18.5

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
18.5.E1	${\displaystyle{\displaystyle T_{n}\left(x\right)=\cos\left(n\theta\right)}}$ \ChebyshevpolyT{n}@{x} = \cos@{n\theta}		ChebyshevT(n, x) = cos(n*theta)	ChebyshevT[n, x] == Cos[n*\[Theta]]	Failure	Failure	Failed [90 / 90] Result: .7694569811+.3969495503*I Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 1} Result: 3.747751686+1.159954891*I Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 2} ... skip entries to safe data	Failed [90 / 90] Result: Complex[0.7694569809427748, 0.3969495502290325] Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[3.747751685467572, 1.1599548913509004] Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
18.5.E2	${\displaystyle{\displaystyle U_{n}\left(x\right)=\ifrac{(\sin(n+1)\theta)}{\sin\theta}}}$ \ChebyshevpolyU{n}@{x} = \ifrac{(\sin@@{(n+1)\theta})}{\sin@@{\theta}}		ChebyshevU(n, x) = (sin((n + 1)*theta))/(sin(theta))	ChebyshevU[n, x] == Divide[Sin[(n + 1)*\[Theta]],Sin[\[Theta]]]	Failure	Failure	Failed [90 / 90] Result: 1.538913962+.7938991006*I Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 1} Result: 7.495503373+2.319909783*I Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 2} ... skip entries to safe data	Failed [90 / 90] Result: Complex[1.5389139618855496, 0.7938991004580651] Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[7.495503370935143, 2.3199097827018003] Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
18.5.E6	${\displaystyle{\displaystyle L^{(\alpha)}_{n}\left(\frac{1}{x}\right)=\frac{(-1)^{n}}{n!}x^{n+\alpha+1}e^{\ifrac{1}{x}}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{n}}\left(x^{-\alpha-1}e^{-\ifrac{1}{x}}\right)}}$ \LaguerrepolyL[\alpha]{n}@{\frac{1}{x}} = \frac{(-1)^{n}}{n!}x^{n+\alpha+1}e^{\ifrac{1}{x}}\deriv[n]{}{x}\left(x^{-\alpha-1}e^{-\ifrac{1}{x}}\right)		LaguerreL(n, alpha, (1)/(x)) = ((- 1)^(n))/(factorial(n))*(x)^(n + alpha + 1)* exp((1)/(x))*diff((x)^(- alpha - 1)* exp(-(1)/(x)), [x$(n)])	LaguerreL[n, \[Alpha], Divide[1,x]] == Divide[(- 1)^(n),(n)!]*(x)^(n + \[Alpha]+ 1)* Exp[Divide[1,x]]*D[(x)^(- \[Alpha]- 1)* Exp[-Divide[1,x]], {x, n}]	Missing Macro Error	Failure	-	Failed [24 / 27] Result: Plus[1.8333333333333335, Times[1.9477340410546757, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[-1, , Plus[-1, Times[-1, ], 1], Plus[Times[-1, ], 1], 1.5, []], Times[Plus[-1, Times[-1, ], 1], Plus[, Times[-1, 1], Times[-2, , 1.5], Times[-3, Power[, 2], 1.5], Times[2, 1, 1.5], Times[3, , 1, 1.5], Times[-1, 1.5], Times[2, 1.5, 1.5], Times[2, , 1.5, 1.5]], [Plus[1, ]]], Times[-1, Plus[Times[-1, ], 1, 1.5], Plus[1, , Times[-1, 1], Times[-4, 1.5], Times[-7, , 1.5], Times[-3, Power[, 2], 1.5], Times[4, 1, 1.5], Times[3, , 1, 1.5], Times[2, 1.5, 1.5], Times[, 1.5, 1.5]], [Plus[2, ]]], Times[Plus[2, ], 1.5, Plus[-1, Times[-1, ], 1, 1.5], Plus[Times[-1, ], 1, 1.5], [Plus[3, ]]]], 0], Equal[[-1], 0], Equal[[0], 0], Equal[[1], Times[Power[E, Times[-1, Power[1.5, -1]]], Binomial[Plus[-1, Times[-1, 1.5]], 1]]]}]][2.0]]], {Rule[n, 1], Rule[x, 1.5], Rule[α, 1.5]} Result: Plus[2.2638888888888893, Times[-1.9477340410546757, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[-1, , Plus[-1, Times[-1, ], 2], Plus[Times[-1, ], 2], 1.5, []], Times[Plus[-1, Times[-1, ], 2], Plus[, Times[-1, 2], Times[-2, , 1.5], Times[-3, Power[, 2], 1.5], Times[2, 2, 1.5], Times[3, , 2, 1.5], Times[-1, 1.5], Times[2, 1.5, 1.5], Times[2, , 1.5, 1.5]], [Plus[1, ]]], Times[-1, Plus[Times[-1, ], 2, 1.5], Plus[1, , Times[-1, 2], Times[-4, 1.5], Times[-7, , 1.5], Times[-3, Power[, 2], 1.5], Times[4, 2, 1.5], Times[3, , 2, 1.5], Times[2, 1.5, 1.5], Times[, 1.5, 1.5]], [Plus[2, ]]], Times[Plus[2, ], 1.5, Plus[-1, Times[-1, ], 2, 1.5], Plus[Times[-1, ], 2, 1.5], [Plus[3, ]]]], 0], Equal[[-1], 0], Equal[[0], 0], Equal[[1], Times[Power[E, Times[-1, Power[1.5, -1]]], Binomial[Plus[-1, Times[-1, 1.5]], 2]]]}]][3.0]]], {Rule[n, 2], Rule[x, 1.5], Rule[α, 1.5]} ... skip entries to safe data
18.5.E7	${\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(x\right)=\sum_{\ell=0}^{n}\frac{{\left(n+\alpha+\beta+1\right)_{\ell}}{\left(\alpha+\ell+1\right)_{n-\ell}}}{\ell!\;(n-\ell)!}\left(\frac{x-1}{2}\right)^{\ell}}}$ \JacobipolyP{\alpha}{\beta}{n}@{x} = \sum_{\ell=0}^{n}\frac{\Pochhammersym{n+\alpha+\beta+1}{\ell}\Pochhammersym{\alpha+\ell+1}{n-\ell}}{\ell!\;(n-\ell)!}\left(\frac{x-1}{2}\right)^{\ell}		JacobiP(n, alpha, beta, x) = sum((pochhammer(n + alpha + beta + 1, ell)*pochhammer(alpha + ell + 1, n - ell))/(factorial(ell)*factorial(n - ell))*((x - 1)/(2))^(ell), ell = 0..n)	JacobiP[n, \[Alpha], \[Beta], x] == Sum[Divide[Pochhammer[n + \[Alpha]+ \[Beta]+ 1, \[ScriptL]]*Pochhammer[\[Alpha]+ \[ScriptL]+ 1, n - \[ScriptL]],(\[ScriptL])!*(n - \[ScriptL])!]*(Divide[x - 1,2])^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 81]
18.5.E7	${\displaystyle{\displaystyle\sum_{\ell=0}^{n}\frac{{\left(n+\alpha+\beta+1\right)_{\ell}}{\left(\alpha+\ell+1\right)_{n-\ell}}}{\ell!\;(n-\ell)!}\left(\frac{x-1}{2}\right)^{\ell}=\frac{{\left(\alpha+1\right)_{n}}}{n!}{{}_{2}F_{1}}\left({-n,n+\alpha+\beta+1\atop\alpha+1};\frac{1-x}{2}\right)}}$ \sum_{\ell=0}^{n}\frac{\Pochhammersym{n+\alpha+\beta+1}{\ell}\Pochhammersym{\alpha+\ell+1}{n-\ell}}{\ell!\;(n-\ell)!}\left(\frac{x-1}{2}\right)^{\ell} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}\genhyperF{2}{1}@@{-n,n+\alpha+\beta+1}{\alpha+1}{\frac{1-x}{2}}		sum((pochhammer(n + alpha + beta + 1, ell)*pochhammer(alpha + ell + 1, n - ell))/(factorial(ell)*factorial(n - ell))*((x - 1)/(2))^(ell), ell = 0..n) = (pochhammer(alpha + 1, n))/(factorial(n))*hypergeom([- n , n + alpha + beta + 1], [alpha + 1], (1 - x)/(2))	Sum[Divide[Pochhammer[n + \[Alpha]+ \[Beta]+ 1, \[ScriptL]]*Pochhammer[\[Alpha]+ \[ScriptL]+ 1, n - \[ScriptL]],(\[ScriptL])!*(n - \[ScriptL])!]*(Divide[x - 1,2])^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None] == Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*HypergeometricPFQ[{- n , n + \[Alpha]+ \[Beta]+ 1}, {\[Alpha]+ 1}, Divide[1 - x,2]]	Successful	Successful	-	Successful [Tested: 81]
18.5.E8	${\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(x\right)=2^{-n}\sum_{\ell=0}^{n}\genfrac{(}{)}{0.0pt}{}{n+\alpha}{\ell}\genfrac{(}{)}{0.0pt}{}{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell}}}$ \JacobipolyP{\alpha}{\beta}{n}@{x} = 2^{-n}\sum_{\ell=0}^{n}\binom{n+\alpha}{\ell}\binom{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell}		JacobiP(n, alpha, beta, x) = (2)^(- n)* sum(binomial(n + alpha,ell)*binomial(n + beta,n - ell)*(x - 1)^(n - ell)*(x + 1)^(ell), ell = 0..n)	JacobiP[n, \[Alpha], \[Beta], x] == (2)^(- n)* Sum[Binomial[n + \[Alpha],\[ScriptL]]*Binomial[n + \[Beta],n - \[ScriptL]]*(x - 1)^(n - \[ScriptL])*(x + 1)^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None]	Failure	Failure	Successful [Tested: 81]	Successful [Tested: 81]
18.5.E8	${\displaystyle{\displaystyle 2^{-n}\sum_{\ell=0}^{n}\genfrac{(}{)}{0.0pt}{}{n+\alpha}{\ell}\genfrac{(}{)}{0.0pt}{}{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell}=\frac{{\left(\alpha+1\right)_{n}}}{n!}\left(\frac{x+1}{2}\right)^{n}{{}_{2}F_{1}}\left({-n,-n-\beta\atop\alpha+1};\frac{x-1}{x+1}\right)}}$ 2^{-n}\sum_{\ell=0}^{n}\binom{n+\alpha}{\ell}\binom{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}\left(\frac{x+1}{2}\right)^{n}\genhyperF{2}{1}@@{-n,-n-\beta}{\alpha+1}{\frac{x-1}{x+1}}		(2)^(- n)* sum(binomial(n + alpha,ell)*binomial(n + beta,n - ell)*(x - 1)^(n - ell)*(x + 1)^(ell), ell = 0..n) = (pochhammer(alpha + 1, n))/(factorial(n))*((x + 1)/(2))^(n)* hypergeom([- n , - n - beta], [alpha + 1], (x - 1)/(x + 1))	(2)^(- n)* Sum[Binomial[n + \[Alpha],\[ScriptL]]*Binomial[n + \[Beta],n - \[ScriptL]]*(x - 1)^(n - \[ScriptL])*(x + 1)^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None] == Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*(Divide[x + 1,2])^(n)* HypergeometricPFQ[{- n , - n - \[Beta]}, {\[Alpha]+ 1}, Divide[x - 1,x + 1]]	Failure	Failure	Successful [Tested: 81]	Successful [Tested: 81]
18.5.E9	${\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(x\right)=\frac{{\left(2\lambda\right)_{n}}}{n!}{{}_{2}F_{1}}\left({-n,n+2\lambda\atop\lambda+\tfrac{1}{2}};\frac{1-x}{2}\right)}}$ \ultrasphpoly{\lambda}{n}@{x} = \frac{\Pochhammersym{2\lambda}{n}}{n!}\genhyperF{2}{1}@@{-n,n+2\lambda}{\lambda+\tfrac{1}{2}}{\frac{1-x}{2}}		GegenbauerC(n, lambda, x) = (pochhammer(2*lambda, n))/(factorial(n))*hypergeom([- n , n + 2*lambda], [lambda +(1)/(2)], (1 - x)/(2))	GegenbauerC[n, \[Lambda], x] == Divide[Pochhammer[2*\[Lambda], n],(n)!]*HypergeometricPFQ[{- n , n + 2*\[Lambda]}, {\[Lambda]+Divide[1,2]}, Divide[1 - x,2]]	Successful	Successful	-	Failed [15 / 90] Result: 0.375 Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[λ, -1.5]} Result: 0.4375 Test Values: {Rule[n, 3], Rule[x, 1.5], Rule[λ, -1.5]} ... skip entries to safe data
18.5.E10	${\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(x\right)=\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{(-1)^{\ell}{\left(\lambda\right)_{n-\ell}}}{\ell!\;(n-2\ell)!}(2x)^{n-2\ell}}}$ \ultrasphpoly{\lambda}{n}@{x} = \sum_{\ell=0}^{\floor{n/2}}\frac{(-1)^{\ell}\Pochhammersym{\lambda}{n-\ell}}{\ell!\;(n-2\ell)!}(2x)^{n-2\ell}		GegenbauerC(n, lambda, x) = sum(((- 1)^(ell)* pochhammer(lambda, n - ell))/(factorial(ell)*factorial(n - 2*ell))*(2*x)^(n - 2*ell), ell = 0..floor(n/2))	GegenbauerC[n, \[Lambda], x] == Sum[Divide[(- 1)^\[ScriptL]* Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!*(n - 2*\[ScriptL])!]*(2*x)^(n - 2*\[ScriptL]), {\[ScriptL], 0, Floor[n/2]}, GenerateConditions->None]	Failure	Successful	Manual Skip!	Successful [Tested: 90]
18.5.E10	${\displaystyle{\displaystyle\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{(-1)^{\ell}{\left(\lambda\right)_{n-\ell}}}{\ell!\;(n-2\ell)!}(2x)^{n-2\ell}=(2x)^{n}\frac{{\left(\lambda\right)_{n}}}{n!}{{}_{2}F_{1}}\left({-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop 1-\lambda-n};\frac{1}{x^{2}}\right)}}$ \sum_{\ell=0}^{\floor{n/2}}\frac{(-1)^{\ell}\Pochhammersym{\lambda}{n-\ell}}{\ell!\;(n-2\ell)!}(2x)^{n-2\ell} = (2x)^{n}\frac{\Pochhammersym{\lambda}{n}}{n!}\genhyperF{2}{1}@@{-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}}{1-\lambda-n}{\frac{1}{x^{2}}}		sum(((- 1)^(ell)* pochhammer(lambda, n - ell))/(factorial(ell)*factorial(n - 2*ell))*(2*x)^(n - 2*ell), ell = 0..floor(n/2)) = (2*x)^(n)*(pochhammer(lambda, n))/(factorial(n))*hypergeom([-(1)/(2)*n , -(1)/(2)*n +(1)/(2)], [1 - lambda - n], (1)/((x)^(2)))	Sum[Divide[(- 1)^\[ScriptL]* Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!*(n - 2*\[ScriptL])!]*(2*x)^(n - 2*\[ScriptL]), {\[ScriptL], 0, Floor[n/2]}, GenerateConditions->None] == (2*x)^(n)*Divide[Pochhammer[\[Lambda], n],(n)!]*HypergeometricPFQ[{-Divide[1,2]*n , -Divide[1,2]*n +Divide[1,2]}, {1 - \[Lambda]- n}, Divide[1,(x)^(2)]]	Failure	Failure	Manual Skip!	Failed [3 / 90] Result: Indeterminate Test Values: {Rule[n, 3], Rule[x, 1.5], Rule[λ, -2]} Result: Indeterminate Test Values: {Rule[n, 3], Rule[x, 0.5], Rule[λ, -2]} ... skip entries to safe data
18.5.E11	${\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(\cos\theta\right)=\sum_{\ell=0}^{n}\frac{{\left(\lambda\right)_{\ell}}{\left(\lambda\right)_{n-\ell}}}{\ell!\;(n-\ell)!}\cos\left((n-2\ell)\theta\right)}}$ \ultrasphpoly{\lambda}{n}@{\cos@@{\theta}} = \sum_{\ell=0}^{n}\frac{\Pochhammersym{\lambda}{\ell}\Pochhammersym{\lambda}{n-\ell}}{\ell!\;(n-\ell)!}\cos@{(n-2\ell)\theta}		GegenbauerC(n, lambda, cos(theta)) = sum((pochhammer(lambda, ell)*pochhammer(lambda, n - ell))/(factorial(ell)*factorial(n - ell))*cos((n - 2*ell)*theta), ell = 0..n)	GegenbauerC[n, \[Lambda], Cos[\[Theta]]] == Sum[Divide[Pochhammer[\[Lambda], \[ScriptL]]*Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!*(n - \[ScriptL])!]*Cos[(n - 2*\[ScriptL])*\[Theta]], {\[ScriptL], 0, n}, GenerateConditions->None]	Failure	Failure	Error	Failed [30 / 300] Result: Indeterminate Test Values: {Rule[n, 1], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, -2]} Result: Indeterminate Test Values: {Rule[n, 2], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, -2]} ... skip entries to safe data
18.5.E11	${\displaystyle{\displaystyle\sum_{\ell=0}^{n}\frac{{\left(\lambda\right)_{\ell}}{\left(\lambda\right)_{n-\ell}}}{\ell!\;(n-\ell)!}\cos\left((n-2\ell)\theta\right)=e^{\mathrm{i}n\theta}\frac{{\left(\lambda\right)_{n}}}{n!}{{}_{2}F_{1}}\left({-n,\lambda\atop 1-\lambda-n};e^{-2\mathrm{i}\theta}\right)}}$ \sum_{\ell=0}^{n}\frac{\Pochhammersym{\lambda}{\ell}\Pochhammersym{\lambda}{n-\ell}}{\ell!\;(n-\ell)!}\cos@{(n-2\ell)\theta} = e^{\iunit n\theta}\frac{\Pochhammersym{\lambda}{n}}{n!}\genhyperF{2}{1}@@{-n,\lambda}{1-\lambda-n}{e^{-2\iunit\theta}}		sum((pochhammer(lambda, ell)*pochhammer(lambda, n - ell))/(factorial(ell)*factorial(n - ell))*cos((n - 2*ell)*theta), ell = 0..n) = exp(I*n*theta)*(pochhammer(lambda, n))/(factorial(n))*hypergeom([- n , lambda], [1 - lambda - n], exp(- 2*I*theta))	Sum[Divide[Pochhammer[\[Lambda], \[ScriptL]]*Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!*(n - \[ScriptL])!]*Cos[(n - 2*\[ScriptL])*\[Theta]], {\[ScriptL], 0, n}, GenerateConditions->None] == Exp[I*n*\[Theta]]*Divide[Pochhammer[\[Lambda], n],(n)!]*HypergeometricPFQ[{- n , \[Lambda]}, {1 - \[Lambda]- n}, Exp[- 2*I*\[Theta]]]	Failure	Failure	Error	Failed [30 / 300] Result: Indeterminate Test Values: {Rule[n, 1], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, -2]} Result: Indeterminate Test Values: {Rule[n, 2], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, -2]} ... skip entries to safe data
18.5.E12	${\displaystyle{\displaystyle L^{(\alpha)}_{n}\left(x\right)=\sum_{\ell=0}^{n}\frac{{\left(\alpha+\ell+1\right)_{n-\ell}}}{(n-\ell)!\;\ell!}(-x)^{\ell}}}$ \LaguerrepolyL[\alpha]{n}@{x} = \sum_{\ell=0}^{n}\frac{\Pochhammersym{\alpha+\ell+1}{n-\ell}}{(n-\ell)!\;\ell!}(-x)^{\ell}		LaguerreL(n, alpha, x) = sum((pochhammer(alpha + ell + 1, n - ell))/(factorial(n - ell)*factorial(ell))*(- x)^(ell), ell = 0..n)	LaguerreL[n, \[Alpha], x] == Sum[Divide[Pochhammer[\[Alpha]+ \[ScriptL]+ 1, n - \[ScriptL]],(n - \[ScriptL])!*(\[ScriptL])!]*(- x)^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None]	Missing Macro Error	Successful	-	Successful [Tested: 27]
18.5.E12	${\displaystyle{\displaystyle\sum_{\ell=0}^{n}\frac{{\left(\alpha+\ell+1\right)_{n-\ell}}}{(n-\ell)!\;\ell!}(-x)^{\ell}=\frac{{\left(\alpha+1\right)_{n}}}{n!}{{}_{1}F_{1}}\left({-n\atop\alpha+1};x\right)}}$ \sum_{\ell=0}^{n}\frac{\Pochhammersym{\alpha+\ell+1}{n-\ell}}{(n-\ell)!\;\ell!}(-x)^{\ell} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}\genhyperF{1}{1}@@{-n}{\alpha+1}{x}		sum((pochhammer(alpha + ell + 1, n - ell))/(factorial(n - ell)*factorial(ell))*(- x)^(ell), ell = 0..n) = (pochhammer(alpha + 1, n))/(factorial(n))*hypergeom([- n], [alpha + 1], x)	Sum[Divide[Pochhammer[\[Alpha]+ \[ScriptL]+ 1, n - \[ScriptL]],(n - \[ScriptL])!*(\[ScriptL])!]*(- x)^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None] == Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*HypergeometricPFQ[{- n}, {\[Alpha]+ 1}, x]	Successful	Successful	-	Successful [Tested: 27]
18.5.E13	${\displaystyle{\displaystyle H_{n}\left(x\right)=n!\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{(-1)^{\ell}(2x)^{n-2\ell}}{\ell!\;(n-2\ell)!}}}$ \HermitepolyH{n}@{x} = n!\sum_{\ell=0}^{\floor{n/2}}\frac{(-1)^{\ell}(2x)^{n-2\ell}}{\ell!\;(n-2\ell)!}		HermiteH(n, x) = factorial(n)*sum(((- 1)^(ell)*(2*x)^(n - 2*ell))/(factorial(ell)*factorial(n - 2*ell)), ell = 0..floor(n/2))	HermiteH[n, x] == (n)!*Sum[Divide[(- 1)^\[ScriptL]*(2*x)^(n - 2*\[ScriptL]),(\[ScriptL])!*(n - 2*\[ScriptL])!], {\[ScriptL], 0, Floor[n/2]}, GenerateConditions->None]	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
18.5.E13	${\displaystyle{\displaystyle n!\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{(-1)^{\ell}(2x)^{n-2\ell}}{\ell!\;(n-2\ell)!}=(2x)^{n}{{}_{2}F_{0}}\left({-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop-};-\frac{1}{x^{2}}\right)}}$ n!\sum_{\ell=0}^{\floor{n/2}}\frac{(-1)^{\ell}(2x)^{n-2\ell}}{\ell!\;(n-2\ell)!} = (2x)^{n}\genhyperF{2}{0}@@{-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}}{-}{-\frac{1}{x^{2}}}		factorial(n)*sum(((- 1)^(ell)*(2*x)^(n - 2*ell))/(factorial(ell)*factorial(n - 2*ell)), ell = 0..floor(n/2)) = (2*x)^(n)* hypergeom([-(1)/(2)*n , -(1)/(2)*n +(1)/(2)], [-], -(1)/((x)^(2)))	(n)!*Sum[Divide[(- 1)^\[ScriptL]*(2*x)^(n - 2*\[ScriptL]),(\[ScriptL])!*(n - 2*\[ScriptL])!], {\[ScriptL], 0, Floor[n/2]}, GenerateConditions->None] == (2*x)^(n)* HypergeometricPFQ[{-Divide[1,2]*n , -Divide[1,2]*n +Divide[1,2]}, {-}, -Divide[1,(x)^(2)]]	Error	Failure	Skip - symbolical successful subtest	Error
18.5#Ex1	${\displaystyle{\displaystyle T_{0}\left(x\right)=1}}$ \ChebyshevpolyT{0}@{x} = 1		ChebyshevT(0, x) = 1	ChebyshevT[0, x] == 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex2	${\displaystyle{\displaystyle T_{1}\left(x\right)=x}}$ \ChebyshevpolyT{1}@{x} = x		ChebyshevT(1, x) = x	ChebyshevT[1, x] == x	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex3	${\displaystyle{\displaystyle T_{2}\left(x\right)=2x^{2}-1}}$ \ChebyshevpolyT{2}@{x} = 2x^{2}-1		ChebyshevT(2, x) = 2*(x)^(2)- 1	ChebyshevT[2, x] == 2*(x)^(2)- 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex4	${\displaystyle{\displaystyle T_{3}\left(x\right)=4x^{3}-3x}}$ \ChebyshevpolyT{3}@{x} = 4x^{3}-3x		ChebyshevT(3, x) = 4*(x)^(3)- 3*x	ChebyshevT[3, x] == 4*(x)^(3)- 3*x	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex5	${\displaystyle{\displaystyle T_{4}\left(x\right)=8x^{4}-8x^{2}+1}}$ \ChebyshevpolyT{4}@{x} = 8x^{4}-8x^{2}+1		ChebyshevT(4, x) = 8*(x)^(4)- 8*(x)^(2)+ 1	ChebyshevT[4, x] == 8*(x)^(4)- 8*(x)^(2)+ 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex6	${\displaystyle{\displaystyle T_{5}\left(x\right)=16x^{5}-20x^{3}+5x}}$ \ChebyshevpolyT{5}@{x} = 16x^{5}-20x^{3}+5x		ChebyshevT(5, x) = 16*(x)^(5)- 20*(x)^(3)+ 5*x	ChebyshevT[5, x] == 16*(x)^(5)- 20*(x)^(3)+ 5*x	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex7	${\displaystyle{\displaystyle T_{6}\left(x\right)=32x^{6}-48x^{4}+18x^{2}-1}}$ \ChebyshevpolyT{6}@{x} = 32x^{6}-48x^{4}+18x^{2}-1		ChebyshevT(6, x) = 32*(x)^(6)- 48*(x)^(4)+ 18*(x)^(2)- 1	ChebyshevT[6, x] == 32*(x)^(6)- 48*(x)^(4)+ 18*(x)^(2)- 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex8	${\displaystyle{\displaystyle U_{0}\left(x\right)=1}}$ \ChebyshevpolyU{0}@{x} = 1		ChebyshevU(0, x) = 1	ChebyshevU[0, x] == 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex9	${\displaystyle{\displaystyle U_{1}\left(x\right)=2x}}$ \ChebyshevpolyU{1}@{x} = 2x		ChebyshevU(1, x) = 2*x	ChebyshevU[1, x] == 2*x	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex10	${\displaystyle{\displaystyle U_{2}\left(x\right)=4x^{2}-1}}$ \ChebyshevpolyU{2}@{x} = 4x^{2}-1		ChebyshevU(2, x) = 4*(x)^(2)- 1	ChebyshevU[2, x] == 4*(x)^(2)- 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex11	${\displaystyle{\displaystyle U_{3}\left(x\right)=8x^{3}-4x}}$ \ChebyshevpolyU{3}@{x} = 8x^{3}-4x		ChebyshevU(3, x) = 8*(x)^(3)- 4*x	ChebyshevU[3, x] == 8*(x)^(3)- 4*x	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex12	${\displaystyle{\displaystyle U_{4}\left(x\right)=16x^{4}-12x^{2}+1}}$ \ChebyshevpolyU{4}@{x} = 16x^{4}-12x^{2}+1		ChebyshevU(4, x) = 16*(x)^(4)- 12*(x)^(2)+ 1	ChebyshevU[4, x] == 16*(x)^(4)- 12*(x)^(2)+ 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex13	${\displaystyle{\displaystyle U_{5}\left(x\right)=32x^{5}-32x^{3}+6x}}$ \ChebyshevpolyU{5}@{x} = 32x^{5}-32x^{3}+6x		ChebyshevU(5, x) = 32*(x)^(5)- 32*(x)^(3)+ 6*x	ChebyshevU[5, x] == 32*(x)^(5)- 32*(x)^(3)+ 6*x	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex14	${\displaystyle{\displaystyle U_{6}\left(x\right)=64x^{6}-80x^{4}+24x^{2}-1}}$ \ChebyshevpolyU{6}@{x} = 64x^{6}-80x^{4}+24x^{2}-1		ChebyshevU(6, x) = 64*(x)^(6)- 80*(x)^(4)+ 24*(x)^(2)- 1	ChebyshevU[6, x] == 64*(x)^(6)- 80*(x)^(4)+ 24*(x)^(2)- 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex15	${\displaystyle{\displaystyle P_{0}\left(x\right)=1}}$ \LegendrepolyP{0}@{x} = 1		LegendreP(0, x) = 1	LegendreP[0, x] == 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex16	${\displaystyle{\displaystyle P_{1}\left(x\right)=x}}$ \LegendrepolyP{1}@{x} = x		LegendreP(1, x) = x	LegendreP[1, x] == x	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex17	${\displaystyle{\displaystyle P_{2}\left(x\right)=\tfrac{3}{2}x^{2}-\tfrac{1}{2}}}$ \LegendrepolyP{2}@{x} = \tfrac{3}{2}x^{2}-\tfrac{1}{2}		LegendreP(2, x) = (3)/(2)*(x)^(2)-(1)/(2)	LegendreP[2, x] == Divide[3,2]*(x)^(2)-Divide[1,2]	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex18	${\displaystyle{\displaystyle P_{3}\left(x\right)=\tfrac{5}{2}x^{3}-\tfrac{3}{2}x}}$ \LegendrepolyP{3}@{x} = \tfrac{5}{2}x^{3}-\tfrac{3}{2}x		LegendreP(3, x) = (5)/(2)*(x)^(3)-(3)/(2)*x	LegendreP[3, x] == Divide[5,2]*(x)^(3)-Divide[3,2]*x	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex19	${\displaystyle{\displaystyle P_{4}\left(x\right)=\tfrac{35}{8}x^{4}-\tfrac{15}{4}x^{2}+\tfrac{3}{8}}}$ \LegendrepolyP{4}@{x} = \tfrac{35}{8}x^{4}-\tfrac{15}{4}x^{2}+\tfrac{3}{8}		LegendreP(4, x) = (35)/(8)*(x)^(4)-(15)/(4)*(x)^(2)+(3)/(8)	LegendreP[4, x] == Divide[35,8]*(x)^(4)-Divide[15,4]*(x)^(2)+Divide[3,8]	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex20	${\displaystyle{\displaystyle P_{5}\left(x\right)=\tfrac{63}{8}x^{5}-\tfrac{35}{4}x^{3}+\tfrac{15}{8}x}}$ \LegendrepolyP{5}@{x} = \tfrac{63}{8}x^{5}-\tfrac{35}{4}x^{3}+\tfrac{15}{8}x		LegendreP(5, x) = (63)/(8)*(x)^(5)-(35)/(4)*(x)^(3)+(15)/(8)*x	LegendreP[5, x] == Divide[63,8]*(x)^(5)-Divide[35,4]*(x)^(3)+Divide[15,8]*x	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex21	${\displaystyle{\displaystyle P_{6}\left(x\right)=\tfrac{231}{16}x^{6}-\tfrac{315}{16}x^{4}+\tfrac{105}{16}x^{2}-\tfrac{5}{16}}}$ \LegendrepolyP{6}@{x} = \tfrac{231}{16}x^{6}-\tfrac{315}{16}x^{4}+\tfrac{105}{16}x^{2}-\tfrac{5}{16}		LegendreP(6, x) = (231)/(16)*(x)^(6)-(315)/(16)*(x)^(4)+(105)/(16)*(x)^(2)-(5)/(16)	LegendreP[6, x] == Divide[231,16]*(x)^(6)-Divide[315,16]*(x)^(4)+Divide[105,16]*(x)^(2)-Divide[5,16]	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex22	${\displaystyle{\displaystyle L_{0}\left(x\right)=1}}$ \LaguerrepolyL[]{0}@{x} = 1		LaguerreL(0, x) = 1	LaguerreL[0, x] == 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex23	${\displaystyle{\displaystyle L_{1}\left(x\right)=-x+1}}$ \LaguerrepolyL[]{1}@{x} = -x+1		LaguerreL(1, x) = - x + 1	LaguerreL[1, x] == - x + 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex24	${\displaystyle{\displaystyle L_{2}\left(x\right)=\tfrac{1}{2}x^{2}-2x+1}}$ \LaguerrepolyL[]{2}@{x} = \tfrac{1}{2}x^{2}-2x+1		LaguerreL(2, x) = (1)/(2)*(x)^(2)- 2*x + 1	LaguerreL[2, x] == Divide[1,2]*(x)^(2)- 2*x + 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex25	${\displaystyle{\displaystyle L_{3}\left(x\right)=-\tfrac{1}{6}x^{3}+\tfrac{3}{2}x^{2}-3x+1}}$ \LaguerrepolyL[]{3}@{x} = -\tfrac{1}{6}x^{3}+\tfrac{3}{2}x^{2}-3x+1		LaguerreL(3, x) = -(1)/(6)*(x)^(3)+(3)/(2)*(x)^(2)- 3*x + 1	LaguerreL[3, x] == -Divide[1,6]*(x)^(3)+Divide[3,2]*(x)^(2)- 3*x + 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex26	${\displaystyle{\displaystyle L_{4}\left(x\right)=\tfrac{1}{24}x^{4}-\tfrac{2}{3}x^{3}+3x^{2}-4x+1}}$ \LaguerrepolyL[]{4}@{x} = \tfrac{1}{24}x^{4}-\tfrac{2}{3}x^{3}+3x^{2}-4x+1		LaguerreL(4, x) = (1)/(24)*(x)^(4)-(2)/(3)*(x)^(3)+ 3*(x)^(2)- 4*x + 1	LaguerreL[4, x] == Divide[1,24]*(x)^(4)-Divide[2,3]*(x)^(3)+ 3*(x)^(2)- 4*x + 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex27	${\displaystyle{\displaystyle L_{5}\left(x\right)=-\tfrac{1}{120}x^{5}+\tfrac{5}{24}x^{4}-\tfrac{5}{3}x^{3}+5x^{2}-5x+1}}$ \LaguerrepolyL[]{5}@{x} = -\tfrac{1}{120}x^{5}+\tfrac{5}{24}x^{4}-\tfrac{5}{3}x^{3}+5x^{2}-5x+1		LaguerreL(5, x) = -(1)/(120)*(x)^(5)+(5)/(24)*(x)^(4)-(5)/(3)*(x)^(3)+ 5*(x)^(2)- 5*x + 1	LaguerreL[5, x] == -Divide[1,120]*(x)^(5)+Divide[5,24]*(x)^(4)-Divide[5,3]*(x)^(3)+ 5*(x)^(2)- 5*x + 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex28	${\displaystyle{\displaystyle L_{6}\left(x\right)=\tfrac{1}{720}x^{6}-\tfrac{1}{20}x^{5}+\tfrac{5}{8}x^{4}-\tfrac{10}{3}x^{3}+\tfrac{15}{2}x^{2}-6x+1}}$ \LaguerrepolyL[]{6}@{x} = \tfrac{1}{720}x^{6}-\tfrac{1}{20}x^{5}+\tfrac{5}{8}x^{4}-\tfrac{10}{3}x^{3}+\tfrac{15}{2}x^{2}-6x+1		LaguerreL(6, x) = (1)/(720)*(x)^(6)-(1)/(20)*(x)^(5)+(5)/(8)*(x)^(4)-(10)/(3)*(x)^(3)+(15)/(2)*(x)^(2)- 6*x + 1	LaguerreL[6, x] == Divide[1,720]*(x)^(6)-Divide[1,20]*(x)^(5)+Divide[5,8]*(x)^(4)-Divide[10,3]*(x)^(3)+Divide[15,2]*(x)^(2)- 6*x + 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex29	${\displaystyle{\displaystyle H_{0}\left(x\right)=1}}$ \HermitepolyH{0}@{x} = 1		HermiteH(0, x) = 1	HermiteH[0, x] == 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex30	${\displaystyle{\displaystyle H_{1}\left(x\right)=2x}}$ \HermitepolyH{1}@{x} = 2x		HermiteH(1, x) = 2*x	HermiteH[1, x] == 2*x	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex31	${\displaystyle{\displaystyle H_{2}\left(x\right)=4x^{2}-2}}$ \HermitepolyH{2}@{x} = 4x^{2}-2		HermiteH(2, x) = 4*(x)^(2)- 2	HermiteH[2, x] == 4*(x)^(2)- 2	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex32	${\displaystyle{\displaystyle H_{3}\left(x\right)=8x^{3}-12x}}$ \HermitepolyH{3}@{x} = 8x^{3}-12x		HermiteH(3, x) = 8*(x)^(3)- 12*x	HermiteH[3, x] == 8*(x)^(3)- 12*x	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex33	${\displaystyle{\displaystyle H_{4}\left(x\right)=16x^{4}-48x^{2}+12}}$ \HermitepolyH{4}@{x} = 16x^{4}-48x^{2}+12		HermiteH(4, x) = 16*(x)^(4)- 48*(x)^(2)+ 12	HermiteH[4, x] == 16*(x)^(4)- 48*(x)^(2)+ 12	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex34	${\displaystyle{\displaystyle H_{5}\left(x\right)=32x^{5}-160x^{3}+120x}}$ \HermitepolyH{5}@{x} = 32x^{5}-160x^{3}+120x		HermiteH(5, x) = 32*(x)^(5)- 160*(x)^(3)+ 120*x	HermiteH[5, x] == 32*(x)^(5)- 160*(x)^(3)+ 120*x	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex35	${\displaystyle{\displaystyle H_{6}\left(x\right)=64x^{6}-480x^{4}+720x^{2}-120}}$ \HermitepolyH{6}@{x} = 64x^{6}-480x^{4}+720x^{2}-120		HermiteH(6, x) = 64*(x)^(6)- 480*(x)^(4)+ 720*(x)^(2)- 120	HermiteH[6, x] == 64*(x)^(6)- 480*(x)^(4)+ 720*(x)^(2)- 120	Successful	Successful	-	Successful [Tested: 3]