)

D[Hypergeometric2F1[a, b, c, z], {z, n}] == Divide[Pochhammer[a, n]*Pochhammer[b, n],Pochhammer[c, n]]* Hypergeometric2F1[a + n, b + n, c + n, z]
Successful Successful - Successful [Tested: 300]
15.5.E3 ( z ⁢ d d z ⁡ z ) n ⁢ ( z a - 1 ⁢ F ⁡ ( a , b ; c ; z ) ) = ( a ) n ⁢ z a + n - 1 ⁢ F ⁡ ( a + n , b ; c ; z ) superscript 𝑧 derivative 𝑧 𝑧 𝑛 superscript 𝑧 𝑎 1 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 Pochhammer 𝑎 𝑛 superscript 𝑧 𝑎 𝑛 1 Gauss-hypergeometric-F 𝑎 𝑛 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}% \left(z^{a-1}F\left(a,b;c;z\right)\right)={\left(a\right)_{n}}z^{a+n-1}F\left(% a+n,b;c;z\right)}}
\left(z\deriv{}{z}z\right)^{n}\left(z^{a-1}\hyperF@{a}{b}{c}{z}\right) = \Pochhammersym{a}{n}z^{a+n-1}\hyperF@{a+n}{b}{c}{z}		 

(z*diff(z, z))^(n)*((z)^(a - 1)* hypergeom([a, b], [c], z)) = pochhammer(a, n)*(z)^(a + n - 1)* hypergeom([a + n, b], [c], z)

(z*D[z, z])^(n)*((z)^(a - 1)* Hypergeometric2F1[a, b, c, z]) == Pochhammer[a, n]*(z)^(a + n - 1)* Hypergeometric2F1[a + n, b, c, z]
Failure Failure Manual Skip!
Failed [298 / 300]
Result: Complex[2.047155237894918, -4.15915132240068]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[n, 1], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-0.9084280791008837, -0.4608118321937779]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[n, 2], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
15.5.E4 d n d z n ⁡ ( z c - 1 ⁢ F ⁡ ( a , b ; c ; z ) ) = ( c - n ) n ⁢ z c - n - 1 ⁢ F ⁡ ( a , b ; c - n ; z ) derivative 𝑧 𝑛 superscript 𝑧 𝑐 1 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 Pochhammer 𝑐 𝑛 𝑛 superscript 𝑧 𝑐 𝑛 1 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑛 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^% {c-1}F\left(a,b;c;z\right)\right)={\left(c-n\right)_{n}}z^{c-n-1}F\left(a,b;c-% n;z\right)}}
\deriv[n]{}{z}\left(z^{c-1}\hyperF@{a}{b}{c}{z}\right) = \Pochhammersym{c-n}{n}z^{c-n-1}\hyperF@{a}{b}{c-n}{z}		 

diff((z)^(c - 1)* hypergeom([a, b], [c], z), [z$(n)]) = pochhammer(c - n, n)*(z)^(c - n - 1)* hypergeom([a, b], [c - n], z)

D[(z)^(c - 1)* Hypergeometric2F1[a, b, c, z], {z, n}] == Pochhammer[c - n, n]*(z)^(c - n - 1)* Hypergeometric2F1[a, b, c - n, z]
Failure Aborted Skipped - Because timed out
Failed [300 / 300]
Result: Plus[Complex[-10.313412337740687, -15.40985641083086], Times[Complex[-2.9282032302755074, -10.928203230275509], DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-1, Plus[, -1.5], Plus[, -1.5], Plus[-1, Times[-1, ], 1], Plus[Times[-1, ], 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], []], Times[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, , Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-3, Power[, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-3, Power[, 3], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, , -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-2, <syntaxhighlight lang=mathematica>Result: Plus[Complex[123.08315470740952, 79.99762770469566], Times[Complex[-31.999999999999993, -32.0], DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-1, Plus[, -1.5], Plus[, -1.5], Plus[-1, Times[-1, ], 2], Plus[Times[-1, ], 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], []], Times[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, , Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-3, Power[, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-3, Power[, 3], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, , -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-2, Power[, 2], -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, , -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-2, Power[, 2], -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, , -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-3, , -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-2, Power[, 2], -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, , -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, , -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[3, , 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[3, Power[, 2], 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, , -1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, , -1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1.5, -1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], [Plus[1, ]]], Times[-1, 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], Times[3, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[6, , Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[3, Power[, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-3, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-3, , 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, 2, Times[Rational[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[-1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Times[Binomial[Plus[-1, -1.5], 2], Hypergeometric2F1[-1.5, -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]], Equal[[2], Times[Binomial[Plus[-1, -1.5], 2], Plus[Hypergeometric2F1[-1.5, -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1.5, -1.5, 2, Power[Plus[Power[-1.5, 2], Times[-1, -1.5, 2]], -1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Hypergeometric2F1[Plus[1, -1.5], Plus[1, -1.5], Plus[1, -1.5], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]]}]][3.0]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[n, 2], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
15.5.E5 ( z ⁢ d d z ⁡ z ) n ⁢ ( z c - a - 1 ⁢ ( 1 - z ) a + b - c ⁢ F ⁡ ( a , b ; c ; z ) ) = ( c - a ) n ⁢ z c - a + n - 1 ⁢ ( 1 - z ) a - n + b - c ⁢ F ⁡ ( a - n , b ; c ; z ) superscript 𝑧 derivative 𝑧 𝑧 𝑛 superscript 𝑧 𝑐 𝑎 1 superscript 1 𝑧 𝑎 𝑏 𝑐 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 Pochhammer 𝑐 𝑎 𝑛 superscript 𝑧 𝑐 𝑎 𝑛 1 superscript 1 𝑧 𝑎 𝑛 𝑏 𝑐 Gauss-hypergeometric-F 𝑎 𝑛 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}% \left(z^{c-a-1}(1-z)^{a+b-c}F\left(a,b;c;z\right)\right)={\left(c-a\right)_{n}% }z^{c-a+n-1}(1-z)^{a-n+b-c}\*F\left(a-n,b;c;z\right)}}
\left(z\deriv{}{z}z\right)^{n}\left(z^{c-a-1}(1-z)^{a+b-c}\hyperF@{a}{b}{c}{z}\right) = \Pochhammersym{c-a}{n}z^{c-a+n-1}(1-z)^{a-n+b-c}\*\hyperF@{a-n}{b}{c}{z}		 

(z*diff(z, z))^(n)*((z)^(c - a - 1)*(1 - z)^(a + b - c)* hypergeom([a, b], [c], z)) = pochhammer(c - a, n)*(z)^(c - a + n - 1)*(1 - z)^(a - n + b - c)* hypergeom([a - n, b], [c], z)

(z*D[z, z])^(n)*((z)^(c - a - 1)*(1 - z)^(a + b - c)* Hypergeometric2F1[a, b, c, z]) == Pochhammer[c - a, n]*(z)^(c - a + n - 1)*(1 - z)^(a - n + b - c)* Hypergeometric2F1[a - n, b, c, z]
Failure Failure Skipped - Because timed out
Failed [298 / 300]
Result: Complex[0.9999999999999999, -5.551115123125783*^-17]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[n, 1], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[0.4330127018922193, 0.24999999999999992]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[n, 2], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
15.5.E6 d n d z n ⁡ ( ( 1 - z ) a + b - c ⁢ F ⁡ ( a , b ; c ; z ) ) = ( c - a ) n ⁢ ( c - b ) n ( c ) n ⁢ ( 1 - z ) a + b - c - n ⁢ F ⁡ ( a , b ; c + n ; z ) derivative 𝑧 𝑛 superscript 1 𝑧 𝑎 𝑏 𝑐 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 Pochhammer 𝑐 𝑎 𝑛 Pochhammer 𝑐 𝑏 𝑛 Pochhammer 𝑐 𝑛 superscript 1 𝑧 𝑎 𝑏 𝑐 𝑛 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑛 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left((1% -z)^{a+b-c}F\left(a,b;c;z\right)\right)=\frac{{\left(c-a\right)_{n}}{\left(c-b% \right)_{n}}}{{\left(c\right)_{n}}}(1-z)^{a+b-c-n}\*F\left(a,b;c+n;z\right)}}
\deriv[n]{}{z}\left((1-z)^{a+b-c}\hyperF@{a}{b}{c}{z}\right) = \frac{\Pochhammersym{c-a}{n}\Pochhammersym{c-b}{n}}{\Pochhammersym{c}{n}}(1-z)^{a+b-c-n}\*\hyperF@{a}{b}{c+n}{z}		 

diff((1 - z)^(a + b - c)* hypergeom([a, b], [c], z), [z$(n)]) = (pochhammer(c - a, n)*pochhammer(c - b, n))/(pochhammer(c, n))*(1 - z)^(a + b - c - n)* hypergeom([a, b], [c + n], z)

D[(1 - z)^(a + b - c)* Hypergeometric2F1[a, b, c, z], {z, n}] == Divide[Pochhammer[c - a, n]*Pochhammer[c - b, n],Pochhammer[c, n]]*(1 - z)^(a + b - c - n)* Hypergeometric2F1[a, b, c + n, z]
Failure Aborted Skipped - Because timed out
Failed [300 / 300]
Result: Plus[Complex[0.0, 0.0], Times[Complex[-1.6799040046341822, -2.8501979384465357], DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-1, Plus[, -1.5], Plus[, -1.5], Plus[-1, Times[-1, ], 1], Plus[Times[-1, ], 1], Plus[-1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], []], Times[-1, Plus[-1, Times[-1, ], 1], Plus[Times[-1, ], Times[-2, Power[, 2]], Times[-2, Power[, 3]], Times[-1, , -1.5], Times[-2, Power[, 2], -1.5], Times[-1, , -1.5], Times[-2, Power[, 2], -1.5], Times[-1, , -1.5, -1.5], Times[-1, -1.5], Times[-1, , -1.5], Times[-1, -1.5, -1.5], Times[-1, , -1.5, -1.5], Times[-1, -1.5, -1.5], Times[-1, , -1.5, -1.5], Power[-1.5, 2], Times[, Power[-1.5, 2]], Times[, 1], Times[2, Power[, 2], 1], Times[, -1.5, 1], Times[, -1.5, 1], Times[-1.5, -1.5, 1], Times[-1.5, 1], Times[, -1.5, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[4, , Times[Rational[1,<syntaxhighlight lang=mathematica>Result: Plus[Complex[0.0, 0.0], Times[Complex[1.2497428237239117, 10.604878809262228], DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-1, Plus[, -1.5], Plus[, -1.5], Plus[-1, Times[-1, ], 2], Plus[Times[-1, ], 2], Plus[-1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], []], Times[-1, Plus[-1, Times[-1, ], 2], Plus[Times[-1, ], Times[-2, Power[, 2]], Times[-2, Power[, 3]], Times[-1, , -1.5], Times[-2, Power[, 2], -1.5], Times[-1, , -1.5], Times[-2, Power[, 2], -1.5], Times[-1, , -1.5, -1.5], Times[-1, -1.5], Times[-1, , -1.5], Times[-1, -1.5, -1.5], Times[-1, , -1.5, -1.5], Times[-1, -1.5, -1.5], Times[-1, , -1.5, -1.5], Power[-1.5, 2], Times[, Power[-1.5, 2]], Times[, 2], Times[2, Power[, 2], 2], Times[, -1.5, 2], Times[, -1.5, 2], Times[-1.5, -1.5, 2], Times[-1.5, 2], Times[, -1.5, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[4, , Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[5, Power[, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[3, Power[, 3], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[5, , -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[4, Power[, 2], -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[Power[-1.5, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[, Power[-1.5, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[5, , -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[4, Power[, 2], -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[3, , -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[Power[-1.5, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[, Power[-1.5, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-3, , -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-2, Power[, 2], -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, , -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, , -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-3, , 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-3, Power[, 2], 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-2, , -1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-2, , -1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, -1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], [Plus[1, ]]], Times[Plus[1, ], Plus[-1, Times[-1, ], Times[-1, -1.5], Times[-1, -1.5], -1.5, 2], Plus[Times[-1, ], Times[-1, Power[, 2]], Times[-1, -1.5], Times[-1, , -1.5], Times[, 2], Times[-1.5, 2], Times[5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[7, , Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[3, Power[, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[3, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, , -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[3, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, , -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-2, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, , -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-3, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-3, , 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], [Plus[2, ]]], Times[Plus[1, ], Plus[2, ], Plus[1, , -1.5, -1.5, Times[-1, -1.5], Times[-1, 2]], Plus[2, , -1.5, -1.5, Times[-1, -1.5], Times[-1, 2]], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Times[Binomial[Plus[-1.5, -1.5, Times[-1, -1.5]], 2], Hypergeometric2F1[-1.5, -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]], Equal[[2], Times[Binomial[Plus[-1.5, -1.5, Times[-1, -1.5]], 2], Plus[Hypergeometric2F1[-1.5, -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1.5, -1.5, Power[-1.5, -1], Power[Plus[1, -1.5, -1.5, Times[-1, -1.5], Times[-1, 2]], -1], 2, Plus[-1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Hypergeometric2F1[Plus[1, -1.5], Plus[1, -1.5], Plus[1, -1.5], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]]}]][3.0]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[n, 2], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
15.5.E7 ( ( 1 - z ) ⁢ d d z ⁡ ( 1 - z ) ) n ⁢ ( ( 1 - z ) a - 1 ⁢ F ⁡ ( a , b ; c ; z ) ) = ( - 1 ) n ⁢ ( a ) n ⁢ ( c - b ) n ( c ) n ⁢ ( 1 - z ) a + n - 1 ⁢ F ⁡ ( a + n , b ; c + n ; z ) superscript 1 𝑧 derivative 𝑧 1 𝑧 𝑛 superscript 1 𝑧 𝑎 1 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 superscript 1 𝑛 Pochhammer 𝑎 𝑛 Pochhammer 𝑐 𝑏 𝑛 Pochhammer 𝑐 𝑛 superscript 1 𝑧 𝑎 𝑛 1 Gauss-hypergeometric-F 𝑎 𝑛 𝑏 𝑐 𝑛 𝑧 {\displaystyle{\displaystyle\left((1-z)\frac{\mathrm{d}}{\mathrm{d}z}(1-z)% \right)^{n}\left((1-z)^{a-1}F\left(a,b;c;z\right)\right)=(-1)^{n}\frac{{\left(% a\right)_{n}}{\left(c-b\right)_{n}}}{{\left(c\right)_{n}}}(1-z)^{a+n-1}\*F% \left(a+n,b;c+n;z\right)}}
\left((1-z)\deriv{}{z}(1-z)\right)^{n}\left((1-z)^{a-1}\hyperF@{a}{b}{c}{z}\right) = (-1)^{n}\frac{\Pochhammersym{a}{n}\Pochhammersym{c-b}{n}}{\Pochhammersym{c}{n}}(1-z)^{a+n-1}\*\hyperF@{a+n}{b}{c+n}{z}		 

((1 - z)*diff(1 - z, z))^(n)*((1 - z)^(a - 1)* hypergeom([a, b], [c], z)) = (- 1)^(n)*(pochhammer(a, n)*pochhammer(c - b, n))/(pochhammer(c, n))*(1 - z)^(a + n - 1)* hypergeom([a + n, b], [c + n], z)

((1 - z)*D[1 - z, z])^(n)*((1 - z)^(a - 1)* Hypergeometric2F1[a, b, c, z]) == (- 1)^(n)*Divide[Pochhammer[a, n]*Pochhammer[c - b, n],Pochhammer[c, n]]*(1 - z)^(a + n - 1)* Hypergeometric2F1[a + n, b, c + n, z]
Failure Failure Skipped - Because timed out
Failed [300 / 300]
Result: Complex[-0.9999999999999999, 5.551115123125783*^-17]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[n, 1], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[0.5669872981077805, -0.24999999999999994]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[n, 2], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
15.5.E8 ( ( 1 - z ) ⁢ d d z ⁡ ( 1 - z ) ) n ⁢ ( z c - 1 ⁢ ( 1 - z ) b - c ⁢ F ⁡ ( a , b ; c ; z ) ) = ( c - n ) n ⁢ z c - n - 1 ⁢ ( 1 - z ) b - c + n ⁢ F ⁡ ( a - n , b ; c - n ; z ) superscript 1 𝑧 derivative 𝑧 1 𝑧 𝑛 superscript 𝑧 𝑐 1 superscript 1 𝑧 𝑏 𝑐 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 Pochhammer 𝑐 𝑛 𝑛 superscript 𝑧 𝑐 𝑛 1 superscript 1 𝑧 𝑏 𝑐 𝑛 Gauss-hypergeometric-F 𝑎 𝑛 𝑏 𝑐 𝑛 𝑧 {\displaystyle{\displaystyle\left((1-z)\frac{\mathrm{d}}{\mathrm{d}z}(1-z)% \right)^{n}\left(z^{c-1}(1-z)^{b-c}F\left(a,b;c;z\right)\right)={\left(c-n% \right)_{n}}z^{c-n-1}(1-z)^{b-c+n}\*F\left(a-n,b;c-n;z\right)}}
\left((1-z)\deriv{}{z}(1-z)\right)^{n}\left(z^{c-1}(1-z)^{b-c}\hyperF@{a}{b}{c}{z}\right) = \Pochhammersym{c-n}{n}z^{c-n-1}(1-z)^{b-c+n}\*\hyperF@{a-n}{b}{c-n}{z}		 

((1 - z)*diff(1 - z, z))^(n)*((z)^(c - 1)*(1 - z)^(b - c)* hypergeom([a, b], [c], z)) = pochhammer(c - n, n)*(z)^(c - n - 1)*(1 - z)^(b - c + n)* hypergeom([a - n, b], [c - n], z)

((1 - z)*D[1 - z, z])^(n)*((z)^(c - 1)*(1 - z)^(b - c)* Hypergeometric2F1[a, b, c, z]) == Pochhammer[c - n, n]*(z)^(c - n - 1)*(1 - z)^(b - c + n)* Hypergeometric2F1[a - n, b, c - n, z]
Failure Failure Skipped - Because timed out
Failed [299 / 300]
Result: Complex[-7.039508221073909, -1.0669744439111815]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[n, 1], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[28.125871703124346, -23.36453828137185]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[n, 2], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
15.5.E9 d n d z n ⁡ ( z c - 1 ⁢ ( 1 - z ) a + b - c ⁢ F ⁡ ( a , b ; c ; z ) ) = ( c - n ) n ⁢ z c - n - 1 ⁢ ( 1 - z ) a + b - c - n ⁢ F ⁡ ( a - n , b - n ; c - n ; z ) derivative 𝑧 𝑛 superscript 𝑧 𝑐 1 superscript 1 𝑧 𝑎 𝑏 𝑐 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 Pochhammer 𝑐 𝑛 𝑛 superscript 𝑧 𝑐 𝑛 1 superscript 1 𝑧 𝑎 𝑏 𝑐 𝑛 Gauss-hypergeometric-F 𝑎 𝑛 𝑏 𝑛 𝑐 𝑛 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^% {c-1}(1-z)^{a+b-c}F\left(a,b;c;z\right)\right)={\left(c-n\right)_{n}}z^{c-n-1}% (1-z)^{a+b-c-n}\*F\left(a-n,b-n;c-n;z\right)}}
\deriv[n]{}{z}\left(z^{c-1}(1-z)^{a+b-c}\hyperF@{a}{b}{c}{z}\right) = \Pochhammersym{c-n}{n}z^{c-n-1}(1-z)^{a+b-c-n}\*\hyperF@{a-n}{b-n}{c-n}{z}		 

diff((z)^(c - 1)*(1 - z)^(a + b - c)* hypergeom([a, b], [c], z), [z$(n)]) = pochhammer(c - n, n)*(z)^(c - n - 1)*(1 - z)^(a + b - c - n)* hypergeom([a - n, b - n], [c - n], z)

D[(z)^(c - 1)*(1 - z)^(a + b - c)* Hypergeometric2F1[a, b, c, z], {z, n}] == Pochhammer[c - n, n]*(z)^(c - n - 1)*(1 - z)^(a + b - c - n)* Hypergeometric2F1[a - n, b - n, c - n, z]
Failure Aborted Skipped - Because timed out
Failed [300 / 300]
Result: Plus[Complex[-7.320508075688771, -27.32050807568877], DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[-1, Times[-1, ], -1.5], Plus[-1, Times[-1, ], -1.5], []], Times[Plus[1, ], Plus[-2, Times[-1, ], -1.5, Times[3, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, , Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], [Plus[1, ]]], Times[Plus[1, ], Plus[2, ], Plus[-1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], [Plus[2, ]]]], 0], Equal[[0], Times[Power[Plus[1, Times[-1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Plus[-1.5, -1.5, Times[-1, -1.5]]], Power[Tim<syntaxhighlight lang=mathematica>Result: Plus[Complex[139.99999999999997, 139.99999999999997], Times[2.0, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[-1, Times[-1, ], -1.5], Plus[-1, Times[-1, ], -1.5], []], Times[Plus[1, ], Plus[-2, Times[-1, ], -1.5, Times[3, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, , Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], [Plus[1, ]]], Times[Plus[1, ], Plus[2, ], Plus[-1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], [Plus[2, ]]]], 0], Equal[[0], Times[Power[Plus[1, Times[-1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Plus[-1.5, -1.5, Times[-1, -1.5]]], Power[Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[-1, -1.5]], Hypergeometric2F1[-1.5, -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]], Equal[[1], Times[Power[Plus[1, Times[-1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Plus[-1.5, -1.5, Times[-1, -1.5]]], Power[Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[-2, -1.5]], Plus[Times[Power[Plus[-1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], -1], Plus[1, Times[-1, -1.5], Times[-1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Hypergeometric2F1[-1.5, -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Times[-1.5, -1.5, Power[-1.5, -1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Hypergeometric2F1[Plus[1, -1.5], Plus[1, -1.5], Plus[1, -1.5], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]]}]][2.0]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[n, 2], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
15.5.E10 ( z ⁢ d d z ⁡ z ) n = z n ⁢ d n d z n ⁡ z n superscript 𝑧 derivative 𝑧 𝑧 𝑛 superscript 𝑧 𝑛 derivative 𝑧 𝑛 superscript 𝑧 𝑛 {\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}		 

(z*diff(z, z))^(n) = (z)^(n)* diff((z)^(n), [z$(n)])

(z*D[z, z])^(n) == (z)^(n)* D[(z)^(n), {z, n}]
Failure Failure
Failed [7 / 7]
Result: -.1616869430e-8-5.000000005*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, n = 3}


Result: -5.000000005+.1616869430e-8*I
Test Values: {z = -1/2+1/2*I*3^(1/2), n = 3}

... skip entries to safe data
Failed [7 / 7]
Result: Complex[0.0, -0.625]
Test Values: {Rule[n, 3], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: -0.625
Test Values: {Rule[n, 3], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
15.5.E11 ( c - a ) ⁢ F ⁡ ( a - 1 , b ; c ; z ) + ( 2 ⁢ a - c + ( b - a ) ⁢ z ) ⁢ F ⁡ ( a , b ; c ; z ) + a ⁢ ( z - 1 ) ⁢ F ⁡ ( a + 1 , b ; c ; z ) = 0 𝑐 𝑎 Gauss-hypergeometric-F 𝑎 1 𝑏 𝑐 𝑧 2 𝑎 𝑐 𝑏 𝑎 𝑧 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 𝑎 𝑧 1 Gauss-hypergeometric-F 𝑎 1 𝑏 𝑐 𝑧 0 {\displaystyle{\displaystyle(c-a)F\left(a-1,b;c;z\right)+\left(2a-c+(b-a)z% \right)F\left(a,b;c;z\right)+a(z-1)F\left(a+1,b;c;z\right)=0}}
(c-a)\hyperF@{a-1}{b}{c}{z}+\left(2a-c+(b-a)z\right)\hyperF@{a}{b}{c}{z}+a(z-1)\hyperF@{a+1}{b}{c}{z} = 0		 

(c - a)*hypergeom([a - 1, b], [c], z)+(2*a - c +(b - a)*z)*hypergeom([a, b], [c], z)+ a*(z - 1)*hypergeom([a + 1, b], [c], z) = 0

(c - a)*Hypergeometric2F1[a - 1, b, c, z]+(2*a - c +(b - a)*z)*Hypergeometric2F1[a, b, c, z]+ a*(z - 1)*Hypergeometric2F1[a + 1, b, c, z] == 0
Successful Successful -
Failed [42 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
15.5.E12 ( b - a ) ⁢ F ⁡ ( a , b ; c ; z ) + a ⁢ F ⁡ ( a + 1 , b ; c ; z ) - b ⁢ F ⁡ ( a , b + 1 ; c ; z ) = 0 𝑏 𝑎 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 𝑎 Gauss-hypergeometric-F 𝑎 1 𝑏 𝑐 𝑧 𝑏 Gauss-hypergeometric-F 𝑎 𝑏 1 𝑐 𝑧 0 {\displaystyle{\displaystyle(b-a)F\left(a,b;c;z\right)+aF\left(a+1,b;c;z\right% )-bF\left(a,b+1;c;z\right)=0}}
(b-a)\hyperF@{a}{b}{c}{z}+a\hyperF@{a+1}{b}{c}{z}-b\hyperF@{a}{b+1}{c}{z} = 0		 

(b - a)*hypergeom([a, b], [c], z)+ a*hypergeom([a + 1, b], [c], z)- b*hypergeom([a, b + 1], [c], z) = 0

(b - a)*Hypergeometric2F1[a, b, c, z]+ a*Hypergeometric2F1[a + 1, b, c, z]- b*Hypergeometric2F1[a, b + 1, c, z] == 0
Successful Successful -
Failed [42 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
15.5.E13 ( c - a - b ) ⁢ F ⁡ ( a , b ; c ; z ) + a ⁢ ( 1 - z ) ⁢ F ⁡ ( a + 1 , b ; c ; z ) - ( c - b ) ⁢ F ⁡ ( a , b - 1 ; c ; z ) = 0 𝑐 𝑎 𝑏 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 𝑎 1 𝑧 Gauss-hypergeometric-F 𝑎 1 𝑏 𝑐 𝑧 𝑐 𝑏 Gauss-hypergeometric-F 𝑎 𝑏 1 𝑐 𝑧 0 {\displaystyle{\displaystyle(c-a-b)F\left(a,b;c;z\right)+a(1-z)F\left(a+1,b;c;% z\right)-(c-b)F\left(a,b-1;c;z\right)=0}}
(c-a-b)\hyperF@{a}{b}{c}{z}+a(1-z)\hyperF@{a+1}{b}{c}{z}-(c-b)\hyperF@{a}{b-1}{c}{z} = 0		 

(c - a - b)*hypergeom([a, b], [c], z)+ a*(1 - z)*hypergeom([a + 1, b], [c], z)-(c - b)*hypergeom([a, b - 1], [c], z) = 0

(c - a - b)*Hypergeometric2F1[a, b, c, z]+ a*(1 - z)*Hypergeometric2F1[a + 1, b, c, z]-(c - b)*Hypergeometric2F1[a, b - 1, c, z] == 0
Successful Successful -
Failed [49 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
15.5.E14 c ⁢ ( a + ( b - c ) ⁢ z ) ⁢ F ⁡ ( a , b ; c ; z ) - a ⁢ c ⁢ ( 1 - z ) ⁢ F ⁡ ( a + 1 , b ; c ; z ) + ( c - a ) ⁢ ( c - b ) ⁢ z ⁢ F ⁡ ( a , b ; c + 1 ; z ) = 0 𝑐 𝑎 𝑏 𝑐 𝑧 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 𝑎 𝑐 1 𝑧 Gauss-hypergeometric-F 𝑎 1 𝑏 𝑐 𝑧 𝑐 𝑎 𝑐 𝑏 𝑧 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 1 𝑧 0 {\displaystyle{\displaystyle c\left(a+(b-c)z\right)F\left(a,b;c;z\right)-ac(1-% z)F\left(a+1,b;c;z\right)+(c-a)(c-b)zF\left(a,b;c+1;z\right)=0}}
c\left(a+(b-c)z\right)\hyperF@{a}{b}{c}{z}-ac(1-z)\hyperF@{a+1}{b}{c}{z}+(c-a)(c-b)z\hyperF@{a}{b}{c+1}{z} = 0		 

c*(a +(b - c)*z)*hypergeom([a, b], [c], z)- a*c*(1 - z)*hypergeom([a + 1, b], [c], z)+(c - a)*(c - b)*z*hypergeom([a, b], [c + 1], z) = 0

c*(a +(b - c)*z)*Hypergeometric2F1[a, b, c, z]- a*c*(1 - z)*Hypergeometric2F1[a + 1, b, c, z]+(c - a)*(c - b)*z*Hypergeometric2F1[a, b, c + 1, z] == 0
Successful Successful -
Failed [49 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
15.5.E15 ( c - a - 1 ) ⁢ F ⁡ ( a , b ; c ; z ) + a ⁢ F ⁡ ( a + 1 , b ; c ; z ) - ( c - 1 ) ⁢ F ⁡ ( a , b ; c - 1 ; z ) = 0 𝑐 𝑎 1 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 𝑎 Gauss-hypergeometric-F 𝑎 1 𝑏 𝑐 𝑧 𝑐 1 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 1 𝑧 0 {\displaystyle{\displaystyle(c-a-1)F\left(a,b;c;z\right)+aF\left(a+1,b;c;z% \right)-(c-1)F\left(a,b;c-1;z\right)=0}}
(c-a-1)\hyperF@{a}{b}{c}{z}+a\hyperF@{a+1}{b}{c}{z}-(c-1)\hyperF@{a}{b}{c-1}{z} = 0		 

(c - a - 1)*hypergeom([a, b], [c], z)+ a*hypergeom([a + 1, b], [c], z)-(c - 1)*hypergeom([a, b], [c - 1], z) = 0

(c - a - 1)*Hypergeometric2F1[a, b, c, z]+ a*Hypergeometric2F1[a + 1, b, c, z]-(c - 1)*Hypergeometric2F1[a, b, c - 1, z] == 0
Successful Successful -
Failed [42 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
15.5.E16 c ⁢ ( 1 - z ) ⁢ F ⁡ ( a , b ; c ; z ) - c ⁢ F ⁡ ( a - 1 , b ; c ; z ) + ( c - b ) ⁢ z ⁢ F ⁡ ( a , b ; c + 1 ; z ) = 0 𝑐 1 𝑧 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 𝑐 Gauss-hypergeometric-F 𝑎 1 𝑏 𝑐 𝑧 𝑐 𝑏 𝑧 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 1 𝑧 0 {\displaystyle{\displaystyle c(1-z)F\left(a,b;c;z\right)-cF\left(a-1,b;c;z% \right)+(c-b)zF\left(a,b;c+1;z\right)=0}}
c(1-z)\hyperF@{a}{b}{c}{z}-c\hyperF@{a-1}{b}{c}{z}+(c-b)z\hyperF@{a}{b}{c+1}{z} = 0		 

c*(1 - z)*hypergeom([a, b], [c], z)- c*hypergeom([a - 1, b], [c], z)+(c - b)*z*hypergeom([a, b], [c + 1], z) = 0

c*(1 - z)*Hypergeometric2F1[a, b, c, z]- c*Hypergeometric2F1[a - 1, b, c, z]+(c - b)*z*Hypergeometric2F1[a, b, c + 1, z] == 0
Successful Successful -
Failed [49 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
15.5.E17 ( a - 1 + ( b + 1 - c ) ⁢ z ) ⁢ F ⁡ ( a , b ; c ; z ) + ( c - a ) ⁢ F ⁡ ( a - 1 , b ; c ; z ) - ( c - 1 ) ⁢ ( 1 - z ) ⁢ F ⁡ ( a , b ; c - 1 ; z ) = 0 𝑎 1 𝑏 1 𝑐 𝑧 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 𝑐 𝑎 Gauss-hypergeometric-F 𝑎 1 𝑏 𝑐 𝑧 𝑐 1 1 𝑧 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 1 𝑧 0 {\displaystyle{\displaystyle\left(a-1+(b+1-c)z\right)F\left(a,b;c;z\right)+(c-% a)F\left(a-1,b;c;z\right)-(c-1)(1-z)F\left(a,b;c-1;z\right)=0}}
\left(a-1+(b+1-c)z\right)\hyperF@{a}{b}{c}{z}+(c-a)\hyperF@{a-1}{b}{c}{z}-(c-1)(1-z)\hyperF@{a}{b}{c-1}{z} = 0		 

(a - 1 +(b + 1 - c)*z)*hypergeom([a, b], [c], z)+(c - a)*hypergeom([a - 1, b], [c], z)-(c - 1)*(1 - z)*hypergeom([a, b], [c - 1], z) = 0

(a - 1 +(b + 1 - c)*z)*Hypergeometric2F1[a, b, c, z]+(c - a)*Hypergeometric2F1[a - 1, b, c, z]-(c - 1)*(1 - z)*Hypergeometric2F1[a, b, c - 1, z] == 0
Successful Successful -
Failed [42 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
15.5.E18 c ⁢ ( c - 1 ) ⁢ ( z - 1 ) ⁢ F ⁡ ( a , b ; c - 1 ; z ) + c ⁢ ( c - 1 - ( 2 ⁢ c - a - b - 1 ) ⁢ z ) ⁢ F ⁡ ( a , b ; c ; z ) + ( c - a ) ⁢ ( c - b ) ⁢ z ⁢ F ⁡ ( a , b ; c + 1 ; z ) = 0 𝑐 𝑐 1 𝑧 1 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 1 𝑧 𝑐 𝑐 1 2 𝑐 𝑎 𝑏 1 𝑧 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 𝑐 𝑎 𝑐 𝑏 𝑧 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 1 𝑧 0 {\displaystyle{\displaystyle c(c-1)(z-1)F\left(a,b;c-1;z\right)+{c\left(c-1-(2% c-a-b-1)z\right)}F\left(a,b;c;z\right)+(c-a)(c-b)zF\left(a,b;c+1;z\right)=0}}
c(c-1)(z-1)\hyperF@{a}{b}{c-1}{z}+{c\left(c-1-(2c-a-b-1)z\right)}\hyperF@{a}{b}{c}{z}+(c-a)(c-b)z\hyperF@{a}{b}{c+1}{z} = 0		 

c*(c - 1)*(z - 1)*hypergeom([a, b], [c - 1], z)+c*(c - 1 -(2*c - a - b - 1)*z)*hypergeom([a, b], [c], z)+(c - a)*(c - b)*z*hypergeom([a, b], [c + 1], z) = 0

c*(c - 1)*(z - 1)*Hypergeometric2F1[a, b, c - 1, z]+c*(c - 1 -(2*c - a - b - 1)*z)*Hypergeometric2F1[a, b, c, z]+(c - a)*(c - b)*z*Hypergeometric2F1[a, b, c + 1, z] == 0
Successful Successful -
Failed [49 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
15.5.E19 z ⁢ ( 1 - z ) ⁢ ( a + 1 ) ⁢ ( b + 1 ) ⁢ F ⁡ ( a + 2 , b + 2 ; c + 2 ; z ) + ( c - ( a + b + 1 ) ⁢ z ) ⁢ ( c + 1 ) ⁢ F ⁡ ( a + 1 , b + 1 ; c + 1 ; z ) - c ⁢ ( c + 1 ) ⁢ F ⁡ ( a , b ; c ; z ) = 0 𝑧 1 𝑧 𝑎 1 𝑏 1 Gauss-hypergeometric-F 𝑎 2 𝑏 2 𝑐 2 𝑧 𝑐 𝑎 𝑏 1 𝑧 𝑐 1 Gauss-hypergeometric-F 𝑎 1 𝑏 1 𝑐 1 𝑧 𝑐 𝑐 1 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 0 {\displaystyle{\displaystyle{z(1-z)(a+1)(b+1)}F\left(a+2,b+2;c+2;z\right)+{(c-% (a+b+1)z)(c+1)}F\left(a+1,b+1;c+1;z\right)-{c(c+1)}F\left(a,b;c;z\right)=0}}
{z(1-z)(a+1)(b+1)}\hyperF@{a+2}{b+2}{c+2}{z}+{(c-(a+b+1)z)(c+1)}\hyperF@{a+1}{b+1}{c+1}{z}-{c(c+1)}\hyperF@{a}{b}{c}{z} = 0		 

z*(1 - z)*(a + 1)*(b + 1)*hypergeom([a + 2, b + 2], [c + 2], z)+(c -(a + b + 1)*z)*(c + 1)*hypergeom([a + 1, b + 1], [c + 1], z)-c*(c + 1)*hypergeom([a, b], [c], z) = 0

z*(1 - z)*(a + 1)*(b + 1)*Hypergeometric2F1[a + 2, b + 2, c + 2, z]+(c -(a + b + 1)*z)*(c + 1)*Hypergeometric2F1[a + 1, b + 1, c + 1, z]-c*(c + 1)*Hypergeometric2F1[a, b, c, z] == 0
Successful Successful -
Failed [42 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
15.5.E20 z ⁢ ( 1 - z ) ⁢ ( d F ⁡ ( a , b ; c ; z ) / d z ) = ( c - a ) ⁢ F ⁡ ( a - 1 , b ; c ; z ) + ( a - c + b ⁢ z ) ⁢ F ⁡ ( a , b ; c ; z ) 𝑧 1 𝑧 derivative Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 𝑧 𝑐 𝑎 Gauss-hypergeometric-F 𝑎 1 𝑏 𝑐 𝑧 𝑎 𝑐 𝑏 𝑧 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle z(1-z)\left(\ifrac{\mathrm{d}F\left(a,b;c;z\right% )}{\mathrm{d}z}\right)=(c-a)F\left(a-1,b;c;z\right)+(a-c+bz)F\left(a,b;c;z% \right)}}
z(1-z)\left(\ideriv{\hyperF@{a}{b}{c}{z}}{z}\right) = (c-a)\hyperF@{a-1}{b}{c}{z}+(a-c+bz)\hyperF@{a}{b}{c}{z}		 

z*(1 - z)*(diff(hypergeom([a, b], [c], z), z)) = (c - a)*hypergeom([a - 1, b], [c], z)+(a - c + b*z)*hypergeom([a, b], [c], z)

z*(1 - z)*(D[Hypergeometric2F1[a, b, c, z], z]) == (c - a)*Hypergeometric2F1[a - 1, b, c, z]+(a - c + b*z)*Hypergeometric2F1[a, b, c, z]
Successful Successful -
Failed [42 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
15.5.E20 ( c - a ) ⁢ F ⁡ ( a - 1 , b ; c ; z ) + ( a - c + b ⁢ z ) ⁢ F ⁡ ( a , b ; c ; z ) = ( c - b ) ⁢ F ⁡ ( a , b - 1 ; c ; z ) + ( b - c + a ⁢ z ) ⁢ F ⁡ ( a , b ; c ; z ) 𝑐 𝑎 Gauss-hypergeometric-F 𝑎 1 𝑏 𝑐 𝑧 𝑎 𝑐 𝑏 𝑧 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 𝑐 𝑏 Gauss-hypergeometric-F 𝑎 𝑏 1 𝑐 𝑧 𝑏 𝑐 𝑎 𝑧 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle(c-a)F\left(a-1,b;c;z\right)+(a-c+bz)F\left(a,b;c;% z\right)=(c-b)F\left(a,b-1;c;z\right)+(b-c+az)F\left(a,b;c;z\right)}}
(c-a)\hyperF@{a-1}{b}{c}{z}+(a-c+bz)\hyperF@{a}{b}{c}{z} = (c-b)\hyperF@{a}{b-1}{c}{z}+(b-c+az)\hyperF@{a}{b}{c}{z}		 

(c - a)*hypergeom([a - 1, b], [c], z)+(a - c + b*z)*hypergeom([a, b], [c], z) = (c - b)*hypergeom([a, b - 1], [c], z)+(b - c + a*z)*hypergeom([a, b], [c], z)

(c - a)*Hypergeometric2F1[a - 1, b, c, z]+(a - c + b*z)*Hypergeometric2F1[a, b, c, z] == (c - b)*Hypergeometric2F1[a, b - 1, c, z]+(b - c + a*z)*Hypergeometric2F1[a, b, c, z]
Successful Successful -
Failed [49 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
15.5.E21 c ⁢ ( 1 - z ) ⁢ ( d F ⁡ ( a , b ; c ; z ) / d z ) = ( c - a ) ⁢ ( c - b ) ⁢ F ⁡ ( a , b ; c + 1 ; z ) + c ⁢ ( a + b - c ) ⁢ F ⁡ ( a , b ; c ; z ) 𝑐 1 𝑧 derivative Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 𝑧 𝑐 𝑎 𝑐 𝑏 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 1 𝑧 𝑐 𝑎 𝑏 𝑐 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle c(1-z)\left(\ifrac{\mathrm{d}F\left(a,b;c;z\right% )}{\mathrm{d}z}\right)=(c-a)(c-b)F\left(a,b;c+1;z\right)+c(a+b-c)F\left(a,b;c;% z\right)}}
c(1-z)\left(\ideriv{\hyperF@{a}{b}{c}{z}}{z}\right) = (c-a)(c-b)\hyperF@{a}{b}{c+1}{z}+c(a+b-c)\hyperF@{a}{b}{c}{z}		 

c*(1 - z)*(diff(hypergeom([a, b], [c], z), z)) = (c - a)*(c - b)*hypergeom([a, b], [c + 1], z)+ c*(a + b - c)*hypergeom([a, b], [c], z)
 
c*(1 - z)*(D[Hypergeometric2F1[a, b, c, z], z]) == (c - a)*(c - b)*Hypergeometric2F1[a, b, c + 1, z]+ c*(a + b - c)*Hypergeometric2F1[a, b, c, z]
 Successful Successful -
Failed [49 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
Retrieved from "https://lct.wmflabs.org/w/index.php?title=15.5&oldid=58301"