Results of Orthogonal Polynomials

DLMF	Formula	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
18.1.E1	${\displaystyle{\displaystyle C^{(0)}_{n}\left(x\right)=\frac{2}{n}T_{n}\left(x% \right)}}$	`GegenbauerC(n, 0, x)=(2)/(n)*ChebyshevT(n, x)`	`GegenbauerC[n, 0, x]=Divide[2,n]*ChebyshevT[n, x]`	Failure	Failure	Successful	Fail `-0.6666666666666666 <- {Rule[n, 3], Rule[x, 1]}` `-17.333333333333332 <- {Rule[n, 3], Rule[x, 2]}` `-66.0 <- {Rule[n, 3], Rule[x, 3]}`
18.1.E1	${\displaystyle{\displaystyle\frac{2}{n}T_{n}\left(x\right)=\frac{2(n-1)!}{{% \left(\tfrac{1}{2}\right)_{n}}}P^{(-\frac{1}{2},-\frac{1}{2})}_{n}\left(x% \right)}}$	`(2)/(n)ChebyshevT(n, x)=(2factorial(n - 1))/(pochhammer((1)/(2), n))*JacobiP(n, -(1)/(2), -(1)/(2), x)`	`Divide[2,n]ChebyshevT[n, x]=Divide[2(n - 1)!,Pochhammer[Divide[1,2], n]]*JacobiP[n, -Divide[1,2], -Divide[1,2], x]`	Successful	Successful	-	-
18.1.E2	${\displaystyle{\displaystyle G_{n}\left(p,q,x\right)=\frac{n!}{{\left(n+p% \right)_{n}}}P^{(p-q,q-1)}_{n}\left(2x-1\right)}}$	`JacobiP(n, p-q, q-1, 2(x)-1)((n)!)/pochhammer(n+p, n)=(factorial(n))/(pochhammer(n + p, n))JacobiP(n, p - q, q - 1, 2x - 1)`	`Error`	Successful	Error	-	-
18.2.E1	${\displaystyle{\displaystyle\int_{a}^{b}p_{n}(x)p_{m}(x)w(x)\mathrm{d}x=0}}$	`int(p[n](x) p[m](x) w*(x), x = a..b)= 0`	`Integrate[Subscript[p, n](x) Subscript[p, m](x) w*(x), {x, a, b}]= 0`	Failure	Failure	Skip	Successful
18.3.E2	${\displaystyle{\displaystyle x_{N+1,n}=\cos\left((n-\tfrac{1}{2})\pi/(N+1)% \right)}}$	`x[N + 1 , n]= cos((n -(1)/(2))* Pi/(N + 1))`	`Subscript[x, N + 1 , n]= Cos[(n -Divide[1,2])* Pi/(N + 1)]`	Failure	Failure	Fail `.4933988023+1.280284738I <- {N = 2^(1/2)+I2^(1/2), x[N+1,n] = 2^(1/2)+I2^(1/2), n = 1}` `1.251822237+.4629104109I <- {N = 2^(1/2)+I2^(1/2), x[N+1,n] = 2^(1/2)+I2^(1/2), n = 2}` `3.059241197+.132349918I <- {N = 2^(1/2)+I2^(1/2), x[N+1,n] = 2^(1/2)+I2^(1/2), n = 3}` `.4933988023-1.548142386I <- {N = 2^(1/2)+I2^(1/2), x[N+1,n] = 2^(1/2)-I2^(1/2), n = 1}` ... skip entries to safe data	Successful
18.5.E1	${\displaystyle{\displaystyle T_{n}\left(x\right)=\cos\left(n\theta\right)}}$	`ChebyshevT(n, x)= cos(n*theta)`	`ChebyshevT[n, x]= Cos[n*\[Theta]]`	Failure	Failure	Fail `.6603260076+1.911393109I <- {theta = 2^(1/2)+I2^(1/2), n = 1, x = 1}` `1.660326008+1.911393109I <- {theta = 2^(1/2)+I2^(1/2), n = 1, x = 2}` `2.660326008+1.911393109I <- {theta = 2^(1/2)+I2^(1/2), n = 1, x = 3}` `9.076090394+2.597002114I <- {theta = 2^(1/2)+I2^(1/2), n = 2, x = 1}` ... skip entries to safe data	Fail `Complex[0.6603260083052754, 1.9113931101642103] <- {Rule[n, 1], Rule[x, 1], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[1.6603260083052755, 1.9113931101642103] <- {Rule[n, 1], Rule[x, 2], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[2.6603260083052755, 1.9113931101642103] <- {Rule[n, 1], Rule[x, 3], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[9.076090401898771, 2.5970021097090865] <- {Rule[n, 2], Rule[x, 1], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` ... skip entries to safe data
18.5.E2	${\displaystyle{\displaystyle U_{n}\left(x\right)=\ifrac{(\sin(n+1)\theta)}{% \sin\theta}}}$	`ChebyshevU(n, x)=(sin((n + 1)* theta))/(sin(theta))`	`ChebyshevU[n, x]=Divide[Sin[(n + 1)* \[Theta]],Sin[\[Theta]]]`	Failure	Failure	Fail `1.320652015+3.822786219I <- {theta = 2^(1/2)+I2^(1/2), n = 1, x = 1}` `3.320652015+3.822786219I <- {theta = 2^(1/2)+I2^(1/2), n = 1, x = 2}` `5.320652015+3.822786219I <- {theta = 2^(1/2)+I2^(1/2), n = 1, x = 3}` `18.15218079+5.194004229I <- {theta = 2^(1/2)+I2^(1/2), n = 2, x = 1}` ... skip entries to safe data	Fail `Complex[1.3206520166105502, 3.82278622032842] <- {Rule[n, 1], Rule[x, 1], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[3.32065201661055, 3.82278622032842] <- {Rule[n, 1], Rule[x, 2], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[5.32065201661055, 3.82278622032842] <- {Rule[n, 1], Rule[x, 3], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[18.152180803797542, 5.194004219418172] <- {Rule[n, 2], Rule[x, 1], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` ... skip entries to safe data
18.5.E5	${\displaystyle{\displaystyle p_{n}(x)=\frac{1}{\kappa_{n}w(x)}\frac{{\mathrm{d% }}^{n}}{{\mathrm{d}x}^{n}}\left(w(x)(F(x))^{n}\right)}}$	`p[n](x)=(1)/(kappa[n]w(x))diff(w(x)(F*(x))^(n), [x$(n)])`	`Subscript[p, n](x)=Divide[1,Subscript[\[Kappa], n]w(x)]D[w(x)(F*(x))^(n), {x, n}]`	Failure	Failure	Skip	Skip
18.5.E7	${\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(x\right)=\sum_{\ell=0% }^{n}\frac{{\left(n+\alpha+\beta+1\right)_{\ell}}{\left(\alpha+\ell+1\right)_{% n-\ell}}}{\ell!\;(n-\ell)!}\left(\frac{x-1}{2}\right)^{\ell}}}$	`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}]`	Successful	Successful	-	-
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((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}]=Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]HypergeometricPFQ[{- n , n + \[Alpha]+ \[Beta]+ 1}, {\[Alpha]+ 1}, Divide[1 - x,2]]`	Successful	Successful	-	-
18.5.E8	${\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(x\right)=2^{-n}\sum_{% \ell=0}^{n}\genfrac{(}{)}{0.0pt}{}{n+\alpha}{\ell}\genfrac{(}{)}{0.0pt}{}{n+% \beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell}}}$	`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}]`	Failure	Failure	Skip	Skip
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(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}]=Divide[Pochhammer[\[Alpha]+ 1, n],(n)!](Divide[x + 1,2])^(n)* HypergeometricPFQ[{- n , - n - \[Beta]}, {\[Alpha]+ 1}, Divide[x - 1,x + 1]]`	Failure	Failure	Skip	Skip
18.5.E9	${\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(x\right)=\frac{{\left(2% \lambda\right)_{n}}}{n!}{{}_{2}F_{1}}\left({-n,n+2\lambda\atop\lambda+\tfrac{1% }{2}};\frac{1-x}{2}\right)}}$	`GegenbauerC(n, lambda, x)=(pochhammer(2lambda, 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	-	-
18.5.E10	${\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(x\right)=\sum_{\ell=0}^{% \left\lfloor n/2\right\rfloor}\frac{(-1)^{\ell}{\left(\lambda\right)_{n-\ell}}% }{\ell!\;(n-2\ell)!}(2x)^{n-2\ell}}}$	`GegenbauerC(n, lambda, x)= sum(((- 1)^(ell)* pochhammer(lambda, n - ell))/(factorial(ell)factorial(n - 2ell))(2x)^(n - 2*ell), ell = 0..floor(n/ 2))`	`GegenbauerC[n, \[Lambda], x]= Sum[Divide[(- 1)^(\[ScriptL])* Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!(n - 2\[ScriptL])!](2x)^(n - 2*\[ScriptL]), {\[ScriptL], 0, Floor[n/ 2]}]`	Failure	Successful	Skip	-
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(((- 1)^(ell)* pochhammer(lambda, n - ell))/(factorial(ell)factorial(n - 2ell))(2x)^(n - 2ell), ell = 0..floor(n/ 2))=(2x)^(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])!](2x)^(n - 2\[ScriptL]), {\[ScriptL], 0, Floor[n/ 2]}]=(2x)^(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	Skip	Successful
18.5.E11	${\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(\cos\theta\right)=\sum_{% \ell=0}^{n}\frac{{\left(\lambda\right)_{\ell}}{\left(\lambda\right)_{n-\ell}}}% {\ell!\;(n-\ell)!}\cos\left((n-2\ell)\theta\right)}}$	`GegenbauerC(n, lambda, cos(theta))= sum((pochhammer(lambda, ell)pochhammer(lambda, n - ell))/(factorial(ell)factorial(n - ell))cos((n - 2ell)* 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}]`	Failure	Failure	Skip	Successful
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((pochhammer(lambda, ell)pochhammer(lambda, n - ell))/(factorial(ell)factorial(n - ell))cos((n - 2ell)* theta), ell = 0..n)= exp(Intheta)(pochhammer(lambda, n))/(factorial(n))hypergeom([- n , lambda], [1 - lambda - n], exp(- 2Itheta))`	`Sum[Divide[Pochhammer[\[Lambda], \[ScriptL]]Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!(n - \[ScriptL])!]Cos[(n - 2\[ScriptL])* \[Theta]], {\[ScriptL], 0, n}]= Exp[In\[Theta]]Divide[Pochhammer[\[Lambda], n],(n)!]HypergeometricPFQ[{- n , \[Lambda]}, {1 - \[Lambda]- n}, Exp[- 2I\[Theta]]]`	Failure	Failure	Skip	Skip
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((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}]=Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*HypergeometricPFQ[{- n}, {\[Alpha]+ 1}, x]`	Successful	Successful	-	-
18.5.E13	${\displaystyle{\displaystyle H_{n}\left(x\right)=n!\sum_{\ell=0}^{\left\lfloor n% /2\right\rfloor}\frac{(-1)^{\ell}(2x)^{n-2\ell}}{\ell!\;(n-2\ell)!}}}$	`HermiteH(n, x)= factorial(n)sum(((- 1)^(ell)(2x)^(n - 2ell))/(factorial(ell)factorial(n - 2ell)), ell = 0..floor(n/ 2))`	`HermiteH[n, x]= (n)!Sum[Divide[(- 1)^(\[ScriptL])(2x)^(n - 2\[ScriptL]),(\[ScriptL])!(n - 2\[ScriptL])!], {\[ScriptL], 0, Floor[n/ 2]}]`	Failure	Failure	Skip	Successful
18.5.E13	${\displaystyle{\displaystyle n!\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}% \frac{(-1)^{\ell}(2x)^{n-2\ell}}{\ell!\;(n-2\ell)!}=(2x)^{n}{{}_{2}F_{0}}\left% ({-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop-};-\frac{1}{x^{2}}\right)}}$	`factorial(n)sum(((- 1)^(ell)(2x)^(n - 2ell))/(factorial(ell)factorial(n - 2ell)), ell = 0..floor(n/ 2))=(2x)^(n) hypergeom([-(1)/(2)n , -(1)/(2)n +(1)/(2)], [-], -(1)/((x)^(2)))`	`(n)!Sum[Divide[(- 1)^(\[ScriptL])(2x)^(n - 2\[ScriptL]),(\[ScriptL])!(n - 2\[ScriptL])!], {\[ScriptL], 0, Floor[n/ 2]}]=(2x)^(n) HypergeometricPFQ[{-Divide[1,2]n , -Divide[1,2]n +Divide[1,2]}, {-}, -Divide[1,(x)^(2)]]`	Error	Failure	-	Error
18.5#Ex1	${\displaystyle{\displaystyle T_{0}\left(x\right)=1}}$	`ChebyshevT(0, x)= 1`	`ChebyshevT[0, x]= 1`	Successful	Successful	-	-
18.5#Ex2	${\displaystyle{\displaystyle T_{1}\left(x\right)=x}}$	`ChebyshevT(1, x)= x`	`ChebyshevT[1, x]= x`	Successful	Successful	-	-
18.5#Ex3	${\displaystyle{\displaystyle T_{2}\left(x\right)=2x^{2}-1}}$	`ChebyshevT(2, x)= 2*(x)^(2)- 1`	`ChebyshevT[2, x]= 2*(x)^(2)- 1`	Successful	Successful	-	-
18.5#Ex4	${\displaystyle{\displaystyle T_{3}\left(x\right)=4x^{3}-3x}}$	`ChebyshevT(3, x)= 4(x)^(3)- 3x`	`ChebyshevT[3, x]= 4(x)^(3)- 3x`	Successful	Successful	-	-
18.5#Ex5	${\displaystyle{\displaystyle T_{4}\left(x\right)=8x^{4}-8x^{2}+1}}$	`ChebyshevT(4, x)= 8(x)^(4)- 8(x)^(2)+ 1`	`ChebyshevT[4, x]= 8(x)^(4)- 8(x)^(2)+ 1`	Successful	Successful	-	-
18.5#Ex6	${\displaystyle{\displaystyle T_{5}\left(x\right)=16x^{5}-20x^{3}+5x}}$	`ChebyshevT(5, x)= 16(x)^(5)- 20(x)^(3)+ 5*x`	`ChebyshevT[5, x]= 16(x)^(5)- 20(x)^(3)+ 5*x`	Successful	Successful	-	-
18.5#Ex7	${\displaystyle{\displaystyle T_{6}\left(x\right)=32x^{6}-48x^{4}+18x^{2}-1}}$	`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	-	-
18.5#Ex8	${\displaystyle{\displaystyle U_{0}\left(x\right)=1}}$	`ChebyshevU(0, x)= 1`	`ChebyshevU[0, x]= 1`	Successful	Successful	-	-
18.5#Ex9	${\displaystyle{\displaystyle U_{1}\left(x\right)=2x}}$	`ChebyshevU(1, x)= 2*x`	`ChebyshevU[1, x]= 2*x`	Successful	Successful	-	-
18.5#Ex10	${\displaystyle{\displaystyle U_{2}\left(x\right)=4x^{2}-1}}$	`ChebyshevU(2, x)= 4*(x)^(2)- 1`	`ChebyshevU[2, x]= 4*(x)^(2)- 1`	Successful	Successful	-	-
18.5#Ex11	${\displaystyle{\displaystyle U_{3}\left(x\right)=8x^{3}-4x}}$	`ChebyshevU(3, x)= 8(x)^(3)- 4x`	`ChebyshevU[3, x]= 8(x)^(3)- 4x`	Successful	Successful	-	-
18.5#Ex12	${\displaystyle{\displaystyle U_{4}\left(x\right)=16x^{4}-12x^{2}+1}}$	`ChebyshevU(4, x)= 16(x)^(4)- 12(x)^(2)+ 1`	`ChebyshevU[4, x]= 16(x)^(4)- 12(x)^(2)+ 1`	Successful	Successful	-	-
18.5#Ex13	${\displaystyle{\displaystyle U_{5}\left(x\right)=32x^{5}-32x^{3}+6x}}$	`ChebyshevU(5, x)= 32(x)^(5)- 32(x)^(3)+ 6*x`	`ChebyshevU[5, x]= 32(x)^(5)- 32(x)^(3)+ 6*x`	Successful	Successful	-	-
18.5#Ex14	${\displaystyle{\displaystyle U_{6}\left(x\right)=64x^{6}-80x^{4}+24x^{2}-1}}$	`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	-	-
18.5#Ex15	${\displaystyle{\displaystyle P_{0}\left(x\right)=1}}$	`LegendreP(0, x)= 1`	`LegendreP[0, x]= 1`	Successful	Successful	-	-
18.5#Ex16	${\displaystyle{\displaystyle P_{1}\left(x\right)=x}}$	`LegendreP(1, x)= x`	`LegendreP[1, x]= x`	Successful	Successful	-	-
18.5#Ex17	${\displaystyle{\displaystyle P_{2}\left(x\right)=\tfrac{3}{2}x^{2}-\tfrac{1}{2% }}}$	`LegendreP(2, x)=(3)/(2)*(x)^(2)-(1)/(2)`	`LegendreP[2, x]=Divide[3,2]*(x)^(2)-Divide[1,2]`	Successful	Successful	-	-
18.5#Ex18	${\displaystyle{\displaystyle P_{3}\left(x\right)=\tfrac{5}{2}x^{3}-\tfrac{3}{2% }x}}$	`LegendreP(3, x)=(5)/(2)(x)^(3)-(3)/(2)x`	`LegendreP[3, x]=Divide[5,2](x)^(3)-Divide[3,2]x`	Successful	Successful	-	-
18.5#Ex19	${\displaystyle{\displaystyle P_{4}\left(x\right)=\tfrac{35}{8}x^{4}-\tfrac{15}% {4}x^{2}+\tfrac{3}{8}}}$	`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	-	-
18.5#Ex20	${\displaystyle{\displaystyle P_{5}\left(x\right)=\tfrac{63}{8}x^{5}-\tfrac{35}% {4}x^{3}+\tfrac{15}{8}x}}$	`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	-	-
18.5#Ex21	${\displaystyle{\displaystyle P_{6}\left(x\right)=\tfrac{231}{16}x^{6}-\tfrac{3% 15}{16}x^{4}+\tfrac{105}{16}x^{2}-\tfrac{5}{16}}}$	`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	-	-
18.5#Ex22	${\displaystyle{\displaystyle L_{0}\left(x\right)=1}}$	`LaguerreL(0, x)= 1`	`Error`	Successful	Error	-	-
18.5#Ex23	${\displaystyle{\displaystyle L_{1}\left(x\right)=-x+1}}$	`LaguerreL(1, x)= - x + 1`	`Error`	Successful	Error	-	-
18.5#Ex24	${\displaystyle{\displaystyle L_{2}\left(x\right)=\tfrac{1}{2}x^{2}-2x+1}}$	`LaguerreL(2, x)=(1)/(2)(x)^(2)- 2x + 1`	`Error`	Successful	Error	-	-
18.5#Ex25	${\displaystyle{\displaystyle L_{3}\left(x\right)=-\tfrac{1}{6}x^{3}+\tfrac{3}{% 2}x^{2}-3x+1}}$	`LaguerreL(3, x)= -(1)/(6)(x)^(3)+(3)/(2)(x)^(2)- 3*x + 1`	`Error`	Successful	Error	-	-
18.5#Ex26	${\displaystyle{\displaystyle L_{4}\left(x\right)=\tfrac{1}{24}x^{4}-\tfrac{2}{% 3}x^{3}+3x^{2}-4x+1}}$	`LaguerreL(4, x)=(1)/(24)(x)^(4)-(2)/(3)(x)^(3)+ 3(x)^(2)- 4x + 1`	`Error`	Successful	Error	-	-
18.5#Ex27	${\displaystyle{\displaystyle L_{5}\left(x\right)=-\tfrac{1}{120}x^{5}+\tfrac{5% }{24}x^{4}-\tfrac{5}{3}x^{3}+5x^{2}-5x+1}}$	`LaguerreL(5, x)= -(1)/(120)(x)^(5)+(5)/(24)(x)^(4)-(5)/(3)(x)^(3)+ 5(x)^(2)- 5*x + 1`	`Error`	Successful	Error	-	-
18.5#Ex28	${\displaystyle{\displaystyle L_{6}\left(x\right)=\tfrac{1}{720}x^{6}-\tfrac{1}% {20}x^{5}+\tfrac{5}{8}x^{4}-\tfrac{10}{3}x^{3}+\tfrac{15}{2}x^{2}-6x+1}}$	`LaguerreL(6, x)=(1)/(720)(x)^(6)-(1)/(20)(x)^(5)+(5)/(8)(x)^(4)-(10)/(3)(x)^(3)+(15)/(2)(x)^(2)- 6x + 1`	`Error`	Successful	Error	-	-
18.5#Ex29	${\displaystyle{\displaystyle H_{0}\left(x\right)=1}}$	`HermiteH(0, x)= 1`	`HermiteH[0, x]= 1`	Successful	Successful	-	-
18.5#Ex30	${\displaystyle{\displaystyle H_{1}\left(x\right)=2x}}$	`HermiteH(1, x)= 2*x`	`HermiteH[1, x]= 2*x`	Successful	Successful	-	-
18.5#Ex31	${\displaystyle{\displaystyle H_{2}\left(x\right)=4x^{2}-2}}$	`HermiteH(2, x)= 4*(x)^(2)- 2`	`HermiteH[2, x]= 4*(x)^(2)- 2`	Successful	Successful	-	-
18.5#Ex32	${\displaystyle{\displaystyle H_{3}\left(x\right)=8x^{3}-12x}}$	`HermiteH(3, x)= 8(x)^(3)- 12x`	`HermiteH[3, x]= 8(x)^(3)- 12x`	Successful	Successful	-	-
18.5#Ex33	${\displaystyle{\displaystyle H_{4}\left(x\right)=16x^{4}-48x^{2}+12}}$	`HermiteH(4, x)= 16(x)^(4)- 48(x)^(2)+ 12`	`HermiteH[4, x]= 16(x)^(4)- 48(x)^(2)+ 12`	Successful	Successful	-	-
18.5#Ex34	${\displaystyle{\displaystyle H_{5}\left(x\right)=32x^{5}-160x^{3}+120x}}$	`HermiteH(5, x)= 32(x)^(5)- 160(x)^(3)+ 120*x`	`HermiteH[5, x]= 32(x)^(5)- 160(x)^(3)+ 120*x`	Successful	Successful	-	-
18.5#Ex35	${\displaystyle{\displaystyle H_{6}\left(x\right)=64x^{6}-480x^{4}+720x^{2}-120}}$	`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	-	-
18.6.E2	${\displaystyle{\displaystyle\lim_{\alpha\to\infty}\frac{P^{(\alpha,\beta)}_{n}% \left(x\right)}{P^{(\alpha,\beta)}_{n}\left(1\right)}=\left(\frac{1+x}{2}% \right)^{n}}}$	`limit((JacobiP(n, alpha, beta, x))/(JacobiP(n, alpha, beta, 1)), alpha = infinity)=((1 + x)/(2))^(n)`	`Limit[Divide[JacobiP[n, \[Alpha], \[Beta], x],JacobiP[n, \[Alpha], \[Beta], 1]], \[Alpha] -> Infinity]=(Divide[1 + x,2])^(n)`	Failure	Failure	Skip	Error
18.6.E3	${\displaystyle{\displaystyle\lim_{\beta\to\infty}\frac{P^{(\alpha,\beta)}_{n}% \left(x\right)}{P^{(\alpha,\beta)}_{n}\left(-1\right)}=\left(\frac{1-x}{2}% \right)^{n}}}$	`limit((JacobiP(n, alpha, beta, x))/(JacobiP(n, alpha, beta, - 1)), beta = infinity)=((1 - x)/(2))^(n)`	`Limit[Divide[JacobiP[n, \[Alpha], \[Beta], x],JacobiP[n, \[Alpha], \[Beta], - 1]], \[Beta] -> Infinity]=(Divide[1 - x,2])^(n)`	Failure	Failure	Skip	Skip
18.6.E4	${\displaystyle{\displaystyle\lim_{\lambda\to\infty}\frac{C^{(\lambda)}_{n}% \left(x\right)}{C^{(\lambda)}_{n}\left(1\right)}=x^{n}}}$	`limit((GegenbauerC(n, lambda, x))/(GegenbauerC(n, lambda, 1)), lambda = infinity)= (x)^(n)`	`Limit[Divide[GegenbauerC[n, \[Lambda], x],GegenbauerC[n, \[Lambda], 1]], \[Lambda] -> Infinity]= (x)^(n)`	Failure	Failure	Skip	Error
18.7.E1	${\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(x\right)=\frac{{\left(2% \lambda\right)_{n}}}{{\left(\lambda+\frac{1}{2}\right)_{n}}}P^{(\lambda-\frac{% 1}{2},\lambda-\frac{1}{2})}_{n}\left(x\right)}}$	`GegenbauerC(n, lambda, x)=(pochhammer(2lambda, n))/(pochhammer(lambda +(1)/(2), n))JacobiP(n, lambda -(1)/(2), lambda -(1)/(2), x)`	`GegenbauerC[n, \[Lambda], x]=Divide[Pochhammer[2\[Lambda], n],Pochhammer[\[Lambda]+Divide[1,2], n]]JacobiP[n, \[Lambda]-Divide[1,2], \[Lambda]-Divide[1,2], x]`	Successful	Successful	-	-
18.7.E2	${\displaystyle{\displaystyle P^{(\alpha,\alpha)}_{n}\left(x\right)=\frac{{% \left(\alpha+1\right)_{n}}}{{\left(2\alpha+1\right)_{n}}}C^{(\alpha+\frac{1}{2% })}_{n}\left(x\right)}}$	`JacobiP(n, alpha, alpha, x)=(pochhammer(alpha + 1, n))/(pochhammer(2alpha + 1, n))GegenbauerC(n, alpha +(1)/(2), x)`	`JacobiP[n, \[Alpha], \[Alpha], x]=Divide[Pochhammer[\[Alpha]+ 1, n],Pochhammer[2\[Alpha]+ 1, n]]GegenbauerC[n, \[Alpha]+Divide[1,2], x]`	Successful	Successful	-	-
18.7.E3	${\displaystyle{\displaystyle T_{n}\left(x\right)=\ifrac{P^{(-\frac{1}{2},-% \frac{1}{2})}_{n}\left(x\right)}{P^{(-\frac{1}{2},-\frac{1}{2})}_{n}\left(1% \right)}}}$	`ChebyshevT(n, x)=(JacobiP(n, -(1)/(2), -(1)/(2), x))/(JacobiP(n, -(1)/(2), -(1)/(2), 1))`	`ChebyshevT[n, x]=Divide[JacobiP[n, -Divide[1,2], -Divide[1,2], x],JacobiP[n, -Divide[1,2], -Divide[1,2], 1]]`	Successful	Successful	-	-
18.7.E4	${\displaystyle{\displaystyle U_{n}\left(x\right)=C^{(1)}_{n}\left(x\right)}}$	`ChebyshevU(n, x)= GegenbauerC(n, 1, x)`	`ChebyshevU[n, x]= GegenbauerC[n, 1, x]`	Successful	Successful	-	-
18.7.E4	${\displaystyle{\displaystyle C^{(1)}_{n}\left(x\right)=\ifrac{(n+1)P^{(\frac{1% }{2},\frac{1}{2})}_{n}\left(x\right)}{P^{(\frac{1}{2},\frac{1}{2})}_{n}\left(1% \right)}}}$	`GegenbauerC(n, 1, x)=((n + 1)* JacobiP(n, (1)/(2), (1)/(2), x))/(JacobiP(n, (1)/(2), (1)/(2), 1))`	`GegenbauerC[n, 1, x]=Divide[(n + 1)* JacobiP[n, Divide[1,2], Divide[1,2], x],JacobiP[n, Divide[1,2], Divide[1,2], 1]]`	Successful	Successful	-	-
18.7.E9	${\displaystyle{\displaystyle P_{n}\left(x\right)=C^{(\frac{1}{2})}_{n}\left(x% \right)}}$	`LegendreP(n, x)= GegenbauerC(n, (1)/(2), x)`	`LegendreP[n, x]= GegenbauerC[n, Divide[1,2], x]`	Successful	Successful	-	-
18.7.E9	${\displaystyle{\displaystyle C^{(\frac{1}{2})}_{n}\left(x\right)=P^{(0,0)}_{n}% \left(x\right)}}$	`GegenbauerC(n, (1)/(2), x)= JacobiP(n, 0, 0, x)`	`GegenbauerC[n, Divide[1,2], x]= JacobiP[n, 0, 0, x]`	Successful	Successful	-	-
18.7.E10	${\displaystyle{\displaystyle P^{*}_{n}\left(x\right)=P_{n}\left(2x-1\right)}}$	`LegendreP(n, 2(x) - 1)= LegendreP(n, 2x - 1)`	`Error`	Successful	Error	-	-
18.7.E13	${\displaystyle{\displaystyle\frac{P^{(\alpha,\alpha)}_{2n}\left(x\right)}{P^{(% \alpha,\alpha)}_{2n}\left(1\right)}=\frac{P^{(\alpha,-\frac{1}{2})}_{n}\left(2% x^{2}-1\right)}{P^{(\alpha,-\frac{1}{2})}_{n}\left(1\right)}}}$	`(JacobiP(2n, alpha, alpha, x))/(JacobiP(2n, alpha, alpha, 1))=(JacobiP(n, alpha, -(1)/(2), 2*(x)^(2)- 1))/(JacobiP(n, alpha, -(1)/(2), 1))`	`Divide[JacobiP[2n, \[Alpha], \[Alpha], x],JacobiP[2n, \[Alpha], \[Alpha], 1]]=Divide[JacobiP[n, \[Alpha], -Divide[1,2], 2*(x)^(2)- 1],JacobiP[n, \[Alpha], -Divide[1,2], 1]]`	Failure	Failure	Successful	Successful
18.7.E14	${\displaystyle{\displaystyle\frac{P^{(\alpha,\alpha)}_{2n+1}\left(x\right)}{P^% {(\alpha,\alpha)}_{2n+1}\left(1\right)}=\frac{xP^{(\alpha,\frac{1}{2})}_{n}% \left(2x^{2}-1\right)}{P^{(\alpha,\frac{1}{2})}_{n}\left(1\right)}}}$	`(JacobiP(2n + 1, alpha, alpha, x))/(JacobiP(2n + 1, alpha, alpha, 1))=(xJacobiP(n, alpha, (1)/(2), 2(x)^(2)- 1))/(JacobiP(n, alpha, (1)/(2), 1))`	`Divide[JacobiP[2n + 1, \[Alpha], \[Alpha], x],JacobiP[2n + 1, \[Alpha], \[Alpha], 1]]=Divide[xJacobiP[n, \[Alpha], Divide[1,2], 2(x)^(2)- 1],JacobiP[n, \[Alpha], Divide[1,2], 1]]`	Failure	Failure	Successful	Successful
18.7.E15	${\displaystyle{\displaystyle C^{(\lambda)}_{2n}\left(x\right)=\frac{{\left(% \lambda\right)_{n}}}{{\left(\tfrac{1}{2}\right)_{n}}}P^{(\lambda-\frac{1}{2},-% \frac{1}{2})}_{n}\left(2x^{2}-1\right)}}$	`GegenbauerC(2n, lambda, x)=(pochhammer(lambda, n))/(pochhammer((1)/(2), n))JacobiP(n, lambda -(1)/(2), -(1)/(2), 2*(x)^(2)- 1)`	`GegenbauerC[2n, \[Lambda], x]=Divide[Pochhammer[\[Lambda], n],Pochhammer[Divide[1,2], n]]JacobiP[n, \[Lambda]-Divide[1,2], -Divide[1,2], 2*(x)^(2)- 1]`	Failure	Failure	Successful	Successful
18.7.E16	${\displaystyle{\displaystyle C^{(\lambda)}_{2n+1}\left(x\right)=\frac{{\left(% \lambda\right)_{n+1}}}{{\left(\frac{1}{2}\right)_{n+1}}}xP^{(\lambda-\frac{1}{% 2},\frac{1}{2})}_{n}\left(2x^{2}-1\right)}}$	`GegenbauerC(2n + 1, lambda, x)=(pochhammer(lambda, n + 1))/(pochhammer((1)/(2), n + 1))xJacobiP(n, lambda -(1)/(2), (1)/(2), 2(x)^(2)- 1)`	`GegenbauerC[2n + 1, \[Lambda], x]=Divide[Pochhammer[\[Lambda], n + 1],Pochhammer[Divide[1,2], n + 1]]xJacobiP[n, \[Lambda]-Divide[1,2], Divide[1,2], 2(x)^(2)- 1]`	Failure	Failure	Successful	Successful
18.7.E23	${\displaystyle{\displaystyle\lim_{\alpha\to\infty}\alpha^{-\frac{1}{2}n}P^{(% \alpha,\alpha)}_{n}\left(\alpha^{-\frac{1}{2}}x\right)=\frac{H_{n}\left(x% \right)}{2^{n}n!}}}$	`limit((alpha)^(-(1)/(2)n) JacobiP(n, alpha, alpha, (alpha)^(-(1)/(2))* x), alpha = infinity)=(HermiteH(n, x))/((2)^(n)* factorial(n))`	`Limit[(\[Alpha])^(-Divide[1,2]n) JacobiP[n, \[Alpha], \[Alpha], (\[Alpha])^(-Divide[1,2])* x], \[Alpha] -> Infinity]=Divide[HermiteH[n, x],(2)^(n)* (n)!]`	Failure	Failure	Skip	Error
18.7.E24	${\displaystyle{\displaystyle\lim_{\lambda\to\infty}\lambda^{-\frac{1}{2}n}C^{(% \lambda)}_{n}\left(\lambda^{-\frac{1}{2}}x\right)=\frac{H_{n}\left(x\right)}{n% !}}}$	`limit((lambda)^(-(1)/(2)n) GegenbauerC(n, lambda, (lambda)^(-(1)/(2))* x), lambda = infinity)=(HermiteH(n, x))/(factorial(n))`	`Limit[(\[Lambda])^(-Divide[1,2]n) GegenbauerC[n, \[Lambda], (\[Lambda])^(-Divide[1,2])* x], \[Lambda] -> Infinity]=Divide[HermiteH[n, x],(n)!]`	Failure	Failure	Skip	Skip
18.7.E25	${\displaystyle{\displaystyle\lim_{\lambda\to 0}\frac{1}{\lambda}C^{(\lambda)}_% {n}\left(x\right)=\frac{2}{n}T_{n}\left(x\right)}}$	`limit((1)/(lambda)GegenbauerC(n, lambda, x), lambda = 0)=(2)/(n)ChebyshevT(n, x)`	`Limit[Divide[1,\[Lambda]]GegenbauerC[n, \[Lambda], x], \[Lambda] -> 0]=Divide[2,n]ChebyshevT[n, x]`	Failure	Failure	Skip	Successful
18.9.E3	${\displaystyle{\displaystyle P^{(\alpha,\beta-1)}_{n}\left(x\right)-P^{(\alpha% -1,\beta)}_{n}\left(x\right)=P^{(\alpha,\beta)}_{n-1}\left(x\right)}}$	`JacobiP(n, alpha, beta - 1, x)- JacobiP(n, alpha - 1, beta, x)= JacobiP(n - 1, alpha, beta, x)`	`JacobiP[n, \[Alpha], \[Beta]- 1, x]- JacobiP[n, \[Alpha]- 1, \[Beta], x]= JacobiP[n - 1, \[Alpha], \[Beta], x]`	Failure	Successful	Successful	-
18.9.E4	${\displaystyle{\displaystyle(1-x)P^{(\alpha+1,\beta)}_{n}\left(x\right)+(1+x)P% ^{(\alpha,\beta+1)}_{n}\left(x\right)=2\!P^{(\alpha,\beta)}_{n}\left(x\right)}}$	`(1 - x)* JacobiP(n, alpha + 1, beta, x)+(1 + x)* JacobiP(n, alpha, beta + 1, x)= 2*JacobiP(n, alpha, beta, x)`	`(1 - x)* JacobiP[n, \[Alpha]+ 1, \[Beta], x]+(1 + x)* JacobiP[n, \[Alpha], \[Beta]+ 1, x]= 2*JacobiP[n, \[Alpha], \[Beta], x]`	Failure	Successful	Successful	-
18.9.E5	${\displaystyle{\displaystyle(2n+\alpha+\beta+1)P^{(\alpha,\beta)}_{n}\left(x% \right)=(n+\alpha+\beta+1)P^{(\alpha,\beta+1)}_{n}\left(x\right)+(n+\alpha)P^{% (\alpha,\beta+1)}_{n-1}\left(x\right)}}$	`(2n + alpha + beta + 1) JacobiP(n, alpha, beta, x)=(n + alpha + beta + 1)* JacobiP(n, alpha, beta + 1, x)+(n + alpha)* JacobiP(n - 1, alpha, beta + 1, x)`	`(2n + \[Alpha]+ \[Beta]+ 1) JacobiP[n, \[Alpha], \[Beta], x]=(n + \[Alpha]+ \[Beta]+ 1)* JacobiP[n, \[Alpha], \[Beta]+ 1, x]+(n + \[Alpha])* JacobiP[n - 1, \[Alpha], \[Beta]+ 1, x]`	Failure	Successful	Successful	-
18.9.E6	${\displaystyle{\displaystyle(n+\tfrac{1}{2}\alpha+\tfrac{1}{2}\beta+1)(1+x)P^{% (\alpha,\beta+1)}_{n}\left(x\right)=(n+1)P^{(\alpha,\beta)}_{n+1}\left(x\right% )+(n+\beta+1)P^{(\alpha,\beta)}_{n}\left(x\right)}}$	`(n +(1)/(2)alpha +(1)/(2)beta + 1)(1 + x) JacobiP(n, alpha, beta + 1, x)=(n + 1)* JacobiP(n + 1, alpha, beta, x)+(n + beta + 1)* JacobiP(n, alpha, beta, x)`	`(n +Divide[1,2]\[Alpha]+Divide[1,2]\[Beta]+ 1)(1 + x) JacobiP[n, \[Alpha], \[Beta]+ 1, x]=(n + 1)* JacobiP[n + 1, \[Alpha], \[Beta], x]+(n + \[Beta]+ 1)* JacobiP[n, \[Alpha], \[Beta], x]`	Failure	Successful	Successful	-
18.9.E7	${\displaystyle{\displaystyle(n+\lambda)C^{(\lambda)}_{n}\left(x\right)=\lambda% \left(C^{(\lambda+1)}_{n}\left(x\right)-C^{(\lambda+1)}_{n-2}\left(x\right)% \right)}}$	`(n + lambda)* GegenbauerC(n, lambda, x)= lambda*(GegenbauerC(n, lambda + 1, x)- GegenbauerC(n - 2, lambda + 1, x))`	`(n + \[Lambda])* GegenbauerC[n, \[Lambda], x]= \[Lambda]*(GegenbauerC[n, \[Lambda]+ 1, x]- GegenbauerC[n - 2, \[Lambda]+ 1, x])`	Successful	Successful	-	-
18.9.E8	${\displaystyle{\displaystyle 4\lambda(n+\lambda+1)(1-x^{2})C^{(\lambda+1)}_{n}% \left(x\right)=-(n+1)(n+2)C^{(\lambda)}_{n+2}\left(x\right)+(n+2\lambda)(n+2% \lambda+1)C^{(\lambda)}_{n}\left(x\right)}}$	`4lambda(n + lambda + 1)(1 - (x)^(2)) GegenbauerC(n, lambda + 1, x)= -(n + 1)(n + 2) GegenbauerC(n + 2, lambda, x)+(n + 2lambda)(n + 2lambda + 1) GegenbauerC(n, lambda, x)`	`4\[Lambda](n + \[Lambda]+ 1)(1 - (x)^(2)) GegenbauerC[n, \[Lambda]+ 1, x]= -(n + 1)(n + 2) GegenbauerC[n + 2, \[Lambda], x]+(n + 2\[Lambda])(n + 2\[Lambda]+ 1) GegenbauerC[n, \[Lambda], x]`	Successful	Successful	-	-
18.9.E9	${\displaystyle{\displaystyle T_{n}\left(x\right)=\tfrac{1}{2}\left(U_{n}\left(% x\right)-U_{n-2}\left(x\right)\right)}}$	`ChebyshevT(n, x)=(1)/(2)*(ChebyshevU(n, x)- ChebyshevU(n - 2, x))`	`ChebyshevT[n, x]=Divide[1,2]*(ChebyshevU[n, x]- ChebyshevU[n - 2, x])`	Successful	Failure	-	Successful
18.9.E10	${\displaystyle{\displaystyle(1-x^{2})U_{n}\left(x\right)=-\tfrac{1}{2}\left(T_% {n+2}\left(x\right)-T_{n}\left(x\right)\right)}}$	`(1 - (x)^(2))* ChebyshevU(n, x)= -(1)/(2)*(ChebyshevT(n + 2, x)- ChebyshevT(n, x))`	`(1 - (x)^(2))* ChebyshevU[n, x]= -Divide[1,2]*(ChebyshevT[n + 2, x]- ChebyshevT[n, x])`	Successful	Failure	-	Successful
18.9.E15	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}P^{(\alpha,\beta)}_{% n}\left(x\right)=\tfrac{1}{2}(n+\alpha+\beta+1)P^{(\alpha+1,\beta+1)}_{n-1}% \left(x\right)}}$	`diff(JacobiP(n, alpha, beta, x), x)=(1)/(2)(n + alpha + beta + 1) JacobiP(n - 1, alpha + 1, beta + 1, x)`	`D[JacobiP[n, \[Alpha], \[Beta], x], x]=Divide[1,2](n + \[Alpha]+ \[Beta]+ 1) JacobiP[n - 1, \[Alpha]+ 1, \[Beta]+ 1, x]`	Failure	Successful	Fail `Float(infinity)+Float(infinity)I <- {alpha = 2^(1/2)+I2^(1/2), beta = 2^(1/2)+I2^(1/2), n = 1, x = 1}` `Float(infinity)+Float(infinity)I <- {alpha = 2^(1/2)+I2^(1/2), beta = 2^(1/2)+I2^(1/2), n = 2, x = 1}` `Float(infinity)+Float(infinity)I <- {alpha = 2^(1/2)+I2^(1/2), beta = 2^(1/2)+I2^(1/2), n = 3, x = 1}` `Float(infinity)+Float(infinity)I <- {alpha = 2^(1/2)+I2^(1/2), beta = 2^(1/2)-I2^(1/2), n = 1, x = 1}` ... skip entries to safe data	-
18.9.E16	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left((1-x)^{\alpha}% (1+x)^{\beta}P^{(\alpha,\beta)}_{n}\left(x\right)\right)=-2(n+1)(1-x)^{\alpha-% 1}(1+x)^{\beta-1}P^{(\alpha-1,\beta-1)}_{n+1}\left(x\right)}}$	`diff((1 - x)^(alpha)(1 + x)^(beta) JacobiP(n, alpha, beta, x), x)= - 2(n + 1)(1 - x)^(alpha - 1)(1 + x)^(beta - 1) JacobiP(n + 1, alpha - 1, beta - 1, x)`	`D[(1 - x)^(\[Alpha])(1 + x)^(\[Beta]) JacobiP[n, \[Alpha], \[Beta], x], x]= - 2(n + 1)(1 - x)^(\[Alpha]- 1)(1 + x)^(\[Beta]- 1) JacobiP[n + 1, \[Alpha]- 1, \[Beta]- 1, x]`	Failure	Successful	Fail `Float(undefined)+Float(undefined)I <- {alpha = 2^(1/2)+I2^(1/2), beta = 2^(1/2)+I2^(1/2), n = 1, x = 1}` `Float(undefined)+Float(undefined)I <- {alpha = 2^(1/2)+I2^(1/2), beta = 2^(1/2)+I2^(1/2), n = 2, x = 1}` `Float(undefined)+Float(undefined)I <- {alpha = 2^(1/2)+I2^(1/2), beta = 2^(1/2)+I2^(1/2), n = 3, x = 1}` `Float(undefined)+Float(undefined)I <- {alpha = 2^(1/2)+I2^(1/2), beta = 2^(1/2)-I2^(1/2), n = 1, x = 1}` ... skip entries to safe data	-
18.9.E17	${\displaystyle{\displaystyle(2n+\alpha+\beta)(1-x^{2})\frac{\mathrm{d}}{% \mathrm{d}x}P^{(\alpha,\beta)}_{n}\left(x\right)=n\left(\alpha-\beta-(2n+% \alpha+\beta)x\right)P^{(\alpha,\beta)}_{n}\left(x\right)+2(n+\alpha)(n+\beta)% P^{(\alpha,\beta)}_{n-1}\left(x\right)}}$	`(2n + alpha + beta)(1 - (x)^(2))* diff(JacobiP(n, alpha, beta, x), x)= n(alpha - beta -(2n + alpha + beta)x) JacobiP(n, alpha, beta, x)+ 2(n + alpha)(n + beta)* JacobiP(n - 1, alpha, beta, x)`	`(2n + \[Alpha]+ \[Beta])(1 - (x)^(2))* D[JacobiP[n, \[Alpha], \[Beta], x], x]= n(\[Alpha]- \[Beta]-(2n + \[Alpha]+ \[Beta])x) JacobiP[n, \[Alpha], \[Beta], x]+ 2(n + \[Alpha])(n + \[Beta])* JacobiP[n - 1, \[Alpha], \[Beta], x]`	Failure	Successful	Successful	-
18.9.E18	${\displaystyle{\displaystyle(2n+\alpha+\beta+2)(1-x^{2})\frac{\mathrm{d}}{% \mathrm{d}x}P^{(\alpha,\beta)}_{n}\left(x\right)=(n+\alpha+\beta+1)\left(% \alpha-\beta+(2n+\alpha+\beta+2)x\right)P^{(\alpha,\beta)}_{n}\left(x\right)-2% (n+1)(n+\alpha+\beta+1)P^{(\alpha,\beta)}_{n+1}\left(x\right)}}$	`(2n + alpha + beta + 2)(1 - (x)^(2))* diff(JacobiP(n, alpha, beta, x), x)=(n + alpha + beta + 1)(alpha - beta +(2n + alpha + beta + 2)x) JacobiP(n, alpha, beta, x)- 2(n + 1)(n + alpha + beta + 1)* JacobiP(n + 1, alpha, beta, x)`	`(2n + \[Alpha]+ \[Beta]+ 2)(1 - (x)^(2))* D[JacobiP[n, \[Alpha], \[Beta], x], x]=(n + \[Alpha]+ \[Beta]+ 1)(\[Alpha]- \[Beta]+(2n + \[Alpha]+ \[Beta]+ 2)x) JacobiP[n, \[Alpha], \[Beta], x]- 2(n + 1)(n + \[Alpha]+ \[Beta]+ 1)* JacobiP[n + 1, \[Alpha], \[Beta], x]`	Failure	Successful	Successful	-
18.9.E19	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}C^{(\lambda)}_{n}% \left(x\right)=2\lambda C^{(\lambda+1)}_{n-1}\left(x\right)}}$	`diff(GegenbauerC(n, lambda, x), x)= 2lambdaGegenbauerC(n - 1, lambda + 1, x)`	`D[GegenbauerC[n, \[Lambda], x], x]= 2\[Lambda]GegenbauerC[n - 1, \[Lambda]+ 1, x]`	Successful	Successful	-	-
18.9.E20	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left((1-x^{2})^{% \lambda-\frac{1}{2}}C^{(\lambda)}_{n}\left(x\right)\right)=-\frac{(n+1)(n+2% \lambda-1)}{2(\lambda-1)}{(1-x^{2})^{\lambda-\frac{3}{2}}}C^{(\lambda-1)}_{n+1% }\left(x\right)}}$	`diff((1 - (x)^(2))^(lambda -(1)/(2))* GegenbauerC(n, lambda, x), x)= -((n + 1)(n + 2lambda - 1))/(2(lambda - 1))(1 - (x)^(2))^(lambda -(3)/(2))*GegenbauerC(n + 1, lambda - 1, x)`	`D[(1 - (x)^(2))^(\[Lambda]-Divide[1,2])* GegenbauerC[n, \[Lambda], x], x]= -Divide[(n + 1)(n + 2\[Lambda]- 1),2(\[Lambda]- 1)](1 - (x)^(2))^(\[Lambda]-Divide[3,2])*GegenbauerC[n + 1, \[Lambda]- 1, x]`	Failure	Successful	Fail `Float(infinity)+Float(infinity)I <- {lambda = 2^(1/2)+I2^(1/2), n = 1, x = 1}` `Float(infinity)+Float(infinity)I <- {lambda = 2^(1/2)+I2^(1/2), n = 2, x = 1}` `Float(infinity)+Float(infinity)I <- {lambda = 2^(1/2)+I2^(1/2), n = 3, x = 1}` `Float(infinity)+Float(infinity)I <- {lambda = 2^(1/2)-I2^(1/2), n = 1, x = 1}` ... skip entries to safe data	-
18.9.E21	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}T_{n}\left(x\right)=% nU_{n-1}\left(x\right)}}$	`diff(ChebyshevT(n, x), x)= n*ChebyshevU(n - 1, x)`	`D[ChebyshevT[n, x], x]= n*ChebyshevU[n - 1, x]`	Successful	Successful	-	-
18.9.E22	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left((1-x^{2})^{% \frac{1}{2}}U_{n}\left(x\right)\right)=-(n+1){(1-x^{2})^{-\frac{1}{2}}}T_{n+1}% \left(x\right)}}$	`diff((1 - (x)^(2))^((1)/(2))* ChebyshevU(n, x), x)= -(n + 1)(1 - (x)^(2))^(-(1)/(2))ChebyshevT(n + 1, x)`	`D[(1 - (x)^(2))^(Divide[1,2])* ChebyshevU[n, x], x]= -(n + 1)(1 - (x)^(2))^(-Divide[1,2])ChebyshevT[n + 1, x]`	Successful	Successful	-	-
18.9.E25	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}H_{n}\left(x\right)=% 2nH_{n-1}\left(x\right)}}$	`diff(HermiteH(n, x), x)= 2nHermiteH(n - 1, x)`	`D[HermiteH[n, x], x]= 2nHermiteH[n - 1, x]`	Successful	Successful	-	-
18.9.E26	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left(e^{-x^{2}}H_{n% }\left(x\right)\right)=-e^{-x^{2}}H_{n+1}\left(x\right)}}$	`diff(exp(- (x)^(2))HermiteH(n, x), x)= - exp(- (x)^(2))HermiteH(n + 1, x)`	`D[Exp[- (x)^(2)]HermiteH[n, x], x]= - Exp[- (x)^(2)]HermiteH[n + 1, x]`	Successful	Successful	-	-
18.10.E1	${\displaystyle{\displaystyle\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta% \right)}{P^{(\alpha,\alpha)}_{n}\left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}% _{n}\left(\cos\theta\right)}{C^{(\alpha+\frac{1}{2})}_{n}\left(1\right)}}}$	`(JacobiP(n, alpha, alpha, cos(theta)))/(JacobiP(n, alpha, alpha, 1))=(GegenbauerC(n, alpha +(1)/(2), cos(theta)))/(GegenbauerC(n, alpha +(1)/(2), 1))`	`Divide[JacobiP[n, \[Alpha], \[Alpha], Cos[\[Theta]]],JacobiP[n, \[Alpha], \[Alpha], 1]]=Divide[GegenbauerC[n, \[Alpha]+Divide[1,2], Cos[\[Theta]]],GegenbauerC[n, \[Alpha]+Divide[1,2], 1]]`	Successful	Successful	-	-
18.10.E1	${\displaystyle{\displaystyle\frac{C^{(\alpha+\frac{1}{2})}_{n}\left(\cos\theta% \right)}{C^{(\alpha+\frac{1}{2})}_{n}\left(1\right)}=\frac{2^{\alpha+\frac{1}{% 2}}\Gamma\left(\alpha+1\right)}{\pi^{\frac{1}{2}}\Gamma\left(\alpha+\frac{1}{2% }\right)}(\sin\theta)^{-2\alpha}\int_{0}^{\theta}\frac{\cos\left((n+\alpha+% \tfrac{1}{2})\phi\right)}{(\cos\phi-\cos\theta)^{-\alpha+\frac{1}{2}}}\mathrm{% d}\phi}}$	`(GegenbauerC(n, alpha +(1)/(2), cos(theta)))/(GegenbauerC(n, alpha +(1)/(2), 1))=((2)^(alpha +(1)/(2))* GAMMA(alpha + 1))/((Pi)^((1)/(2))* GAMMA(alpha +(1)/(2)))(sin(theta))^(- 2alpha)* int((cos((n + alpha +(1)/(2))* phi))/((cos(phi)- cos(theta))^(- alpha +(1)/(2))), phi = 0..theta)`	`Divide[GegenbauerC[n, \[Alpha]+Divide[1,2], Cos[\[Theta]]],GegenbauerC[n, \[Alpha]+Divide[1,2], 1]]=Divide[(2)^(\[Alpha]+Divide[1,2])* Gamma[\[Alpha]+ 1],(Pi)^(Divide[1,2])* Gamma[\[Alpha]+Divide[1,2]]](Sin[\[Theta]])^(- 2\[Alpha])* Integrate[Divide[Cos[(n + \[Alpha]+Divide[1,2])* \[Phi]],(Cos[\[Phi]]- Cos[\[Theta]])^(- \[Alpha]+Divide[1,2])], {\[Phi], 0, \[Theta]}]`	Failure	Failure	Skip	Successful
18.10.E2	${\displaystyle{\displaystyle P_{n}\left(\cos\theta\right)=\frac{2^{\frac{1}{2}% }}{\pi}\int_{0}^{\theta}\frac{\cos\left((n+\tfrac{1}{2})\phi\right)}{(\cos\phi% -\cos\theta)^{\frac{1}{2}}}\mathrm{d}\phi}}$	`LegendreP(n, cos(theta))=((2)^((1)/(2)))/(Pi)int((cos((n +(1)/(2)) phi))/((cos(phi)- cos(theta))^((1)/(2))), phi = 0..theta)`	`LegendreP[n, Cos[\[Theta]]]=Divide[(2)^(Divide[1,2]),Pi]Integrate[Divide[Cos[(n +Divide[1,2]) \[Phi]],(Cos[\[Phi]]- Cos[\[Theta]])^(Divide[1,2])], {\[Phi], 0, \[Theta]}]`	Failure	Failure	Skip	Successful
18.10.E5	${\displaystyle{\displaystyle P_{n}\left(\cos\theta\right)=\frac{1}{\pi}\int_{0% }^{\pi}(\cos\theta+i\sin\theta\cos\phi)^{n}\mathrm{d}\phi}}$	`LegendreP(n, cos(theta))=(1)/(Pi)int((cos(theta)+ Isin(theta)*cos(phi))^(n), phi = 0..Pi)`	`LegendreP[n, Cos[\[Theta]]]=Divide[1,Pi]Integrate[(Cos[\[Theta]]+ ISin[\[Theta]]*Cos[\[Phi]])^(n), {\[Phi], 0, Pi}]`	Failure	Failure	Skip	Error
18.10.E7	${\displaystyle{\displaystyle H_{n}\left(x\right)=\frac{2^{n}}{\pi^{\frac{1}{2}% }}\int_{-\infty}^{\infty}(x+it)^{n}e^{-t^{2}}\mathrm{d}t}}$	`HermiteH(n, x)=((2)^(n))/((Pi)^((1)/(2)))int((x + It)^(n)* exp(- (t)^(2)), t = - infinity..infinity)`	`HermiteH[n, x]=Divide[(2)^(n),(Pi)^(Divide[1,2])]Integrate[(x + It)^(n)* Exp[- (t)^(2)], {t, - Infinity, Infinity}]`	Failure	Failure	Skip	Skip
18.10.E10	${\displaystyle{\displaystyle H_{n}\left(x\right)=\frac{(-2i)^{n}e^{x^{2}}}{\pi% ^{\frac{1}{2}}}\int_{-\infty}^{\infty}e^{-t^{2}}t^{n}e^{2ixt}\mathrm{d}t}}$	`HermiteH(n, x)=((- 2I)^(n) exp((x)^(2)))/((Pi)^((1)/(2)))int(exp(- (t)^(2))(t)^(n)* exp(2Ix*t), t = - infinity..infinity)`	`HermiteH[n, x]=Divide[(- 2I)^(n) Exp[(x)^(2)],(Pi)^(Divide[1,2])]Integrate[Exp[- (t)^(2)](t)^(n)* Exp[2Ix*t], {t, - Infinity, Infinity}]`	Failure	Failure	Skip	Successful
18.10.E10	${\displaystyle{\displaystyle\frac{(-2i)^{n}e^{x^{2}}}{\pi^{\frac{1}{2}}}\int_{% -\infty}^{\infty}e^{-t^{2}}t^{n}e^{2ixt}\mathrm{d}t=\frac{2^{n+1}}{\pi^{\frac{% 1}{2}}}e^{x^{2}}\int_{0}^{\infty}e^{-t^{2}}t^{n}\cos\left(2xt-\tfrac{1}{2}n\pi% \right)\mathrm{d}t}}$	`((- 2I)^(n) exp((x)^(2)))/((Pi)^((1)/(2)))int(exp(- (t)^(2))(t)^(n)* exp(2Ixt), t = - infinity..infinity)=((2)^(n + 1))/((Pi)^((1)/(2)))exp((x)^(2))int(exp(- (t)^(2))(t)^(n)* cos(2xt -(1)/(2)nPi), t = 0..infinity)`	`Divide[(- 2I)^(n) Exp[(x)^(2)],(Pi)^(Divide[1,2])]Integrate[Exp[- (t)^(2)](t)^(n)* Exp[2Ixt], {t, - Infinity, Infinity}]=Divide[(2)^(n + 1),(Pi)^(Divide[1,2])]Exp[(x)^(2)]Integrate[Exp[- (t)^(2)](t)^(n)* Cos[2xt -Divide[1,2]nPi], {t, 0, Infinity}]`	Successful	Failure	-	Error
18.11.E1	${\displaystyle{\displaystyle\mathsf{P}^{m}_{n}\left(x\right)={\left(\tfrac{1}{% 2}\right)_{m}}(-2)^{m}(1-x^{2})^{\frac{1}{2}m}C^{(m+\frac{1}{2})}_{n-m}\left(x% \right)}}$	`LegendreP(n, m, x)= pochhammer((1)/(2), m)(- 2)^(m)(1 - (x)^(2))^((1)/(2)m) GegenbauerC(n - m, m +(1)/(2), x)`	`LegendreP[n, m, x]= Pochhammer[Divide[1,2], m](- 2)^(m)(1 - (x)^(2))^(Divide[1,2]m) GegenbauerC[n - m, m +Divide[1,2], x]`	Failure	Failure	Skip	Successful
18.11.E1	${\displaystyle{\displaystyle{\left(\tfrac{1}{2}\right)_{m}}(-2)^{m}(1-x^{2})^{% \frac{1}{2}m}C^{(m+\frac{1}{2})}_{n-m}\left(x\right)={\left(n+1\right)_{m}}(-2% )^{-m}(1-x^{2})^{\frac{1}{2}m}P^{(m,m)}_{n-m}\left(x\right)}}$	`pochhammer((1)/(2), m)(- 2)^(m)(1 - (x)^(2))^((1)/(2)m) GegenbauerC(n - m, m +(1)/(2), x)= pochhammer(n + 1, m)(- 2)^(- m)(1 - (x)^(2))^((1)/(2)m) JacobiP(n - m, m, m, x)`	`Pochhammer[Divide[1,2], m](- 2)^(m)(1 - (x)^(2))^(Divide[1,2]m) GegenbauerC[n - m, m +Divide[1,2], x]= Pochhammer[n + 1, m](- 2)^(- m)(1 - (x)^(2))^(Divide[1,2]m) JacobiP[n - m, m, m, x]`	Failure	Failure	Skip	Successful
18.11.E2	${\displaystyle{\displaystyle\frac{{\left(\alpha+1\right)_{n}}}{n!}M\left(-n,% \alpha+1,x\right)=\frac{(-1)^{n}}{n!}U\left(-n,\alpha+1,x\right)}}$	`(pochhammer(alpha + 1, n))/(factorial(n))KummerM(- n, alpha + 1, x)=((- 1)^(n))/(factorial(n))KummerU(- n, alpha + 1, x)`	`Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]Hypergeometric1F1[- n, \[Alpha]+ 1, x]=Divide[(- 1)^(n),(n)!]HypergeometricU[- n, \[Alpha]+ 1, x]`	Failure	Failure	Successful	Successful
18.11.E2	${\displaystyle{\displaystyle\frac{(-1)^{n}}{n!}U\left(-n,\alpha+1,x\right)=% \frac{{\left(\alpha+1\right)_{n}}}{n!}x^{-\frac{1}{2}(\alpha+1)}e^{\frac{1}{2}% x}M_{n+\frac{1}{2}(\alpha+1),\frac{1}{2}\alpha}\left(x\right)}}$	`((- 1)^(n))/(factorial(n))KummerU(- n, alpha + 1, x)=(pochhammer(alpha + 1, n))/(factorial(n))(x)^(-(1)/(2)(alpha + 1)) exp((1)/(2)x)WhittakerM(n +(1)/(2)(alpha + 1), (1)/(2)alpha, x)`	`Divide[(- 1)^(n),(n)!]HypergeometricU[- n, \[Alpha]+ 1, x]=Divide[Pochhammer[\[Alpha]+ 1, n],(n)!](x)^(-Divide[1,2](\[Alpha]+ 1)) Exp[Divide[1,2]x]WhittakerM[n +Divide[1,2](\[Alpha]+ 1), Divide[1,2]\[Alpha], x]`	Failure	Failure	Skip	Successful
18.11.E2	${\displaystyle{\displaystyle\frac{{\left(\alpha+1\right)_{n}}}{n!}x^{-\frac{1}% {2}(\alpha+1)}e^{\frac{1}{2}x}M_{n+\frac{1}{2}(\alpha+1),\frac{1}{2}\alpha}% \left(x\right)=\frac{(-1)^{n}}{n!}x^{-\frac{1}{2}(\alpha+1)}e^{\frac{1}{2}x}W_% {n+\frac{1}{2}(\alpha+1),\frac{1}{2}\alpha}\left(x\right)}}$	`(pochhammer(alpha + 1, n))/(factorial(n))(x)^(-(1)/(2)(alpha + 1))* exp((1)/(2)x)WhittakerM(n +(1)/(2)(alpha + 1), (1)/(2)alpha, x)=((- 1)^(n))/(factorial(n))(x)^(-(1)/(2)(alpha + 1))* exp((1)/(2)x)WhittakerW(n +(1)/(2)(alpha + 1), (1)/(2)alpha, x)`	`Divide[Pochhammer[\[Alpha]+ 1, n],(n)!](x)^(-Divide[1,2](\[Alpha]+ 1))* Exp[Divide[1,2]x]WhittakerM[n +Divide[1,2](\[Alpha]+ 1), Divide[1,2]\[Alpha], x]=Divide[(- 1)^(n),(n)!](x)^(-Divide[1,2](\[Alpha]+ 1))* Exp[Divide[1,2]x]WhittakerW[n +Divide[1,2](\[Alpha]+ 1), Divide[1,2]\[Alpha], x]`	Failure	Failure	Skip	Successful
18.11.E3	${\displaystyle{\displaystyle H_{n}\left(x\right)=2^{n}U\left(-\tfrac{1}{2}n,% \tfrac{1}{2},x^{2}\right)}}$	`HermiteH(n, x)= (2)^(n)* KummerU(-(1)/(2)*n, (1)/(2), (x)^(2))`	`HermiteH[n, x]= (2)^(n)* HypergeometricU[-Divide[1,2]*n, Divide[1,2], (x)^(2)]`	Failure	Failure	Successful	Successful
18.11.E3	${\displaystyle{\displaystyle 2^{n}U\left(-\tfrac{1}{2}n,\tfrac{1}{2},x^{2}% \right)=2^{n}xU\left(-\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{3}{2},x^{2}\right)}}$	`(2)^(n)* KummerU(-(1)/(2)n, (1)/(2), (x)^(2))= (2)^(n) xKummerU(-(1)/(2)n +(1)/(2), (3)/(2), (x)^(2))`	`(2)^(n)* HypergeometricU[-Divide[1,2]n, Divide[1,2], (x)^(2)]= (2)^(n) xHypergeometricU[-Divide[1,2]n +Divide[1,2], Divide[3,2], (x)^(2)]`	Failure	Failure	Successful	Successful
18.11.E3	${\displaystyle{\displaystyle 2^{n}xU\left(-\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{3% }{2},x^{2}\right)=2^{\frac{1}{2}n}e^{\frac{1}{2}x^{2}}U\left(-n-\tfrac{1}{2},2% ^{\frac{1}{2}}x\right)}}$	`(2)^(n)* xKummerU(-(1)/(2)n +(1)/(2), (3)/(2), (x)^(2))= (2)^((1)/(2)n) exp((1)/(2)(x)^(2))CylinderU(- n -(1)/(2), (2)^((1)/(2))* x)`	`(2)^(n)* xHypergeometricU[-Divide[1,2]n +Divide[1,2], Divide[3,2], (x)^(2)]= (2)^(Divide[1,2]n) Exp[Divide[1,2](x)^(2)]ParabolicCylinderD[-- n -Divide[1,2] - 1/2, (2)^(Divide[1,2])* x]`	Failure	Failure	Skip	Successful
18.11.E4	${\displaystyle{\displaystyle 2^{\frac{1}{2}n}U\left(-\tfrac{1}{2}n,\tfrac{1}{2% },\tfrac{1}{2}x^{2}\right)=2^{\frac{1}{2}(n-1)}xU\left(-\tfrac{1}{2}n+\tfrac{1% }{2},\tfrac{3}{2},\tfrac{1}{2}x^{2}\right)}}$	`(2)^((1)/(2)n) KummerU(-(1)/(2)n, (1)/(2), (1)/(2)(x)^(2))= (2)^((1)/(2)(n - 1)) xKummerU(-(1)/(2)n +(1)/(2), (3)/(2), (1)/(2)*(x)^(2))`	`(2)^(Divide[1,2]n) HypergeometricU[-Divide[1,2]n, Divide[1,2], Divide[1,2](x)^(2)]= (2)^(Divide[1,2](n - 1)) xHypergeometricU[-Divide[1,2]n +Divide[1,2], Divide[3,2], Divide[1,2]*(x)^(2)]`	Failure	Failure	Successful	Successful
18.11.E4	${\displaystyle{\displaystyle 2^{\frac{1}{2}(n-1)}xU\left(-\tfrac{1}{2}n+\tfrac% {1}{2},\tfrac{3}{2},\tfrac{1}{2}x^{2}\right)=e^{\tfrac{1}{4}x^{2}}U\left(-n-% \tfrac{1}{2},x\right)}}$	`(2)^((1)/(2)(n - 1)) xKummerU(-(1)/(2)n +(1)/(2), (3)/(2), (1)/(2)(x)^(2))= exp((1)/(4)(x)^(2))*CylinderU(- n -(1)/(2), x)`	`(2)^(Divide[1,2](n - 1)) xHypergeometricU[-Divide[1,2]n +Divide[1,2], Divide[3,2], Divide[1,2](x)^(2)]= Exp[Divide[1,4](x)^(2)]*ParabolicCylinderD[-- n -Divide[1,2] - 1/2, x]`	Failure	Failure	Skip	Successful
18.11.E5	${\displaystyle{\displaystyle\lim_{n\to\infty}\frac{1}{n^{\alpha}}P^{(\alpha,% \beta)}_{n}\left(1-\frac{z^{2}}{2n^{2}}\right)=\lim_{n\to\infty}\frac{1}{n^{% \alpha}}P^{(\alpha,\beta)}_{n}\left(\cos\frac{z}{n}\right)}}$	`limit((1)/((n)^(alpha))JacobiP(n, alpha, beta, 1 -((z)^(2))/(2(n)^(2))), n = infinity)= limit((1)/((n)^(alpha))*JacobiP(n, alpha, beta, cos((z)/(n))), n = infinity)`	`Limit[Divide[1,(n)^(\[Alpha])]JacobiP[n, \[Alpha], \[Beta], 1 -Divide[(z)^(2),2(n)^(2)]], n -> Infinity]= Limit[Divide[1,(n)^(\[Alpha])]*JacobiP[n, \[Alpha], \[Beta], Cos[Divide[z,n]]], n -> Infinity]`	Failure	Failure	Skip	Fail `Complex[0.0, 2.8284271247461903] <- {Rule[Limit[Times[Power[n, Times[-1, α]], JacobiP[n, α, β, Plus[1, Times[Rational[-1, 2], Power[n, -2], Power[z, 2]]]]], Rule[n, DirectedInfinity[1]]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Limit[Times[Power[n, Times[-1, α]], JacobiP[n, α, β, Cos[Times[Power[n, -1], z]]]], Rule[n, DirectedInfinity[1]]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}` `Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[Limit[Times[Power[n, Times[-1, α]], JacobiP[n, α, β, Plus[1, Times[Rational[-1, 2], Power[n, -2], Power[z, 2]]]]], Rule[n, DirectedInfinity[1]]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Limit[Times[Power[n, Times[-1, α]], JacobiP[n, α, β, Cos[Times[Power[n, -1], z]]]], Rule[n, DirectedInfinity[1]]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}` `2.8284271247461903 <- {Rule[Limit[Times[Power[n, Times[-1, α]], JacobiP[n, α, β, Plus[1, Times[Rational[-1, 2], Power[n, -2], Power[z, 2]]]]], Rule[n, DirectedInfinity[1]]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Limit[Times[Power[n, Times[-1, α]], JacobiP[n, α, β, Cos[Times[Power[n, -1], z]]]], Rule[n, DirectedInfinity[1]]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}` `Complex[0.0, -2.8284271247461903] <- {Rule[Limit[Times[Power[n, Times[-1, α]], JacobiP[n, α, β, Plus[1, Times[Rational[-1, 2], Power[n, -2], Power[z, 2]]]]], Rule[n, DirectedInfinity[1]]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Limit[Times[Power[n, Times[-1, α]], JacobiP[n, α, β, Cos[Times[Power[n, -1], z]]]], Rule[n, DirectedInfinity[1]]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` ... skip entries to safe data
18.11.E5	${\displaystyle{\displaystyle\lim_{n\to\infty}\frac{1}{n^{\alpha}}P^{(\alpha,% \beta)}_{n}\left(\cos\frac{z}{n}\right)=\frac{2^{\alpha}}{z^{\alpha}}J_{\alpha% }\left(z\right)}}$	`limit((1)/((n)^(alpha))JacobiP(n, alpha, beta, cos((z)/(n))), n = infinity)=((2)^(alpha))/((z)^(alpha))BesselJ(alpha, z)`	`Limit[Divide[1,(n)^(\[Alpha])]JacobiP[n, \[Alpha], \[Beta], Cos[Divide[z,n]]], n -> Infinity]=Divide[(2)^(\[Alpha]),(z)^(\[Alpha])]BesselJ[\[Alpha], z]`	Failure	Failure	Skip	Skip
18.11.E7	${\displaystyle{\displaystyle\lim_{n\to\infty}\frac{(-1)^{n}n^{\frac{1}{2}}}{2^% {2n}n!}H_{2n}\left(\frac{z}{2n^{\frac{1}{2}}}\right)=\frac{1}{\pi^{\frac{1}{2}% }}\cos z}}$	`limit(((- 1)^(n)* (n)^((1)/(2)))/((2)^(2n) factorial(n))HermiteH(2n, (z)/(2(n)^((1)/(2)))), n = infinity)=(1)/((Pi)^((1)/(2)))cos(z)`	`Limit[Divide[(- 1)^(n)* (n)^(Divide[1,2]),(2)^(2n) (n)!]HermiteH[2n, Divide[z,2(n)^(Divide[1,2])]], n -> Infinity]=Divide[1,(Pi)^(Divide[1,2])]Cos[z]`	Failure	Failure	Skip	-
18.11.E8	${\displaystyle{\displaystyle\lim_{n\to\infty}\frac{(-1)^{n}}{2^{2n}n!}H_{2n+1}% \left(\frac{z}{2n^{\frac{1}{2}}}\right)=\frac{2}{\pi^{\frac{1}{2}}}\sin z}}$	`limit(((- 1)^(n))/((2)^(2n) factorial(n))HermiteH(2n + 1, (z)/(2(n)^((1)/(2)))), n = infinity)=(2)/((Pi)^((1)/(2)))sin(z)`	`Limit[Divide[(- 1)^(n),(2)^(2n) (n)!]HermiteH[2n + 1, Divide[z,2(n)^(Divide[1,2])]], n -> Infinity]=Divide[2,(Pi)^(Divide[1,2])]Sin[z]`	Failure	Failure	Skip	Skip
18.12.E1	${\displaystyle{\displaystyle\frac{2^{\alpha+\beta}}{R(1+R-z)^{\alpha}(1+R+z)^{% \beta}}=\sum_{n=0}^{\infty}P^{(\alpha,\beta)}_{n}\left(x\right)z^{n}}}$	`((2)^(alpha + beta))/(R(1 + R - z)^(alpha)(1 + R + z)^(beta))= sum(JacobiP(n, alpha, beta, x)*(z)^(n), n = 0..infinity)`	`Divide[(2)^(\[Alpha]+ \[Beta]),R(1 + R - z)^(\[Alpha])(1 + R + z)^(\[Beta])]= Sum[JacobiP[n, \[Alpha], \[Beta], x]*(z)^(n), {n, 0, Infinity}]`	Failure	Failure	Skip	Skip
18.12.E3	${\displaystyle{\displaystyle(1+z)^{-\alpha-\beta-1}\*{{}_{2}F_{1}}\left({% \tfrac{1}{2}(\alpha+\beta+1),\tfrac{1}{2}(\alpha+\beta+2)\atop\beta+1};\frac{2% (x+1)z}{(1+z)^{2}}\right)=\sum_{n=0}^{\infty}\frac{{\left(\alpha+\beta+1\right% )_{n}}}{{\left(\beta+1\right)_{n}}}P^{(\alpha,\beta)}_{n}\left(x\right)z^{n}}}$	`(1 + z)^(- alpha - beta - 1)* hypergeom([(1)/(2)(alpha + beta + 1),(1)/(2)(alpha + beta + 2)], [beta + 1], (2(x + 1) z)/((1 + z)^(2)))= sum((pochhammer(alpha + beta + 1, n))/(pochhammer(beta + 1, n))JacobiP(n, alpha, beta, x)(z)^(n), n = 0..infinity)`	`(1 + z)^(- \[Alpha]- \[Beta]- 1)* HypergeometricPFQ[{Divide[1,2](\[Alpha]+ \[Beta]+ 1),Divide[1,2](\[Alpha]+ \[Beta]+ 2)}, {\[Beta]+ 1}, Divide[2(x + 1) z,(1 + z)^(2)]]= Sum[Divide[Pochhammer[\[Alpha]+ \[Beta]+ 1, n],Pochhammer[\[Beta]+ 1, n]]JacobiP[n, \[Alpha], \[Beta], x](z)^(n), {n, 0, Infinity}]`	Failure	Successful	Skip	-
18.12.E4	${\displaystyle{\displaystyle(1-2xz+z^{2})^{-\lambda}=\sum_{n=0}^{\infty}C^{(% \lambda)}_{n}\left(x\right)z^{n}}}$	`(1 - 2xz + (z)^(2))^(- lambda)= sum(GegenbauerC(n, lambda, x)*(z)^(n), n = 0..infinity)`	`(1 - 2xz + (z)^(2))^(- \[Lambda])= Sum[GegenbauerC[n, \[Lambda], x]*(z)^(n), {n, 0, Infinity}]`	Failure	Successful	Skip	-
18.12.E4	${\displaystyle{\displaystyle\sum_{n=0}^{\infty}C^{(\lambda)}_{n}\left(x\right)% z^{n}=\sum_{n=0}^{\infty}\frac{{\left(2\lambda\right)_{n}}}{{\left(\lambda+% \tfrac{1}{2}\right)_{n}}}P^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}_{n}% \left(x\right)z^{n}}}$	`sum(GegenbauerC(n, lambda, x)(z)^(n), n = 0..infinity)= sum((pochhammer(2lambda, n))/(pochhammer(lambda +(1)/(2), n))JacobiP(n, lambda -(1)/(2), lambda -(1)/(2), x)(z)^(n), n = 0..infinity)`	`Sum[GegenbauerC[n, \[Lambda], x](z)^(n), {n, 0, Infinity}]= Sum[Divide[Pochhammer[2\[Lambda], n],Pochhammer[\[Lambda]+Divide[1,2], n]]JacobiP[n, \[Lambda]-Divide[1,2], \[Lambda]-Divide[1,2], x](z)^(n), {n, 0, Infinity}]`	Successful	Failure	-	Skip
18.12.E5	${\displaystyle{\displaystyle\frac{1-xz}{(1-2xz+z^{2})^{\lambda+1}}=\sum_{n=0}^% {\infty}\frac{n+2\lambda}{2\lambda}C^{(\lambda)}_{n}\left(x\right)z^{n}}}$	`(1 - xz)/((1 - 2xz + (z)^(2))^(lambda + 1))= sum((n + 2lambda)/(2lambda)GegenbauerC(n, lambda, x)*(z)^(n), n = 0..infinity)`	`Divide[1 - xz,(1 - 2xz + (z)^(2))^(\[Lambda]+ 1)]= Sum[Divide[n + 2\[Lambda],2\[Lambda]]GegenbauerC[n, \[Lambda], x]*(z)^(n), {n, 0, Infinity}]`	Failure	Failure	Skip	Skip
18.12.E6	${\displaystyle{\displaystyle\Gamma\left(\lambda+\tfrac{1}{2}\right)e^{z\cos% \theta}(\tfrac{1}{2}z\sin\theta)^{\frac{1}{2}-\lambda}J_{\lambda-\frac{1}{2}}% \left(z\sin\theta\right)=\sum_{n=0}^{\infty}\frac{C^{(\lambda)}_{n}\left(\cos% \theta\right)}{{\left(2\lambda\right)_{n}}}z^{n}}}$	`GAMMA(lambda +(1)/(2))exp(zcos(theta))((1)/(2)zsin(theta))^((1)/(2)- lambda) BesselJ(lambda -(1)/(2), zsin(theta))= sum((GegenbauerC(n, lambda, cos(theta)))/(pochhammer(2lambda, n))*(z)^(n), n = 0..infinity)`	`Gamma[\[Lambda]+Divide[1,2]]Exp[zCos[\[Theta]]](Divide[1,2]zSin[\[Theta]])^(Divide[1,2]- \[Lambda]) BesselJ[\[Lambda]-Divide[1,2], zSin[\[Theta]]]= Sum[Divide[GegenbauerC[n, \[Lambda], Cos[\[Theta]]],Pochhammer[2\[Lambda], n]]*(z)^(n), {n, 0, Infinity}]`	Failure	Successful	Skip	-
18.12.E7	${\displaystyle{\displaystyle\frac{1-z^{2}}{1-2xz+z^{2}}=1+2\sum_{n=1}^{\infty}% T_{n}\left(x\right)z^{n}}}$	`(1 - (z)^(2))/(1 - 2xz + (z)^(2))= 1 + 2sum(ChebyshevT(n, x)(z)^(n), n = 1..infinity)`	`Divide[1 - (z)^(2),1 - 2xz + (z)^(2)]= 1 + 2Sum[ChebyshevT[n, x](z)^(n), {n, 1, Infinity}]`	Failure	Successful	Skip	-
18.12.E8	${\displaystyle{\displaystyle\frac{1-xz}{1-2xz+z^{2}}=\sum_{n=0}^{\infty}T_{n}% \left(x\right)z^{n}}}$	`(1 - xz)/(1 - 2xz + (z)^(2))= sum(ChebyshevT(n, x)(z)^(n), n = 0..infinity)`	`Divide[1 - xz,1 - 2xz + (z)^(2)]= Sum[ChebyshevT[n, x](z)^(n), {n, 0, Infinity}]`	Failure	Successful	Skip	-
18.12.E9	${\displaystyle{\displaystyle-\ln\left(1-2xz+z^{2}\right)=2\sum_{n=1}^{\infty}% \frac{T_{n}\left(x\right)}{n}z^{n}}}$	`- ln(1 - 2xz + (z)^(2))= 2sum((ChebyshevT(n, x))/(n)(z)^(n), n = 1..infinity)`	`- Log[1 - 2xz + (z)^(2)]= 2Sum[Divide[ChebyshevT[n, x],n](z)^(n), {n, 1, Infinity}]`	Failure	Failure	Skip	Successful
18.12.E10	${\displaystyle{\displaystyle\frac{1}{1-2xz+z^{2}}=\sum_{n=0}^{\infty}U_{n}% \left(x\right)z^{n}}}$	`(1)/(1 - 2xz + (z)^(2))= sum(ChebyshevU(n, x)*(z)^(n), n = 0..infinity)`	`Divide[1,1 - 2xz + (z)^(2)]= Sum[ChebyshevU[n, x]*(z)^(n), {n, 0, Infinity}]`	Failure	Successful	Skip	-
18.12.E11	${\displaystyle{\displaystyle\frac{1}{\sqrt{1-2xz+z^{2}}}=\sum_{n=0}^{\infty}P_% {n}\left(x\right)z^{n}}}$	`(1)/(sqrt(1 - 2xz + (z)^(2)))= sum(LegendreP(n, x)*(z)^(n), n = 0..infinity)`	`Divide[1,Sqrt[1 - 2xz + (z)^(2)]]= Sum[LegendreP[n, x]*(z)^(n), {n, 0, Infinity}]`	Failure	Successful	Skip	-
18.12.E12	${\displaystyle{\displaystyle e^{xz}J_{0}\left(z\sqrt{1-x^{2}}\right)=\sum_{n=0% }^{\infty}\frac{P_{n}\left(x\right)}{n!}z^{n}}}$	`exp(xz)BesselJ(0, zsqrt(1 - (x)^(2)))= sum((LegendreP(n, x))/(factorial(n))(z)^(n), n = 0..infinity)`	`Exp[xz]BesselJ[0, zSqrt[1 - (x)^(2)]]= Sum[Divide[LegendreP[n, x],(n)!](z)^(n), {n, 0, Infinity}]`	Failure	Successful	Skip	-
18.12.E15	${\displaystyle{\displaystyle e^{2xz-z^{2}}=\sum_{n=0}^{\infty}\frac{H_{n}\left% (x\right)}{n!}z^{n}}}$	`exp(2xz - (z)^(2))= sum((HermiteH(n, x))/(factorial(n))*(z)^(n), n = 0..infinity)`	`Exp[2xz - (z)^(2)]= Sum[Divide[HermiteH[n, x],(n)!]*(z)^(n), {n, 0, Infinity}]`	Error	Successful	-	-
18.14.E1	${\displaystyle{\displaystyle\|P^{(\alpha,\beta)}_{n}\left(x\right)\|<=P^{(\alpha% ,\beta)}_{n}\left(1\right)}}$	`abs(JacobiP(n, alpha, beta, x))< = JacobiP(n, alpha, beta, 1)`	`Abs[JacobiP[n, \[Alpha], \[Beta], x]]< = JacobiP[n, \[Alpha], \[Beta], 1]`	Failure	Failure	Fail `3.528680673 <= 1.500000000+0.I <- {beta = 2^(1/2)+I2^(1/2), n = 1, x = 2, alpha = 1/2}` `5.595865303 <= 1.500000000+0.I <- {beta = 2^(1/2)+I2^(1/2), n = 1, x = 3, alpha = 1/2}` `11.98055245 <= 1.875000000+0.I <- {beta = 2^(1/2)+I2^(1/2), n = 2, x = 2, alpha = 1/2}` `29.86867781 <= 1.875000000+0.I <- {beta = 2^(1/2)+I2^(1/2), n = 2, x = 3, alpha = 1/2}` ... skip entries to safe data	Successful
18.14.E1	${\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(1\right)=\frac{{\left% (\alpha+1\right)_{n}}}{n!}}}$	`JacobiP(n, alpha, beta, 1)=(pochhammer(alpha + 1, n))/(factorial(n))`	`JacobiP[n, \[Alpha], \[Beta], 1]=Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]`	Successful	Successful	-	-
18.14.E2	${\displaystyle{\displaystyle\|P^{(\alpha,\beta)}_{n}\left(x\right)\|<=\|P^{(% \alpha,\beta)}_{n}\left(-1\right)\|=\frac{{\left(\beta+1\right)_{n}}}{n!}}}$	`abs(JacobiP(n, alpha, beta, x))< =abs(JacobiP(n, alpha, beta, - 1))=(pochhammer(beta + 1, n))/(factorial(n))`	`Abs[JacobiP[n, \[Alpha], \[Beta], x]]< =Abs[JacobiP[n, \[Alpha], \[Beta], - 1]]=Divide[Pochhammer[\[Beta]+ 1, n],(n)!]`	Failure	Failure	Error	Fail `1.5 <- {Rule[n, 1], Rule[β, Rational[1, 2]]}` `1.875 <- {Rule[n, 2], Rule[β, Rational[1, 2]]}` `2.1875 <- {Rule[n, 3], Rule[β, Rational[1, 2]]}`
18.14.E4	${\displaystyle{\displaystyle\|C^{(\lambda)}_{n}\left(x\right)\|<=C^{(\lambda)}_{% n}\left(1\right)}}$	`abs(GegenbauerC(n, lambda, x))< = GegenbauerC(n, lambda, 1)`	`Abs[GegenbauerC[n, \[Lambda], x]]< = GegenbauerC[n, \[Lambda], 1]`	Failure	Failure	Fail `2.000000000 <= 1.000000000 <- {n = 1, x = 2, lambda = 1/2}` `3.000000000 <= 1.000000000 <- {n = 1, x = 3, lambda = 1/2}` `5.500000000 <= 1.000000000 <- {n = 2, x = 2, lambda = 1/2}` `13.00000000 <= 1.000000000 <- {n = 2, x = 3, lambda = 1/2}` ... skip entries to safe data	Successful
18.14.E4	${\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(1\right)=\frac{{\left(2% \lambda\right)_{n}}}{n!}}}$	`GegenbauerC(n, lambda, 1)=(pochhammer(2*lambda, n))/(factorial(n))`	`GegenbauerC[n, \[Lambda], 1]=Divide[Pochhammer[2*\[Lambda], n],(n)!]`	Successful	Successful	-	-
18.14.E5	${\displaystyle{\displaystyle\|C^{(\lambda)}_{2m}\left(x\right)\|<=\|C^{(\lambda)}% _{2m}\left(0\right)\|=\left\|\frac{{\left(\lambda\right)_{m}}}{m!}\right\|}}$	`abs(GegenbauerC(2m, lambda, x))< =abs(GegenbauerC(2m, lambda, 0))=abs((pochhammer(lambda, m))/(factorial(m)))`	`Abs[GegenbauerC[2m, \[Lambda], x]]< =Abs[GegenbauerC[2m, \[Lambda], 0]]=Abs[Divide[Pochhammer[\[Lambda], m],(m)!]]`	Failure	Failure	Error	Successful
18.14.E6	${\displaystyle{\displaystyle\|C^{(\lambda)}_{2m+1}\left(x\right)\|<\frac{-2{% \left(\lambda\right)_{m+1}}}{\left((2m+1)(2\lambda+2m+1)\right)^{\frac{1}{2}}m% !}}}$	`abs(GegenbauerC(2m + 1, lambda, x))<(- 2pochhammer(lambda, m + 1))/(((2m + 1)(2lambda + 2m + 1))^((1)/(2))* factorial(m))`	`Abs[GegenbauerC[2m + 1, \[Lambda], x]]<Divide[- 2Pochhammer[\[Lambda], m + 1],((2m + 1)(2\[Lambda]+ 2m + 1))^(Divide[1,2])* (m)!]`	Failure	Failure	Skip	Successful
18.14.E9	${\displaystyle{\displaystyle\frac{1}{(2^{n}n!)^{\frac{1}{2}}}e^{-\frac{1}{2}x^% {2}}\|H_{n}\left(x\right)\|<=1}}$	`(1)/(((2)^(n)* factorial(n))^((1)/(2)))exp(-(1)/(2)(x)^(2))*abs(HermiteH(n, x))< = 1`	`Divide[1,((2)^(n)* (n)!)^(Divide[1,2])]Exp[-Divide[1,2](x)^(2)]*Abs[HermiteH[n, x]]< = 1`	Failure	Failure	Skip	Successful
18.14.E10	${\displaystyle{\displaystyle(P_{n}\left(x\right))^{2}>=P_{n-1}\left(x\right)P_% {n+1}\left(x\right)}}$	`(LegendreP(n, x))^(2)> = LegendreP(n - 1, x)*LegendreP(n + 1, x)`	`(LegendreP[n, x])^(2)> = LegendreP[n - 1, x]*LegendreP[n + 1, x]`	Failure	Failure	Skip	Successful
18.14.E13	${\displaystyle{\displaystyle(H_{n}\left(x\right))^{2}>=H_{n-1}\left(x\right)H_% {n+1}\left(x\right)}}$	`(HermiteH(n, x))^(2)> = HermiteH(n - 1, x)*HermiteH(n + 1, x)`	`(HermiteH[n, x])^(2)> = HermiteH[n - 1, x]*HermiteH[n + 1, x]`	Failure	Failure	Skip	Successful
18.14#Ex1	${\displaystyle{\displaystyle\|P^{(\alpha,\beta)}_{n}\left(x_{n,0}\right)\|>\|P^{(% \alpha,\beta)}_{n}\left(x_{n,1}\right)\|}}$	`abs(JacobiP(n, alpha, beta, x[n , 0]))>abs(JacobiP(n, alpha, beta, x[n , 1]))`	`Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 0]]]>Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 1]]]`	Failure	Failure	Skip	Successful
18.14#Ex2	${\displaystyle{\displaystyle\|P^{(\alpha,\beta)}_{n}\left(x_{n,n}\right)\|>\|P^{(% \alpha,\beta)}_{n}\left(x_{n,n-1}\right)\|}}$	`abs(JacobiP(n, alpha, beta, x[n , n]))>abs(JacobiP(n, alpha, beta, x[n , n - 1]))`	`Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , n]]]>Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , n - 1]]]`	Failure	Failure	Skip	Successful
18.14#Ex3	${\displaystyle{\displaystyle\|P^{(\alpha,\beta)}_{n}\left(x_{n,0}\right)\|<\|P^{(% \alpha,\beta)}_{n}\left(x_{n,1}\right)\|}}$	`abs(JacobiP(n, alpha, beta, x[n , 0]))<abs(JacobiP(n, alpha, beta, x[n , 1]))`	`Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 0]]]<Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 1]]]`	Failure	Failure	Skip	Successful
18.14#Ex4	${\displaystyle{\displaystyle\|P^{(\alpha,\beta)}_{n}\left(x_{n,n}\right)\|<\|P^{(% \alpha,\beta)}_{n}\left(x_{n,n-1}\right)\|}}$	`abs(JacobiP(n, alpha, beta, x[n , n]))<abs(JacobiP(n, alpha, beta, x[n , n - 1]))`	`Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , n]]]<Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , n - 1]]]`	Failure	Failure	Skip	Successful
18.14.E18	${\displaystyle{\displaystyle\|P^{(\alpha,\beta)}_{n}\left(x_{n,0}\right)\|<\|P^{(% \alpha,\beta)}_{n}\left(x_{n,1}\right)\|}}$	`abs(JacobiP(n, alpha, beta, x[n , 0]))<abs(JacobiP(n, alpha, beta, x[n , 1]))`	`Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 0]]]<Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 1]]]`	Failure	Failure	Skip	Successful
18.14.E19	${\displaystyle{\displaystyle\|P^{(\alpha,\beta)}_{n}\left(x_{n,0}\right)\|>\|P^{(% \alpha,\beta)}_{n}\left(x_{n,1}\right)\|}}$	`abs(JacobiP(n, alpha, beta, x[n , 0]))>abs(JacobiP(n, alpha, beta, x[n , 1]))`	`Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 0]]]>Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 1]]]`	Failure	Failure	Skip	Successful
18.14.E20	${\displaystyle{\displaystyle\left\|\frac{P^{(\alpha,\beta)}_{n}\left(x_{n,n-m}% \right)}{P^{(\alpha,\beta)}_{n}\left(1\right)}\right\|>\left\|\frac{P^{(\alpha,% \beta)}_{n+1}\left(x_{n+1,n-m+1}\right)}{P^{(\alpha,\beta)}_{n+1}\left(1\right% )}\right\|}}$	`abs((JacobiP(n, alpha, beta, x[n , n - m]))/(JacobiP(n, alpha, beta, 1)))>abs((JacobiP(n + 1, alpha, beta, x[n + 1 , n - m + 1]))/(JacobiP(n + 1, alpha, beta, 1)))`	`Abs[Divide[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , n - m]],JacobiP[n, \[Alpha], \[Beta], 1]]]>Abs[Divide[JacobiP[n + 1, \[Alpha], \[Beta], Subscript[x, n + 1 , n - m + 1]],JacobiP[n + 1, \[Alpha], \[Beta], 1]]]`	Failure	Failure	Skip	Skip
18.15.E6	${\displaystyle{\displaystyle(\sin\tfrac{1}{2}\theta)^{\alpha+\frac{1}{2}}(\cos% \tfrac{1}{2}\theta)^{\beta+\frac{1}{2}}P^{(\alpha,\beta)}_{n}\left(\cos\theta% \right)=\frac{\Gamma\left(n+\alpha+1\right)}{2^{\frac{1}{2}}\rho^{\alpha}n!}\*% \left(\theta^{\frac{1}{2}}J_{\alpha}\left(\rho\theta\right)\sum_{m=0}^{M}% \dfrac{A_{m}(\theta)}{\rho^{2m}}+\theta^{\frac{3}{2}}J_{\alpha+1}\left(\rho% \theta\right)\sum_{m=0}^{M-1}\dfrac{B_{m}(\theta)}{\rho^{2m+1}}+\varepsilon_{M% }(\rho,\theta)\right)}}$	`(sin((1)/(2)theta))^(alpha +(1)/(2))(cos((1)/(2)theta))^(beta +(1)/(2)) JacobiP(n, alpha, beta, cos(theta))=(GAMMA(n + alpha + 1))/((2)^((1)/(2))* (rho)^(alpha)* factorial(n))((theta)^((1)/(2)) BesselJ(alpha, rhotheta)sum((A[m](theta))/((rho)^(2m)), m = 0..M)+ (theta)^((3)/(2))* BesselJ(alpha + 1, rhotheta)sum((B[m](theta))/((rho)^(2m + 1)), m = 0..M - 1)+ varepsilon[M]*(rho , theta))`	(Sin[Divide[1,2]\[Theta]])^(\[Alpha]+Divide[1,2])(Cos[Divide[1,2]\[Theta]])^(\[Beta]+Divide[1,2]) JacobiP[n, \[Alpha], \[Beta], Cos[\[Theta]]]=Divide[Gamma[n + \[Alpha]+ 1],(2)^(Divide[1,2])* (\[Rho])^(\[Alpha])* (n)!]((\[Theta])^(Divide[1,2]) BesselJ[\[Alpha], \[Rho]\[Theta]]Sum[Divide[Subscript[A, m](\[Theta]),(\[Rho])^(2m)], {m, 0, M}]+ (\[Theta])^(Divide[3,2])* BesselJ[\[Alpha]+ 1, \[Rho]\[Theta]]Sum[Divide[Subscript[B, m](\[Theta]),(\[Rho])^(2m + 1)], {m, 0, M - 1}]+ Subscript[\[CurlyEpsilon], M]*(\[Rho], \[Theta]))	Failure	Failure	Skip	Error
18.15.E28	${\displaystyle{\displaystyle H_{n}\left(x\right)=2^{\frac{1}{4}(\mu^{2}-1)}e^{% \frac{1}{2}\mu^{2}t^{2}}U\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)}}$	`HermiteH(n, x)= (2)^((1)/(4)((mu)^(2)- 1)) exp((1)/(2)(mu)^(2) (t)^(2))CylinderU(-(1)/(2)(mu)^(2), mutsqrt(2))`	`HermiteH[n, x]= (2)^(Divide[1,4]((\[Mu])^(2)- 1)) Exp[Divide[1,2](\[Mu])^(2) (t)^(2)]ParabolicCylinderD[--Divide[1,2](\[Mu])^(2) - 1/2, \[Mu]tSqrt[2]]`	Failure	Failure	Fail `2.014197760+.2431743734e-2I <- {mu = 2^(1/2)+I2^(1/2), t = 2^(1/2)+I2^(1/2), n = 1, x = 1}` `4.014197760+.2431743734e-2I <- {mu = 2^(1/2)+I2^(1/2), t = 2^(1/2)+I2^(1/2), n = 1, x = 2}` `6.014197760+.2431743734e-2I <- {mu = 2^(1/2)+I2^(1/2), t = 2^(1/2)+I2^(1/2), n = 1, x = 3}` `2.014197760+.2431743734e-2I <- {mu = 2^(1/2)+I2^(1/2), t = 2^(1/2)+I2^(1/2), n = 2, x = 1}` ... skip entries to safe data	Skip
18.16.E16	${\displaystyle{\displaystyle(2n+1)^{\frac{1}{2}}>x_{n,1}}}$	`(2*n + 1)^((1)/(2))> x[n , 1]`	`(2*n + 1)^(Divide[1,2])> Subscript[x, n , 1]`	Failure	Failure	Successful	Successful
18.16.E16	${\displaystyle{\displaystyle x_{n,1}>x_{n,2}}}$	`x[n , 1]> x[n , 2]`	`Subscript[x, n , 1]> Subscript[x, n , 2]`	Failure	Failure	Fail `1.414213562+1.414213562I < 1.414213562+1.414213562I <- {x[n,1] = 2^(1/2)+I2^(1/2), x[n,2] = 2^(1/2)+I2^(1/2)}` `1.414213562-1.414213562I < 1.414213562-1.414213562I <- {x[n,1] = 2^(1/2)-I2^(1/2), x[n,2] = 2^(1/2)-I2^(1/2)}` `-1.414213562-1.414213562I < -1.414213562-1.414213562I <- {x[n,1] = -2^(1/2)-I2^(1/2), x[n,2] = -2^(1/2)-I2^(1/2)}` `-1.414213562+1.414213562I < -1.414213562+1.414213562I <- {x[n,1] = -2^(1/2)+I2^(1/2), x[n,2] = -2^(1/2)+I2^(1/2)}`	Successful
18.16.E16	${\displaystyle{\displaystyle x_{n,2}>\cdots}}$	`x[n , 2]> ..`	`Subscript[x, n , 2]> ...`	Failure	Failure	Skip	Successful
18.17.E1	${\displaystyle{\displaystyle 2n\int_{0}^{x}(1-y)^{\alpha}(1+y)^{\beta}P^{(% \alpha,\beta)}_{n}\left(y\right)\mathrm{d}y=P^{(\alpha+1,\beta+1)}_{n-1}\left(% 0\right)-(1-x)^{\alpha+1}(1+x)^{\beta+1}P^{(\alpha+1,\beta+1)}_{n-1}\left(x% \right)}}$	`2nint((1 - y)^(alpha)(1 + y)^(beta) JacobiP(n, alpha, beta, y), y = 0..x)= JacobiP(n - 1, alpha + 1, beta + 1, 0)-(1 - x)^(alpha + 1)(1 + x)^(beta + 1) JacobiP(n - 1, alpha + 1, beta + 1, x)`	`2nIntegrate[(1 - y)^(\[Alpha])(1 + y)^(\[Beta]) JacobiP[n, \[Alpha], \[Beta], y], {y, 0, x}]= JacobiP[n - 1, \[Alpha]+ 1, \[Beta]+ 1, 0]-(1 - x)^(\[Alpha]+ 1)(1 + x)^(\[Beta]+ 1) JacobiP[n - 1, \[Alpha]+ 1, \[Beta]+ 1, x]`	Failure	Failure	Skip	Skip
18.17.E2	${\displaystyle{\displaystyle\int_{0}^{x}L_{m}\left(y\right)L_{n}\left(x-y% \right)\mathrm{d}y=\int_{0}^{x}L_{m+n}\left(y\right)\mathrm{d}y}}$	`int(LaguerreL(m, y)*LaguerreL(n, x - y), y = 0..x)= int(LaguerreL(m + n, y), y = 0..x)`	`Error`	Failure	Error	-	-
18.17.E2	${\displaystyle{\displaystyle\int_{0}^{x}L_{m+n}\left(y\right)\mathrm{d}y=L_{m+% n}\left(x\right)-L_{m+n+1}\left(x\right)}}$	`int(LaguerreL(m + n, y), y = 0..x)= LaguerreL(m + n, x)- LaguerreL(m + n + 1, x)`	`Error`	Successful	Error	-	-
18.17.E3	${\displaystyle{\displaystyle\int_{0}^{x}H_{n}\left(y\right)\mathrm{d}y=\frac{1% }{2(n+1)}(H_{n+1}\left(x\right)-H_{n+1}\left(0\right))}}$	`int(HermiteH(n, y), y = 0..x)=(1)/(2(n + 1))(HermiteH(n + 1, x)- HermiteH(n + 1, 0))`	`Integrate[HermiteH[n, y], {y, 0, x}]=Divide[1,2(n + 1)](HermiteH[n + 1, x]- HermiteH[n + 1, 0])`	Failure	Successful	Skip	-
18.17.E4	${\displaystyle{\displaystyle\int_{0}^{x}e^{-y^{2}}H_{n}\left(y\right)\mathrm{d% }y=H_{n-1}\left(0\right)-e^{-x^{2}}H_{n-1}\left(x\right)}}$	`int(exp(- (y)^(2))HermiteH(n, y), y = 0..x)= HermiteH(n - 1, 0)- exp(- (x)^(2))HermiteH(n - 1, x)`	`Integrate[Exp[- (y)^(2)]HermiteH[n, y], {y, 0, x}]= HermiteH[n - 1, 0]- Exp[- (x)^(2)]HermiteH[n - 1, x]`	Failure	Successful	Skip	-
18.17.E6	${\displaystyle{\displaystyle P_{n}\left(\cos\theta_{1}\right)P_{n}\left(\cos% \theta_{2}\right)=\frac{1}{\pi}\int_{0}^{\pi}P_{n}\left(\cos\theta_{1}\cos% \theta_{2}+\sin\theta_{1}\sin\theta_{2}\cos\phi\right)\mathrm{d}\phi}}$	`LegendreP(n, cos(theta[1]))LegendreP(n, cos(theta[2]))=(1)/(Pi)int(LegendreP(n, cos(theta[1])cos(theta[2])+ sin(theta[1])sin(theta[2])*cos(phi)), phi = 0..Pi)`	`LegendreP[n, Cos[Subscript[\[Theta], 1]]]LegendreP[n, Cos[Subscript[\[Theta], 2]]]=Divide[1,Pi]Integrate[LegendreP[n, Cos[Subscript[\[Theta], 1]]Cos[Subscript[\[Theta], 2]]+ Sin[Subscript[\[Theta], 1]]Sin[Subscript[\[Theta], 2]]*Cos[\[Phi]]], {\[Phi], 0, Pi}]`	Failure	Failure	Skip	Skip
18.17.E7	${\displaystyle{\displaystyle\left(P_{n}\left(x\right)\right)^{2}+4\pi^{-2}% \left(\mathsf{Q}_{n}\left(x\right)\right)^{2}=4\pi^{-2}\*\int_{1}^{\infty}Q_{n% }\left(x^{2}+(1-x^{2})t\right)(t^{2}-1)^{-\frac{1}{2}}\mathrm{d}t}}$	`(LegendreP(n, x))^(2)+ 4(Pi)^(- 2)(LegendreQ(n, x))^(2)= 4(Pi)^(- 2) int(LegendreQ(n, (x)^(2)+(1 - (x)^(2))* t)*((t)^(2)- 1)^(-(1)/(2)), t = 1..infinity)`	`(LegendreP[n, x])^(2)+ 4(Pi)^(- 2)(LegendreQ[n, x])^(2)= 4(Pi)^(- 2) Integrate[LegendreQ[n, 0, 3, (x)^(2)+(1 - (x)^(2))* t]*((t)^(2)- 1)^(-Divide[1,2]), {t, 1, Infinity}]`	Failure	Failure	-	-
18.17.E8	${\displaystyle{\displaystyle\left(H_{n}\left(x\right)\right)^{2}+2^{n}(n!)^{2}% e^{x^{2}}\left(V\left(-n-\tfrac{1}{2},2^{\frac{1}{2}}x\right)\right)^{2}=\frac% {2^{n+\frac{3}{2}}n!\,e^{x^{2}}}{\pi}\int_{0}^{\infty}\frac{e^{-(2n+1)t+x^{2}% \tanh t}}{(\sinh 2t)^{\frac{1}{2}}}\mathrm{d}t}}$	`(HermiteH(n, x))^(2)+ (2)^(n)(factorial(n))^(2) exp((x)^(2))(CylinderV(- n -(1)/(2), (2)^((1)/(2)) x))^(2)=((2)^(n +(3)/(2))* factorial(n)exp((x)^(2)))/(Pi)int((exp(-(2n + 1) t + (x)^(2)* tanh(t)))/((sinh(2*t))^((1)/(2))), t = 0..infinity)`	`Error`	Failure	Error	Skip	-
18.17.E9	${\displaystyle{\displaystyle\frac{(1-x)^{\alpha+\mu}P^{(\alpha+\mu,\beta-\mu)}% _{n}\left(x\right)}{\Gamma\left(\alpha+\mu+n+1\right)}=\int_{x}^{1}\frac{(1-y)% ^{\alpha}P^{(\alpha,\beta)}_{n}\left(y\right)}{\Gamma\left(\alpha+n+1\right)}% \frac{(y-x)^{\mu-1}}{\Gamma\left(\mu\right)}\mathrm{d}y}}$	`((1 - x)^(alpha + mu)* JacobiP(n, alpha + mu, beta - mu, x))/(GAMMA(alpha + mu + n + 1))= int(((1 - y)^(alpha)* JacobiP(n, alpha, beta, y))/(GAMMA(alpha + n + 1))*((y - x)^(mu - 1))/(GAMMA(mu)), y = x..1)`	`Divide[(1 - x)^(\[Alpha]+ \[Mu])* JacobiP[n, \[Alpha]+ \[Mu], \[Beta]- \[Mu], x],Gamma[\[Alpha]+ \[Mu]+ n + 1]]= Integrate[Divide[(1 - y)^(\[Alpha])* JacobiP[n, \[Alpha], \[Beta], y],Gamma[\[Alpha]+ n + 1]]*Divide[(y - x)^(\[Mu]- 1),Gamma[\[Mu]]], {y, x, 1}]`	Failure	Failure	Skip	Error
18.17.E10	${\displaystyle{\displaystyle\frac{x^{\beta+\mu}(x+1)^{n}}{\Gamma\left(\beta+% \mu+n+1\right)}P^{(\alpha,\beta+\mu)}_{n}\left(\frac{x-1}{x+1}\right)=\int_{0}% ^{x}\frac{y^{\beta}(y+1)^{n}}{\Gamma\left(\beta+n+1\right)}P^{(\alpha,\beta)}_% {n}\left(\frac{y-1}{y+1}\right)\*\frac{(x-y)^{\mu-1}}{\Gamma\left(\mu\right)}% \mathrm{d}y}}$	`((x)^(beta + mu)(x + 1)^(n))/(GAMMA(beta + mu + n + 1))JacobiP(n, alpha, beta + mu, (x - 1)/(x + 1))= int(((y)^(beta)(y + 1)^(n))/(GAMMA(beta + n + 1))JacobiP(n, alpha, beta, (y - 1)/(y + 1))*((x - y)^(mu - 1))/(GAMMA(mu)), y = 0..x)`	`Divide[(x)^(\[Beta]+ \[Mu])(x + 1)^(n),Gamma[\[Beta]+ \[Mu]+ n + 1]]JacobiP[n, \[Alpha], \[Beta]+ \[Mu], Divide[x - 1,x + 1]]= Integrate[Divide[(y)^(\[Beta])(y + 1)^(n),Gamma[\[Beta]+ n + 1]]JacobiP[n, \[Alpha], \[Beta], Divide[y - 1,y + 1]]*Divide[(x - y)^(\[Mu]- 1),Gamma[\[Mu]]], {y, 0, x}]`	Failure	Failure	Skip	Error
18.17.E11	${\displaystyle{\displaystyle\frac{\Gamma\left(n+\alpha+\beta-\mu+1\right)}{x^{% n+\alpha+\beta-\mu+1}}P^{(\alpha,\beta-\mu)}_{n}\left(1-2x^{-1}\right)=\int_{x% }^{\infty}\frac{\Gamma\left(n+\alpha+\beta+1\right)}{y^{n+\alpha+\beta+1}}P^{(% \alpha,\beta)}_{n}\left(1-2y^{-1}\right)\*\frac{(y-x)^{\mu-1}}{\Gamma\left(\mu% \right)}\mathrm{d}y}}$	`(GAMMA(n + alpha + beta - mu + 1))/((x)^(n + alpha + beta - mu + 1))JacobiP(n, alpha, beta - mu, 1 - 2(x)^(- 1))= int((GAMMA(n + alpha + beta + 1))/((y)^(n + alpha + beta + 1))JacobiP(n, alpha, beta, 1 - 2(y)^(- 1))*((y - x)^(mu - 1))/(GAMMA(mu)), y = x..infinity)`	`Divide[Gamma[n + \[Alpha]+ \[Beta]- \[Mu]+ 1],(x)^(n + \[Alpha]+ \[Beta]- \[Mu]+ 1)]JacobiP[n, \[Alpha], \[Beta]- \[Mu], 1 - 2(x)^(- 1)]= Integrate[Divide[Gamma[n + \[Alpha]+ \[Beta]+ 1],(y)^(n + \[Alpha]+ \[Beta]+ 1)]JacobiP[n, \[Alpha], \[Beta], 1 - 2(y)^(- 1)]*Divide[(y - x)^(\[Mu]- 1),Gamma[\[Mu]]], {y, x, Infinity}]`	Failure	Failure	Skip	Error
18.17.E12	${\displaystyle{\displaystyle\frac{\Gamma\left(\lambda-\mu\right)C^{(\lambda-% \mu)}_{n}\left(x^{-\frac{1}{2}}\right)}{x^{\lambda-\mu+\frac{1}{2}n}}=\int_{x}% ^{\infty}\frac{\Gamma\left(\lambda\right)C^{(\lambda)}_{n}\left(y^{-\frac{1}{2% }}\right)}{y^{\lambda+\frac{1}{2}n}}\frac{(y-x)^{\mu-1}}{\Gamma\left(\mu\right% )}\mathrm{d}y}}$	`(GAMMA(lambda - mu)GegenbauerC(n, lambda - mu, (x)^(-(1)/(2))))/((x)^(lambda - mu +(1)/(2)n))= int((GAMMA(lambda)GegenbauerC(n, lambda, (y)^(-(1)/(2))))/((y)^(lambda +(1)/(2)n))*((y - x)^(mu - 1))/(GAMMA(mu)), y = x..infinity)`	`Divide[Gamma[\[Lambda]- \[Mu]]GegenbauerC[n, \[Lambda]- \[Mu], (x)^(-Divide[1,2])],(x)^(\[Lambda]- \[Mu]+Divide[1,2]n)]= Integrate[Divide[Gamma[\[Lambda]]GegenbauerC[n, \[Lambda], (y)^(-Divide[1,2])],(y)^(\[Lambda]+Divide[1,2]n)]*Divide[(y - x)^(\[Mu]- 1),Gamma[\[Mu]]], {y, x, Infinity}]`	Failure	Failure	Skip	Error
18.17.E13	${\displaystyle{\displaystyle\frac{x^{\frac{1}{2}n}(x-1)^{\lambda+\mu-\frac{1}{% 2}}}{\Gamma\left(\lambda+\mu+\tfrac{1}{2}\right)}\frac{C^{(\lambda+\mu)}_{n}% \left(x^{-\frac{1}{2}}\right)}{C^{(\lambda+\mu)}_{n}\left(1\right)}=\int_{1}^{% x}\frac{y^{\frac{1}{2}n}(y-1)^{\lambda-\frac{1}{2}}}{\Gamma\left(\lambda+% \tfrac{1}{2}\right)}\frac{C^{(\lambda)}_{n}\left(y^{-\frac{1}{2}}\right)}{C^{(% \lambda)}_{n}\left(1\right)}\frac{(x-y)^{\mu-1}}{\Gamma\left(\mu\right)}% \mathrm{d}y}}$	`((x)^((1)/(2)n)(x - 1)^(lambda + mu -(1)/(2)))/(GAMMA(lambda + mu +(1)/(2)))(GegenbauerC(n, lambda + mu, (x)^(-(1)/(2))))/(GegenbauerC(n, lambda + mu, 1))= int(((y)^((1)/(2)n)(y - 1)^(lambda -(1)/(2)))/(GAMMA(lambda +(1)/(2)))(GegenbauerC(n, lambda, (y)^(-(1)/(2))))/(GegenbauerC(n, lambda, 1))*((x - y)^(mu - 1))/(GAMMA(mu)), y = 1..x)`	`Divide[(x)^(Divide[1,2]n)(x - 1)^(\[Lambda]+ \[Mu]-Divide[1,2]),Gamma[\[Lambda]+ \[Mu]+Divide[1,2]]]Divide[GegenbauerC[n, \[Lambda]+ \[Mu], (x)^(-Divide[1,2])],GegenbauerC[n, \[Lambda]+ \[Mu], 1]]= Integrate[Divide[(y)^(Divide[1,2]n)(y - 1)^(\[Lambda]-Divide[1,2]),Gamma[\[Lambda]+Divide[1,2]]]Divide[GegenbauerC[n, \[Lambda], (y)^(-Divide[1,2])],GegenbauerC[n, \[Lambda], 1]]*Divide[(x - y)^(\[Mu]- 1),Gamma[\[Mu]]], {y, 1, x}]`	Failure	Failure	Skip	Error
18.17.E16	${\displaystyle{\displaystyle\int_{-1}^{1}(1-x)^{\alpha}(1+x)^{\beta}P^{(\alpha% ,\beta)}_{n}\left(x\right)e^{ixy}\mathrm{d}x=\frac{(iy)^{n}e^{iy}}{n!}2^{n+% \alpha+\beta+1}\mathrm{B}\left(n+\alpha+1,n+\beta+1\right){{}_{1}F_{1}}\left(n% +\alpha+1;2n+\alpha+\beta+2;-2iy\right)}}$	`int((1 - x)^(alpha)(1 + x)^(beta) JacobiP(n, alpha, beta, x)exp(Ixy), x = - 1..1)=((Iy)^(n)* exp(Iy))/(factorial(n))(2)^(n + alpha + beta + 1)* Beta(n + alpha + 1, n + beta + 1)hypergeom([n + alpha + 1], [2n + alpha + beta + 2], - 2Iy)`	`Integrate[(1 - x)^(\[Alpha])(1 + x)^(\[Beta]) JacobiP[n, \[Alpha], \[Beta], x]Exp[Ixy], {x, - 1, 1}]=Divide[(Iy)^(n)* Exp[Iy],(n)!](2)^(n + \[Alpha]+ \[Beta]+ 1)* Beta[n + \[Alpha]+ 1, n + \[Beta]+ 1]HypergeometricPFQ[{n + \[Alpha]+ 1}, {2n + \[Alpha]+ \[Beta]+ 2}, - 2Iy]`	Failure	Failure	Skip	Error
18.17.E17	${\displaystyle{\displaystyle\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(% \lambda)}_{2n}\left(x\right)\cos\left(xy\right)\mathrm{d}x=\frac{(-1)^{n}\pi% \Gamma\left(2n+2\lambda\right)J_{\lambda+2n}\left(y\right)}{(2n)!\Gamma\left(% \lambda\right)(2y)^{\lambda}}}}$	`int((1 - (x)^(2))^(lambda -(1)/(2))* GegenbauerC(2n, lambda, x)cos(xy), x = 0..1)=((- 1)^(n) PiGAMMA(2n + 2lambda)BesselJ(lambda + 2n, y))/(factorial(2n)GAMMA(lambda)(2*y)^(lambda))`	`Integrate[(1 - (x)^(2))^(\[Lambda]-Divide[1,2])* GegenbauerC[2n, \[Lambda], x]Cos[xy], {x, 0, 1}]=Divide[(- 1)^(n) PiGamma[2n + 2\[Lambda]]BesselJ[\[Lambda]+ 2n, y],(2n)!Gamma[\[Lambda]](2*y)^(\[Lambda])]`	Failure	Failure	Skip	Error
18.17.E18	${\displaystyle{\displaystyle\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(% \lambda)}_{2n+1}\left(x\right)\sin\left(xy\right)\mathrm{d}x=\frac{(-1)^{n}\pi% \Gamma\left(2n+2\lambda+1\right)J_{2n+\lambda+1}\left(y\right)}{(2n+1)!\Gamma% \left(\lambda\right)(2y)^{\lambda}}}}$	`int((1 - (x)^(2))^(lambda -(1)/(2))* GegenbauerC(2n + 1, lambda, x)sin(xy), x = 0..1)=((- 1)^(n) PiGAMMA(2n + 2lambda + 1)BesselJ(2n + lambda + 1, y))/(factorial(2n + 1)GAMMA(lambda)(2*y)^(lambda))`	`Integrate[(1 - (x)^(2))^(\[Lambda]-Divide[1,2])* GegenbauerC[2n + 1, \[Lambda], x]Sin[xy], {x, 0, 1}]=Divide[(- 1)^(n) PiGamma[2n + 2\[Lambda]+ 1]BesselJ[2n + \[Lambda]+ 1, y],(2n + 1)!Gamma[\[Lambda]](2*y)^(\[Lambda])]`	Failure	Failure	Skip	Skip
18.17.E19	${\displaystyle{\displaystyle\int_{-1}^{1}P_{n}\left(x\right)e^{ixy}\mathrm{d}x% =i^{n}\sqrt{\frac{2\pi}{y}}J_{n+\frac{1}{2}}\left(y\right)}}$	`int(LegendreP(n, x)exp(Ixy), x = - 1..1)= (I)^(n)sqrt((2Pi)/(y))BesselJ(n +(1)/(2), y)`	`Integrate[LegendreP[n, x]Exp[Ixy], {x, - 1, 1}]= (I)^(n)Sqrt[Divide[2Pi,y]]BesselJ[n +Divide[1,2], y]`	Failure	Failure	Skip	Error
18.17.E20	${\displaystyle{\displaystyle\int_{0}^{1}P_{n}\left(1-2x^{2}\right)\cos\left(xy% \right)\mathrm{d}x=(-1)^{n}\tfrac{1}{2}\pi J_{n+\frac{1}{2}}\left(\tfrac{1}{2}% y\right)J_{-n-\frac{1}{2}}\left(\tfrac{1}{2}y\right)}}$	`int(LegendreP(n, 1 - 2(x)^(2))cos(xy), x = 0..1)=(- 1)^(n)(1)/(2)PiBesselJ(n +(1)/(2), (1)/(2)y)BesselJ(- n -(1)/(2), (1)/(2)*y)`	`Integrate[LegendreP[n, 1 - 2(x)^(2)]Cos[xy], {x, 0, 1}]=(- 1)^(n)Divide[1,2]PiBesselJ[n +Divide[1,2], Divide[1,2]y]BesselJ[- n -Divide[1,2], Divide[1,2]*y]`	Failure	Failure	Skip	Successful
18.17.E21	${\displaystyle{\displaystyle\int_{0}^{1}P_{n}\left(1-2x^{2}\right)\sin\left(xy% \right)\mathrm{d}x=\tfrac{1}{2}\pi\left(J_{n+\frac{1}{2}}\left(\tfrac{1}{2}y% \right)\right)^{2}}}$	`int(LegendreP(n, 1 - 2(x)^(2))sin(xy), x = 0..1)=(1)/(2)Pi(BesselJ(n +(1)/(2), (1)/(2)y))^(2)`	`Integrate[LegendreP[n, 1 - 2(x)^(2)]Sin[xy], {x, 0, 1}]=Divide[1,2]Pi(BesselJ[n +Divide[1,2], Divide[1,2]y])^(2)`	Failure	Failure	Skip	Successful
18.17.E33	${\displaystyle{\displaystyle\int_{-1}^{1}e^{-(x+1)z}P^{(\alpha,\beta)}_{n}% \left(x\right)(1-x)^{\alpha}(1+x)^{\beta}\mathrm{d}x=\frac{(-1)^{n}2^{\alpha+% \beta+n+1}\Gamma\left(\alpha+n+1\right)\Gamma\left(\beta+n+1\right)}{\Gamma% \left(\alpha+\beta+2n+2\right)n!}z^{n}{{}_{1}F_{1}}\left({\beta+n+1\atop\alpha% +\beta+2n+2};-2z\right)}}$	`int(exp(-(x + 1)* z)JacobiP(n, alpha, beta, x)(1 - x)^(alpha)(1 + x)^(beta), x = - 1..1)=((- 1)^(n) (2)^(alpha + beta + n + 1)* GAMMA(alpha + n + 1)GAMMA(beta + n + 1))/(GAMMA(alpha + beta + 2n + 2)factorial(n))(z)^(n)* hypergeom([beta + n + 1], [alpha + beta + 2n + 2], - 2z)`	`Integrate[Exp[-(x + 1)* z]JacobiP[n, \[Alpha], \[Beta], x](1 - x)^(\[Alpha])(1 + x)^(\[Beta]), {x, - 1, 1}]=Divide[(- 1)^(n) (2)^(\[Alpha]+ \[Beta]+ n + 1)* Gamma[\[Alpha]+ n + 1]Gamma[\[Beta]+ n + 1],Gamma[\[Alpha]+ \[Beta]+ 2n + 2](n)!](z)^(n)* HypergeometricPFQ[{\[Beta]+ n + 1}, {\[Alpha]+ \[Beta]+ 2n + 2}, - 2z]`	Failure	Failure	Skip	Error
18.17.E35	${\displaystyle{\displaystyle\int_{-\infty}^{\infty}e^{-xz}H_{n}\left(x\right)e% ^{-x^{2}}\mathrm{d}x=\pi^{\frac{1}{2}}(-z)^{n}e^{\frac{1}{4}z^{2}}}}$	`int(exp(- xz)HermiteH(n, x)exp(- (x)^(2)), x = - infinity..infinity)= (Pi)^((1)/(2))(- z)^(n)* exp((1)/(4)*(z)^(2))`	`Integrate[Exp[- xz]HermiteH[n, x]Exp[- (x)^(2)], {x, - Infinity, Infinity}]= (Pi)^(Divide[1,2])(- z)^(n)* Exp[Divide[1,4]*(z)^(2)]`	Failure	Failure	Skip	Error
18.17.E36	${\displaystyle{\displaystyle\int_{-1}^{1}(1-x)^{z-1}(1+x)^{\beta}P^{(\alpha,% \beta)}_{n}\left(x\right)\mathrm{d}x=\frac{2^{\beta+z}\Gamma\left(z\right)% \Gamma\left(1+\beta+n\right){\left(1+\alpha-z\right)_{n}}}{n!\Gamma\left(1+% \beta+z+n\right)}}}$	`int((1 - x)^(z - 1)(1 + x)^(beta) JacobiP(n, alpha, beta, x), x = - 1..1)=((2)^(beta + z)* GAMMA(z)GAMMA(1 + beta + n)pochhammer(1 + alpha - z, n))/(factorial(n)*GAMMA(1 + beta + z + n))`	`Integrate[(1 - x)^(z - 1)(1 + x)^(\[Beta]) JacobiP[n, \[Alpha], \[Beta], x], {x, - 1, 1}]=Divide[(2)^(\[Beta]+ z)* Gamma[z]Gamma[1 + \[Beta]+ n]Pochhammer[1 + \[Alpha]- z, n],(n)!*Gamma[1 + \[Beta]+ z + n]]`	Failure	Failure	Skip	Error
18.17.E37	${\displaystyle{\displaystyle\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(% \lambda)}_{n}\left(x\right)x^{z-1}\mathrm{d}x=\frac{\pi\,2^{1-2\lambda-z}% \Gamma\left(n+2\lambda\right)\Gamma\left(z\right)}{n!\Gamma\left(\lambda\right% )\Gamma\left(\frac{1}{2}+\frac{1}{2}n+\lambda+\frac{1}{2}z\right)\Gamma\left(% \frac{1}{2}+\frac{1}{2}z-\frac{1}{2}n\right)}}}$	`int((1 - (x)^(2))^(lambda -(1)/(2))* GegenbauerC(n, lambda, x)(x)^(z - 1), x = 0..1)=(Pi(2)^(1 - 2lambda - z) GAMMA(n + 2lambda)GAMMA(z))/(factorial(n)GAMMA(lambda)GAMMA((1)/(2)+(1)/(2)n + lambda +(1)/(2)z)GAMMA((1)/(2)+(1)/(2)z -(1)/(2)*n))`	`Integrate[(1 - (x)^(2))^(\[Lambda]-Divide[1,2])* GegenbauerC[n, \[Lambda], x](x)^(z - 1), {x, 0, 1}]=Divide[Pi(2)^(1 - 2\[Lambda]- z) Gamma[n + 2\[Lambda]]Gamma[z],(n)!Gamma[\[Lambda]]Gamma[Divide[1,2]+Divide[1,2]n + \[Lambda]+Divide[1,2]z]Gamma[Divide[1,2]+Divide[1,2]z -Divide[1,2]*n]]`	Failure	Failure	Skip	Error
18.17.E38	${\displaystyle{\displaystyle\int_{0}^{1}P_{2n}\left(x\right)x^{z-1}\mathrm{d}x% =\frac{(-1)^{n}{\left(\frac{1}{2}-\frac{1}{2}z\right)_{n}}}{2{\left(\frac{1}{2% }z\right)_{n+1}}}}}$	`int(LegendreP(2n, x)(x)^(z - 1), x = 0..1)=((- 1)^(n)* pochhammer((1)/(2)-(1)/(2)z, n))/(2pochhammer((1)/(2)*z, n + 1))`	`Integrate[LegendreP[2n, x](x)^(z - 1), {x, 0, 1}]=Divide[(- 1)^(n)* Pochhammer[Divide[1,2]-Divide[1,2]z, n],2Pochhammer[Divide[1,2]*z, n + 1]]`	Failure	Failure	Skip	Successful
18.17.E39	${\displaystyle{\displaystyle\int_{0}^{1}P_{2n+1}\left(x\right)x^{z-1}\mathrm{d% }x=\frac{(-1)^{n}{\left(1-\frac{1}{2}z\right)_{n}}}{2{\left(\frac{1}{2}+\frac{% 1}{2}z\right)_{n+1}}}}}$	`int(LegendreP(2n + 1, x)(x)^(z - 1), x = 0..1)=((- 1)^(n)* pochhammer(1 -(1)/(2)z, n))/(2pochhammer((1)/(2)+(1)/(2)*z, n + 1))`	`Integrate[LegendreP[2n + 1, x](x)^(z - 1), {x, 0, 1}]=Divide[(- 1)^(n)* Pochhammer[1 -Divide[1,2]z, n],2Pochhammer[Divide[1,2]+Divide[1,2]*z, n + 1]]`	Failure	Failure	Skip	Successful
18.17.E45	${\displaystyle{\displaystyle(n+\tfrac{1}{2})(1+x)^{\frac{1}{2}}\int_{-1}^{x}(x% -t)^{-\frac{1}{2}}P_{n}\left(t\right)\mathrm{d}t=T_{n}\left(x\right)+T_{n+1}% \left(x\right)}}$	`(n +(1)/(2))(1 + x)^((1)/(2)) int((x - t)^(-(1)/(2))* LegendreP(n, t), t = - 1..x)= ChebyshevT(n, x)+ ChebyshevT(n + 1, x)`	`(n +Divide[1,2])(1 + x)^(Divide[1,2]) Integrate[(x - t)^(-Divide[1,2])* LegendreP[n, t], {t, - 1, x}]= ChebyshevT[n, x]+ ChebyshevT[n + 1, x]`	Failure	Failure	Skip	Skip
18.17.E46	${\displaystyle{\displaystyle(n+\tfrac{1}{2})(1-x)^{\frac{1}{2}}\int_{x}^{1}(t-% x)^{-\frac{1}{2}}P_{n}\left(t\right)\mathrm{d}t=T_{n}\left(x\right)-T_{n+1}% \left(x\right)}}$	`(n +(1)/(2))(1 - x)^((1)/(2)) int((t - x)^(-(1)/(2))* LegendreP(n, t), t = x..1)= ChebyshevT(n, x)- ChebyshevT(n + 1, x)`	`(n +Divide[1,2])(1 - x)^(Divide[1,2]) Integrate[(t - x)^(-Divide[1,2])* LegendreP[n, t], {t, x, 1}]= ChebyshevT[n, x]- ChebyshevT[n + 1, x]`	Failure	Failure	Skip	Successful
18.17.E48	${\displaystyle{\displaystyle\int_{-\infty}^{\infty}H_{m}\left(y\right)e^{-y^{2% }}H_{n}\left(x-y\right)e^{-(x-y)^{2}}\mathrm{d}y=\pi^{\frac{1}{2}}2^{-\frac{1}% {2}(m+n+1)}H_{m+n}\left(2^{-\frac{1}{2}}x\right)e^{-\frac{1}{2}x^{2}}}}$	`int(HermiteH(m, y)exp(- (y)^(2))HermiteH(n, x - y)exp(-(x - y)^(2)), y = - infinity..infinity)= (Pi)^((1)/(2)) (2)^(-(1)/(2)(m + n + 1)) HermiteH(m + n, (2)^(-(1)/(2))* x)exp(-(1)/(2)(x)^(2))`	`Integrate[HermiteH[m, y]Exp[- (y)^(2)]HermiteH[n, x - y]Exp[-(x - y)^(2)], {y, - Infinity, Infinity}]= (Pi)^(Divide[1,2]) (2)^(-Divide[1,2](m + n + 1)) HermiteH[m + n, (2)^(-Divide[1,2])* x]Exp[-Divide[1,2](x)^(2)]`	Failure	Failure	Skip	Error
18.17.E49	${\displaystyle{\displaystyle\int_{-\infty}^{\infty}H_{\ell}\left(x\right)H_{m}% \left(x\right)H_{n}\left(x\right)e^{-x^{2}}\mathrm{d}x=\frac{2^{\frac{1}{2}(% \ell+m+n)}\ell\,!\,m\,!\,n\,!\,\sqrt{\pi}}{(\tfrac{1}{2}\ell+\tfrac{1}{2}m-% \tfrac{1}{2}n)\,!\,(\tfrac{1}{2}m+\tfrac{1}{2}n-\tfrac{1}{2}\ell\,)\,!\,(% \tfrac{1}{2}n+\tfrac{1}{2}\ell-\tfrac{1}{2}m\,)\,!}}}$	`int(HermiteH(ell, x)HermiteH(m, x)HermiteH(n, x)exp(- (x)^(2)), x = - infinity..infinity)=((2)^((1)/(2)(ell + m + n))* factorial(ell)factorial(m)factorial(n)sqrt(Pi))/(factorial((1)/(2)ell +(1)/(2)m -(1)/(2)n)factorial((1)/(2)m +(1)/(2)n -(1)/(2)ell)factorial((1)/(2)n +(1)/(2)ell -(1)/(2)m))`	`Integrate[HermiteH[\[ScriptL], x]HermiteH[m, x]HermiteH[n, x]Exp[- (x)^(2)], {x, - Infinity, Infinity}]=Divide[(2)^(Divide[1,2](\[ScriptL]+ m + n))* (\[ScriptL])!(m)!(n)!Sqrt[Pi],(Divide[1,2]\[ScriptL]+Divide[1,2]m -Divide[1,2]n)!(Divide[1,2]m +Divide[1,2]n -Divide[1,2]\[ScriptL])!(Divide[1,2]n +Divide[1,2]\[ScriptL]-Divide[1,2]m)!]`	Failure	Failure	Skip	Error
18.18.E8	${\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(\cos\theta_{1}\cos\theta_{% 2}+\sin\theta_{1}\sin\theta_{2}\cos\phi\right)=\sum_{\ell=0}^{n}2^{2\ell}(n-% \ell)!\frac{2\lambda+2\ell-1}{2\lambda-1}\frac{({\left(\lambda\right)_{\ell}})% ^{2}}{{\left(2\lambda\right)_{n+\ell}}}(\sin\theta_{1})^{\ell}C^{(\lambda+\ell% )}_{n-\ell}\left(\cos\theta_{1}\right)(\sin\theta_{2})^{\ell}C^{(\lambda+\ell)% }_{n-\ell}\left(\cos\theta_{2}\right)C^{(\lambda-\frac{1}{2})}_{\ell}\left(% \cos\phi\right)}}$	`GegenbauerC(n, lambda, cos(theta[1])cos(theta[2])+ sin(theta[1])sin(theta[2])cos(phi))= sum((2)^(2ell)factorial(n - ell)(2lambda + 2ell - 1)/(2lambda - 1)((pochhammer(lambda, ell))^(2))/(pochhammer(2lambda, n + ell))(sin(theta[1]))^(ell)* GegenbauerC(n - ell, lambda + ell, cos(theta[1]))(sin(theta[2]))^(ell) GegenbauerC(n - ell, lambda + ell, cos(theta[2]))*GegenbauerC(ell, lambda -(1)/(2), cos(phi)), ell = 0..n)`	GegenbauerC[n, \[Lambda], Cos[Subscript[\[Theta], 1]]Cos[Subscript[\[Theta], 2]]+ Sin[Subscript[\[Theta], 1]]Sin[Subscript[\[Theta], 2]]Cos[\[Phi]]]= Sum[(2)^(2\[ScriptL])(n - \[ScriptL])!Divide[2\[Lambda]+ 2\[ScriptL]- 1,2\[Lambda]- 1]Divide[(Pochhammer[\[Lambda], \[ScriptL]])^(2),Pochhammer[2\[Lambda], n + \[ScriptL]]](Sin[Subscript[\[Theta], 1]])^(\[ScriptL])* GegenbauerC[n - \[ScriptL], \[Lambda]+ \[ScriptL], Cos[Subscript[\[Theta], 1]]](Sin[Subscript[\[Theta], 2]])^(\[ScriptL]) GegenbauerC[n - \[ScriptL], \[Lambda]+ \[ScriptL], Cos[Subscript[\[Theta], 2]]]*GegenbauerC[\[ScriptL], \[Lambda]-Divide[1,2], Cos[\[Phi]]], {\[ScriptL], 0, n}]	Failure	Failure	Skip	Error
18.18.E9	${\displaystyle{\displaystyle P_{n}\left(\cos\theta_{1}\cos\theta_{2}+\sin% \theta_{1}\sin\theta_{2}\cos\phi\right)={P_{n}\left(\cos\theta_{1}\right)P_{n}% \left(\cos\theta_{2}\right)+2\sum_{\ell=1}^{n}\frac{(n-\ell)!\;(n+\ell)!}{2^{2% \ell}(n!)^{2}}(\sin\theta_{1})^{\ell}P^{(\ell,\ell)}_{n-\ell}\left(\cos\theta_% {1}\right)(\sin\theta_{2})^{\ell}P^{(\ell,\ell)}_{n-\ell}\left(\cos\theta_{2}% \right)\cos\left(\ell\phi\right)}}}$	`LegendreP(n, cos(theta[1])cos(theta[2])+ sin(theta[1])sin(theta[2])cos(phi))=LegendreP(n, cos(theta[1]))LegendreP(n, cos(theta[2]))+ 2sum((factorial(n - ell)factorial(n + ell))/((2)^(2ell)(factorial(n))^(2))(sin(theta[1]))^(ell) JacobiP(n - ell, ell, ell, cos(theta[1]))(sin(theta[2]))^(ell) JacobiP(n - ell, ell, ell, cos(theta[2]))cos(ellphi), ell = 1..n)`	LegendreP[n, Cos[Subscript[\[Theta], 1]]Cos[Subscript[\[Theta], 2]]+ Sin[Subscript[\[Theta], 1]]Sin[Subscript[\[Theta], 2]]Cos[\[Phi]]]=LegendreP[n, Cos[Subscript[\[Theta], 1]]]LegendreP[n, Cos[Subscript[\[Theta], 2]]]+ 2Sum[Divide[(n - \[ScriptL])!(n + \[ScriptL])!,(2)^(2\[ScriptL])((n)!)^(2)](Sin[Subscript[\[Theta], 1]])^(\[ScriptL]) JacobiP[n - \[ScriptL], \[ScriptL], \[ScriptL], Cos[Subscript[\[Theta], 1]]](Sin[Subscript[\[Theta], 2]])^(\[ScriptL]) JacobiP[n - \[ScriptL], \[ScriptL], \[ScriptL], Cos[Subscript[\[Theta], 2]]]Cos[\[ScriptL]\[Phi]], {\[ScriptL], 1, n}]	Failure	Failure	Skip	Error
18.18.E13	${\displaystyle{\displaystyle H_{n}\left(\lambda x\right)=\lambda^{n}\sum_{\ell% =0}^{\left\lfloor n/2\right\rfloor}\frac{{\left(-n\right)_{2\ell}}}{\ell!}(1-% \lambda^{-2})^{\ell}H_{n-2\ell}\left(x\right)}}$	`HermiteH(n, lambdax)= (lambda)^(n) sum((pochhammer(- n, 2ell))/(factorial(ell))(1 - (lambda)^(- 2))^(ell)* HermiteH(n - 2*ell, x), ell = 0..floor(n/ 2))`	`HermiteH[n, \[Lambda]x]= (\[Lambda])^(n) Sum[Divide[Pochhammer[- n, 2\[ScriptL]],(\[ScriptL])!](1 - (\[Lambda])^(- 2))^(\[ScriptL])* HermiteH[n - 2*\[ScriptL], x], {\[ScriptL], 0, Floor[n/ 2]}]`	Failure	Failure	Skip	Fail `Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[n, 1], Rule[x, 1], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[5.656854249492381, 5.656854249492381] <- {Rule[n, 1], Rule[x, 2], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[8.485281374238571, 8.485281374238571] <- {Rule[n, 1], Rule[x, 3], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[-2.0, 16.0] <- {Rule[n, 2], Rule[x, 1], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` ... skip entries to safe data
18.18.E20	${\displaystyle{\displaystyle(2x)^{n}=\sum_{\ell=0}^{\left\lfloor n/2\right% \rfloor}\frac{{\left(-n\right)_{2\ell}}}{\ell!}H_{n-2\ell}\left(x\right)}}$	`(2x)^(n)= sum((pochhammer(- n, 2ell))/(factorial(ell))HermiteH(n - 2ell, x), ell = 0..floor(n/ 2))`	`(2x)^(n)= Sum[Divide[Pochhammer[- n, 2\[ScriptL]],(\[ScriptL])!]HermiteH[n - 2\[ScriptL], x], {\[ScriptL], 0, Floor[n/ 2]}]`	Failure	Failure	Skip	Fail `2.0 <- {Rule[n, 1], Rule[x, 1]}` `4.0 <- {Rule[n, 1], Rule[x, 2]}` `6.0 <- {Rule[n, 1], Rule[x, 3]}` `4.0 <- {Rule[n, 2], Rule[x, 1]}` ... skip entries to safe data
18.18.E21	${\displaystyle{\displaystyle T_{m}\left(x\right)T_{n}\left(x\right)=\tfrac{1}{% 2}(T_{m+n}\left(x\right)+T_{m-n}\left(x\right))}}$	`ChebyshevT(m, x)ChebyshevT(n, x)=(1)/(2)(ChebyshevT(m + n, x)+ ChebyshevT(m - n, x))`	`ChebyshevT[m, x]ChebyshevT[n, x]=Divide[1,2](ChebyshevT[m + n, x]+ ChebyshevT[m - n, x])`	Failure	Failure	Successful	Successful
18.18.E24	${\displaystyle{\displaystyle b_{n,\ell}=\genfrac{(}{)}{0.0pt}{}{n}{\ell}\frac{% {\left(n+\alpha+\beta+1\right)_{\ell}}{\left(-\beta-n\right)_{n-\ell}}}{2^{% \ell}{\left(\alpha+1\right)_{n}}}}}$	`b[n , ell]=binomial(n,ell)(pochhammer(n + alpha + beta + 1, ell)pochhammer(- beta - n, n - ell))/((2)^(ell)* pochhammer(alpha + 1, n))`	`Subscript[b, n , \[ScriptL]]=Binomial[n,\[ScriptL]]Divide[Pochhammer[n + \[Alpha]+ \[Beta]+ 1, \[ScriptL]]Pochhammer[- \[Beta]- n, n - \[ScriptL]],(2)^(\[ScriptL])* Pochhammer[\[Alpha]+ 1, n]]`	Failure	Failure	Fail `.414213562+1.414213562I <- {alpha = 2^(1/2)+I2^(1/2), beta = 2^(1/2)+I2^(1/2), b[n,ell] = 2^(1/2)+I2^(1/2), ell = 1, n = 1}` `3.722604190+1.233562514I <- {alpha = 2^(1/2)+I2^(1/2), beta = 2^(1/2)+I2^(1/2), b[n,ell] = 2^(1/2)+I2^(1/2), ell = 1, n = 2}` `-2.510958325+1.956166705I <- {alpha = 2^(1/2)+I2^(1/2), beta = 2^(1/2)+I2^(1/2), b[n,ell] = 2^(1/2)+I2^(1/2), ell = 1, n = 3}` `1.414213562+1.414213562I <- {alpha = 2^(1/2)+I2^(1/2), beta = 2^(1/2)+I2^(1/2), b[n,ell] = 2^(1/2)+I2^(1/2), ell = 2, n = 1}` ... skip entries to safe data	Skip
18.18.E25	${\displaystyle{\displaystyle\frac{P^{(\alpha,\beta)}_{n}\left(x\right)}{P^{(% \alpha,\beta)}_{n}\left(1\right)}\frac{P^{(\alpha,\beta)}_{n}\left(y\right)}{P% ^{(\alpha,\beta)}_{n}\left(1\right)}=\sum_{\ell=0}^{n}b_{n,\ell}(x+y)^{\ell}\*% \frac{P^{(\alpha,\beta)}_{\ell}\left(\ifrac{(1+xy)}{(x+y)}\right)}{P^{(\alpha,% \beta)}_{\ell}\left(1\right)}}}$	`(JacobiP(n, alpha, beta, x))/(JacobiP(n, alpha, beta, 1))(JacobiP(n, alpha, beta, y))/(JacobiP(n, alpha, beta, 1))= sum(b[n , ell](x + y)^(ell)(JacobiP(ell, alpha, beta, (1 + xy)/(x + y)))/(JacobiP(ell, alpha, beta, 1)), ell = 0..n)`	`Divide[JacobiP[n, \[Alpha], \[Beta], x],JacobiP[n, \[Alpha], \[Beta], 1]]Divide[JacobiP[n, \[Alpha], \[Beta], y],JacobiP[n, \[Alpha], \[Beta], 1]]= Sum[Subscript[b, n , \[ScriptL]](x + y)^(\[ScriptL])Divide[JacobiP[\[ScriptL], \[Alpha], \[Beta], Divide[1 + xy,x + y]],JacobiP[\[ScriptL], \[Alpha], \[Beta], 1]], {\[ScriptL], 0, n}]`	Failure	Failure	Skip	Skip
18.18.E26	${\displaystyle{\displaystyle\frac{P^{(\alpha,\beta)}_{n}\left(x\right)}{P^{(% \alpha,\beta)}_{n}\left(1\right)}=\sum_{\ell=0}^{n}b_{n,\ell}(x+1)^{\ell}}}$	`(JacobiP(n, alpha, beta, x))/(JacobiP(n, alpha, beta, 1))= sum(b[n , ell]*(x + 1)^(ell), ell = 0..n)`	`Divide[JacobiP[n, \[Alpha], \[Beta], x],JacobiP[n, \[Alpha], \[Beta], 1]]= Sum[Subscript[b, n , \[ScriptL]]*(x + 1)^(\[ScriptL]), {\[ScriptL], 0, n}]`	Failure	Failure	Skip	Skip
18.18.E28	${\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{H_{n}\left(x\right)H_{n}% \left(y\right)}{2^{n}n!}z^{n}=(1-z^{2})^{-\frac{1}{2}}\exp\left(\frac{2xyz-(x^% {2}+y^{2})z^{2}}{1-z^{2}}\right)}}$	`sum((HermiteH(n, x)HermiteH(n, y))/((2)^(n) factorial(n))(z)^(n), n = 0..infinity)=(1 - (z)^(2))^(-(1)/(2)) exp((2xyz -((x)^(2)+ (y)^(2)) (z)^(2))/(1 - (z)^(2)))`	`Sum[Divide[HermiteH[n, x]HermiteH[n, y],(2)^(n) (n)!](z)^(n), {n, 0, Infinity}]=(1 - (z)^(2))^(-Divide[1,2]) Exp[Divide[2xyz -((x)^(2)+ (y)^(2)) (z)^(2),1 - (z)^(2)]]`	Error	Failure	-	Skip
18.18.E29	${\displaystyle{\displaystyle\sum_{\ell=0}^{n}C^{(\lambda)}_{\ell}\left(x\right% )C^{(\mu)}_{n-\ell}\left(x\right)=C^{(\lambda+\mu)}_{n}\left(x\right)}}$	`sum(GegenbauerC(ell, lambda, x)*GegenbauerC(n - ell, mu, x), ell = 0..n)= GegenbauerC(n, lambda + mu, x)`	`Sum[GegenbauerC[\[ScriptL], \[Lambda], x]*GegenbauerC[n - \[ScriptL], \[Mu], x], {\[ScriptL], 0, n}]= GegenbauerC[n, \[Lambda]+ \[Mu], x]`	Failure	Successful	Skip	-
18.18.E30	${\displaystyle{\displaystyle\sum_{\ell=0}^{n}\frac{\ell+2\lambda}{2\lambda}C^{% (\lambda)}_{\ell}\left(x\right)x^{n-\ell}=C^{(\lambda+1)}_{n}\left(x\right)}}$	`sum((ell + 2lambda)/(2lambda)GegenbauerC(ell, lambda, x)(x)^(n - ell), ell = 0..n)= GegenbauerC(n, lambda + 1, x)`	`Sum[Divide[\[ScriptL]+ 2\[Lambda],2\[Lambda]]GegenbauerC[\[ScriptL], \[Lambda], x](x)^(n - \[ScriptL]), {\[ScriptL], 0, n}]= GegenbauerC[n, \[Lambda]+ 1, x]`	Failure	Failure	Skip	Error
18.18.E31	${\displaystyle{\displaystyle\sum_{\ell=0}^{n}T_{\ell}\left(x\right)x^{n-\ell}=% U_{n}\left(x\right)}}$	`sum(ChebyshevT(ell, x)*(x)^(n - ell), ell = 0..n)= ChebyshevU(n, x)`	`Sum[ChebyshevT[\[ScriptL], x]*(x)^(n - \[ScriptL]), {\[ScriptL], 0, n}]= ChebyshevU[n, x]`	Failure	Failure	Skip	Error
18.18.E32	${\displaystyle{\displaystyle 2\sum_{\ell=0}^{n}T_{2\ell}\left(x\right)=1+U_{2n% }\left(x\right)}}$	`2sum(ChebyshevT(2ell, x), ell = 0..n)= 1 + ChebyshevU(2*n, x)`	`2Sum[ChebyshevT[2\[ScriptL], x], {\[ScriptL], 0, n}]= 1 + ChebyshevU[2*n, x]`	Failure	Successful	Skip	-
18.18.E33	${\displaystyle{\displaystyle 2\sum_{\ell=0}^{n}T_{2\ell+1}\left(x\right)=U_{2n% +1}\left(x\right)}}$	`2sum(ChebyshevT(2ell + 1, x), ell = 0..n)= ChebyshevU(2*n + 1, x)`	`2Sum[ChebyshevT[2\[ScriptL]+ 1, x], {\[ScriptL], 0, n}]= ChebyshevU[2*n + 1, x]`	Failure	Successful	Skip	-
18.18.E34	${\displaystyle{\displaystyle 2(1-x^{2})\sum_{\ell=0}^{n}U_{2\ell}\left(x\right% )=1-T_{2n+2}\left(x\right)}}$	`2(1 - (x)^(2)) sum(ChebyshevU(2ell, x), ell = 0..n)= 1 - ChebyshevT(2n + 2, x)`	`2(1 - (x)^(2)) Sum[ChebyshevU[2\[ScriptL], x], {\[ScriptL], 0, n}]= 1 - ChebyshevT[2n + 2, x]`	Failure	Successful	Skip	-
18.18.E35	${\displaystyle{\displaystyle 2(1-x^{2})\sum_{\ell=0}^{n}U_{2\ell+1}\left(x% \right)=x-T_{2n+3}\left(x\right)}}$	`2(1 - (x)^(2)) sum(ChebyshevU(2ell + 1, x), ell = 0..n)= x - ChebyshevT(2n + 3, x)`	`2(1 - (x)^(2)) Sum[ChebyshevU[2\[ScriptL]+ 1, x], {\[ScriptL], 0, n}]= x - ChebyshevT[2n + 3, x]`	Failure	Successful	Skip	-
18.18.E36	${\displaystyle{\displaystyle\sum_{\ell=0}^{n}P_{\ell}\left(x\right)P_{n-\ell}% \left(x\right)=U_{n}\left(x\right)}}$	`sum(LegendreP(ell, x)*LegendreP(n - ell, x), ell = 0..n)= ChebyshevU(n, x)`	`Sum[LegendreP[\[ScriptL], x]*LegendreP[n - \[ScriptL], x], {\[ScriptL], 0, n}]= ChebyshevU[n, x]`	Failure	Successful	Skip	-
18.18.E40	${\displaystyle{\displaystyle\sum_{\ell=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{\ell}H% _{2\ell}\left(x\right)H_{2n-2\ell}\left(y\right)=(-1)^{n}2^{2n}n!L_{n}\left(x^% {2}+y^{2}\right)}}$	`sum(binomial(n,ell)HermiteH(2ell, x)HermiteH(2n - 2ell, y), ell = 0..n)=(- 1)^(n) (2)^(2n) factorial(n)*LaguerreL(n, (x)^(2)+ (y)^(2))`	`Error`	Failure	Error	-	-
18.19.E4	${\displaystyle{\displaystyle h_{n}=\frac{2\pi\Gamma\left(n+a+\overline{a}% \right)\Gamma\left(n+b+\overline{b}\right)\|\Gamma\left(n+a+\overline{b}\right)% \|^{2}}{\left(2n+2\Re(a+b)-1\right)\Gamma\left(n+2\Re(a+b)-1\right)n!}}}$	`h[n]=(2PiGAMMA(n + a + conjugate(a))GAMMA(n + b + conjugate(b))(abs(GAMMA(n + a + conjugate(b))))^(2))/((2n + 2Re(a + b)- 1)* GAMMA(n + 2Re(a + b)- 1)factorial(n))`	`Subscript[h, n]=Divide[2PiGamma[n + a + Conjugate[a]]Gamma[n + b + Conjugate[b]](Abs[Gamma[n + a + Conjugate[b]]])^(2),(2n + 2Re[a + b]- 1)* Gamma[n + 2Re[a + b]- 1](n)!]`	Failure	Failure	Fail `-6.363350679+1.414213562I <- {a = 2^(1/2)+I2^(1/2), b = 2^(1/2)+I2^(1/2), h[n] = 2^(1/2)+I2^(1/2), n = 1}` `-112.1467589+1.414213562I <- {a = 2^(1/2)+I2^(1/2), b = 2^(1/2)+I2^(1/2), h[n] = 2^(1/2)+I2^(1/2), n = 2}` `-2509.272955+1.414213562I <- {a = 2^(1/2)+I2^(1/2), b = 2^(1/2)+I2^(1/2), h[n] = 2^(1/2)+I2^(1/2), n = 3}` `-6.363350679-1.414213562I <- {a = 2^(1/2)+I2^(1/2), b = 2^(1/2)+I2^(1/2), h[n] = 2^(1/2)-I2^(1/2), n = 1}` ... skip entries to safe data	Skip
18.19.E5	${\displaystyle{\displaystyle k_{n}=\frac{{\left(n+2\Re(a+b)-1\right)_{n}}}{n!}}}$	`k[n]=(pochhammer(n + 2*Re(a + b)- 1, n))/(factorial(n))`	`Subscript[k, n]=Divide[Pochhammer[n + 2*Re[a + b]- 1, n],(n)!]`	Failure	Failure	Fail `-4.242640686+1.414213562I <- {a = 2^(1/2)+I2^(1/2), b = 2^(1/2)+I2^(1/2), k[n] = 2^(1/2)+I2^(1/2), n = 1}` `-24.07106780+1.414213562I <- {a = 2^(1/2)+I2^(1/2), b = 2^(1/2)+I2^(1/2), k[n] = 2^(1/2)+I2^(1/2), n = 2}` `-105.2687108+1.414213562I <- {a = 2^(1/2)+I2^(1/2), b = 2^(1/2)+I2^(1/2), k[n] = 2^(1/2)+I2^(1/2), n = 3}` `-4.242640686-1.414213562I <- {a = 2^(1/2)+I2^(1/2), b = 2^(1/2)+I2^(1/2), k[n] = 2^(1/2)-I2^(1/2), n = 1}` ... skip entries to safe data	Successful
18.19#Ex1	${\displaystyle{\displaystyle h_{n}=\frac{2\pi\Gamma\left(n+2\lambda\right)}{(2% \sin\phi)^{2\lambda}n!}}}$	`h[n]=(2PiGAMMA(n + 2lambda))/((2sin(phi))^(2lambda) factorial(n))`	`Subscript[h, n]=Divide[2PiGamma[n + 2\[Lambda]],(2Sin[\[Phi]])^(2\[Lambda]) (n)!]`	Failure	Failure	Fail `1.256815617+1.596021436I <- {lambda = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), h[n] = 2^(1/2)+I2^(1/2), n = 1}` `.8558051203+1.539638353I <- {lambda = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), h[n] = 2^(1/2)+I2^(1/2), n = 2}` `.397217113+1.089609188I <- {lambda = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), h[n] = 2^(1/2)+I2^(1/2), n = 3}` `1.256815617-1.232405688I <- {lambda = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), h[n] = 2^(1/2)-I2^(1/2), n = 1}` ... skip entries to safe data	Skip
18.19#Ex2	${\displaystyle{\displaystyle k_{n}=\frac{(2\sin\phi)^{n}}{n!}}}$	`k[n]=((2*sin(phi))^(n))/(factorial(n))`	`Subscript[k, n]=Divide[(2*Sin[\[Phi]])^(n),(n)!]`	Failure	Failure	Fail `-2.888857518+.8106906210I <- {phi = 2^(1/2)+I2^(1/2), k[n] = 2^(1/2)+I2^(1/2), n = 1}` `-7.661876828-1.182788552I <- {phi = 2^(1/2)+I2^(1/2), k[n] = 2^(1/2)+I2^(1/2), n = 2}` `-11.08169034-4.136690922I <- {phi = 2^(1/2)+I2^(1/2), k[n] = 2^(1/2)+I2^(1/2), n = 3}` `-2.888857518-2.017736503I <- {phi = 2^(1/2)+I2^(1/2), k[n] = 2^(1/2)-I2^(1/2), n = 1}` ... skip entries to safe data	Successful
18.20#Ex2	${\displaystyle{\displaystyle\kappa_{n}={\left(-N\right)_{n}}{\left(\alpha+1% \right)_{n}}}}$	`kappa[n]= pochhammer(- N, n)*pochhammer(alpha + 1, n)`	`Subscript[\[Kappa], n]= Pochhammer[- N, n]*Pochhammer[\[Alpha]+ 1, n]`	Failure	Failure	Fail `2.828427124+6.828427122I <- {N = 2^(1/2)+I2^(1/2), alpha = 2^(1/2)+I2^(1/2), kappa[n] = 2^(1/2)+I2^(1/2), n = 1}` `31.55634916-3.071067807I <- {N = 2^(1/2)+I2^(1/2), alpha = 2^(1/2)+I2^(1/2), kappa[n] = 2^(1/2)+I2^(1/2), n = 2}` `115.3553390-182.3502881I <- {N = 2^(1/2)+I2^(1/2), alpha = 2^(1/2)+I2^(1/2), kappa[n] = 2^(1/2)+I2^(1/2), n = 3}` `2.828427124+3.999999998I <- {N = 2^(1/2)+I2^(1/2), alpha = 2^(1/2)+I2^(1/2), kappa[n] = 2^(1/2)-I2^(1/2), n = 1}` ... skip entries to safe data	Successful
18.22.E8	${\displaystyle{\displaystyle(n+1)p_{n+1}(x)=2\left(x\sin\phi+(n+\lambda)\cos% \phi\right)p_{n}(x)-(n+2\lambda-1)p_{n-1}(x)}}$	`(n + 1)* p[n + 1](x)= 2(xsin(phi)+(n + lambda)cos(phi))* p[n](x)-(n + 2lambda - 1)* p[n - 1]*(x)`	`(n + 1)* Subscript[p, n + 1](x)= 2(xSin[\[Phi]]+(n + \[Lambda])Cos[\[Phi]])* Subscript[p, n](x)-(n + 2\[Lambda]- 1)* Subscript[p, n - 1]*(x)`	Failure	Failure	Skip	Skip
18.22.E14	${\displaystyle{\displaystyle A(x)p_{n}(x+i)-\left(A(x)+C(x)\right)p_{n}(x)+C(x% )p_{n}(x-i)+n(n+2\Re(a+b)-1)p_{n}(x)=0}}$	`A(x) p[n](x + I)-(A(x)+ C(x)) p[n](x)+ C(x)* p[n](x - I)+ n(n + 2Re(a + b)- 1) p[n]*(x)= 0`	`A(x) Subscript[p, n](x + I)-(A(x)+ C(x)) Subscript[p, n](x)+ C(x)* Subscript[p, n](x - I)+ n(n + 2Re[a + b]- 1) Subscript[p, n]*(x)= 0`	Failure	Failure	Skip	Skip
18.22#Ex7	${\displaystyle{\displaystyle A(x)=(x+\mathrm{i}\overline{a})(x+\mathrm{i}% \overline{b})}}$	`A(x)=(x + Iconjugate(a))(x + Iconjugate(b))`	`A(x)=(x + IConjugate[a])(x + IConjugate[b])`	Failure	Failure	Fail `-2.414213562-5.414213560I <- {A = 2^(1/2)+I2^(1/2), a = 2^(1/2)+I2^(1/2), b = 2^(1/2)+I2^(1/2), x = 1}` `-6.828427124-6.828427122I <- {A = 2^(1/2)+I2^(1/2), a = 2^(1/2)+I2^(1/2), b = 2^(1/2)+I2^(1/2), x = 2}` `-13.24264068-8.242640684I <- {A = 2^(1/2)+I2^(1/2), a = 2^(1/2)+I2^(1/2), b = 2^(1/2)+I2^(1/2), x = 3}` `4.414213560-1.414213562I <- {A = 2^(1/2)+I2^(1/2), a = 2^(1/2)+I2^(1/2), b = 2^(1/2)-I2^(1/2), x = 1}` ... skip entries to safe data	Fail `Complex[-2.414213562373094, -5.414213562373095] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1]}` `Complex[-6.82842712474619, -6.82842712474619] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2]}` `Complex[-13.242640687119282, -8.242640687119286] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3]}` `Complex[4.414213562373096, -1.4142135623730947] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 1]}` ... skip entries to safe data
18.22#Ex8	${\displaystyle{\displaystyle C(x)=(x-\mathrm{i}a)(x-\mathrm{i}b)}}$	`C(x)=(x - Ia)(x - Ib)`	`C(x)=(x - Ia)(x - Ib)`	Failure	Failure	Fail `-2.414213562+8.242640684I <- {C = 2^(1/2)+I2^(1/2), a = 2^(1/2)+I2^(1/2), b = 2^(1/2)+I2^(1/2), x = 1}` `-6.828427124+12.48528137I <- {C = 2^(1/2)+I2^(1/2), a = 2^(1/2)+I2^(1/2), b = 2^(1/2)+I2^(1/2), x = 2}` `-13.24264068+16.72792206I <- {C = 2^(1/2)+I2^(1/2), a = 2^(1/2)+I2^(1/2), b = 2^(1/2)+I2^(1/2), x = 3}` `4.414213560+4.242640686I <- {C = 2^(1/2)+I2^(1/2), a = 2^(1/2)+I2^(1/2), b = 2^(1/2)-I2^(1/2), x = 1}` ... skip entries to safe data	Fail `Complex[-2.414213562373094, 8.242640687119286] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1]}` `Complex[-6.82842712474619, 12.48528137423857] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2]}` `Complex[-13.242640687119282, 16.72792206135786] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3]}` `Complex[-2.414213562373094, 5.414213562373095] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 1]}` ... skip entries to safe data
18.22.E17	${\displaystyle{\displaystyle A(x)p_{n}(x+i)-\left(A(x)+C(x)\right)p_{n}(x)+C(x% )p_{n}(x-i)+2n\sin\phi\,p_{n}(x)=0}}$	`A(x) p[n](x + I)-(A(x)+ C(x)) p[n](x)+ C(x)* p[n](x - I)+ 2nsin(phi)p[n]*(x)= 0`	`A(x) Subscript[p, n](x + I)-(A(x)+ C(x)) Subscript[p, n](x)+ C(x)* Subscript[p, n](x - I)+ 2nSin[\[Phi]]Subscript[p, n]*(x)= 0`	Failure	Failure	Skip	Skip
18.22#Ex9	${\displaystyle{\displaystyle A(x)=e^{\mathrm{i}\phi}(x+\mathrm{i}\lambda)}}$	`A(x)= exp(Iphi)(x + Ilambda)`	`A(x)= Exp[I\[Phi]](x + I\[Lambda])`	Failure	Failure	Fail `1.769530126+1.460067411I <- {A = 2^(1/2)+I2^(1/2), lambda = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), x = 1}` `3.145831166+2.634138542I <- {A = 2^(1/2)+I2^(1/2), lambda = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), x = 2}` `4.522132206+3.808209673I <- {A = 2^(1/2)+I2^(1/2), lambda = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), x = 3}` `7.425753628+2.190006980I <- {A = 2^(1/2)+I2^(1/2), lambda = 2^(1/2)+I2^(1/2), phi = 2^(1/2)-I2^(1/2), x = 1}` ... skip entries to safe data	Fail `Complex[1.7695301261666299, 1.4600674117156012] <- {Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[3.145831166712761, 2.6341385429136204] <- {Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[4.522132207258893, 3.808209674111639] <- {Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[7.425753631849798, 2.190006983651361] <- {Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}` ... skip entries to safe data
18.22#Ex10	${\displaystyle{\displaystyle C(x)=e^{-\mathrm{i}\phi}(x-\mathrm{i}\lambda)}}$	`C(x)= exp(- Iphi)(x - Ilambda)`	`C(x)= Exp[- I\[Phi]](x - I\[Lambda])`	Failure	Failure	Fail `5.611500167+12.13011774I <- {C = 2^(1/2)+I2^(1/2), lambda = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), x = 1}` `6.384278266+17.60725995I <- {C = 2^(1/2)+I2^(1/2), lambda = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), x = 2}` `7.157056365+23.08440217I <- {C = 2^(1/2)+I2^(1/2), lambda = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), x = 3}` `1.662297320+2.047585079I <- {C = 2^(1/2)+I2^(1/2), lambda = 2^(1/2)+I2^(1/2), phi = 2^(1/2)-I2^(1/2), x = 1}` ... skip entries to safe data	Fail `Complex[5.611500173592372, 12.13011774491578] <- {Rule[C, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[6.384278274402982, 17.607259958792373] <- {Rule[C, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[7.1570563752135925, 23.084402172668966] <- {Rule[C, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[1.662297321063714, 2.0475850791686696] <- {Rule[C, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}` ... skip entries to safe data
18.25.E2	${\displaystyle{\displaystyle\int_{0}^{\infty}p_{n}(x)p_{m}(x)w(x)\mathrm{d}x=h% _{n}\delta_{n,m}}}$	`int(p[n](x) p[m](x) w(x), x = 0..infinity)= h[n]KroneckerDelta[n, m]`	`Integrate[Subscript[p, n](x) Subscript[p, m](x) w(x), {x, 0, Infinity}]= Subscript[h, n]KroneckerDelta[n, m]`	Failure	Failure	Skip	Skip
18.25.E4	${\displaystyle{\displaystyle w(y^{2})=\frac{1}{2y}\left\|\frac{\prod_{j}\Gamma% \left(a_{j}+iy\right)}{\Gamma\left(2iy\right)}\right\|^{2}}}$	`w((y)^(2))=(1)/(2y)(abs((product(GAMMA(a[j]+ Iy), j = - infinity..infinity))/(GAMMA(2Iy))))^(2)`	`w((y)^(2))=Divide[1,2y](Abs[Divide[Product[Gamma[Subscript[a, j]+ Iy], {j, - Infinity, Infinity}],Gamma[2Iy]]])^(2)`	Failure	Failure	Skip	Skip
18.25.E5	${\displaystyle{\displaystyle h_{n}=\frac{n!\,2\pi\prod_{j<\ell}\Gamma\left(n+a% _{j}+a_{\ell}\right)}{(2n-1+\sum_{j}a_{j})\Gamma\left(n-1+\sum_{j}a_{j}\right)% }}}$	`h[n]=(factorial(n)2Piproduct(GAMMA(n + a[j]+ a[ell]), j = - infinity..ell - 1))/((2n - 1 + sum(a[j], j = - infinity..infinity))* GAMMA(n - 1 + sum(a[j], j = - infinity..infinity)))`	`Subscript[h, n]=Divide[(n)!2PiProduct[Gamma[n + Subscript[a, j]+ Subscript[a, \[ScriptL]]], {j, - Infinity, \[ScriptL] - 1}],(2n - 1 + Sum[Subscript[a, j], {j, - Infinity, Infinity}])* Gamma[n - 1 + Sum[Subscript[a, j], {j, - Infinity, Infinity}]]]`	Error	Error	-	-
18.25.E7	${\displaystyle{\displaystyle w(y^{2})=\frac{1}{2y}\left\|\frac{\prod_{j}\Gamma% \left(a_{j}+iy\right)}{\Gamma\left(2iy\right)}\right\|^{2}}}$	`w((y)^(2))=(1)/(2y)(abs((product(GAMMA(a[j]+ Iy), j = - infinity..infinity))/(GAMMA(2Iy))))^(2)`	`w((y)^(2))=Divide[1,2y](Abs[Divide[Product[Gamma[Subscript[a, j]+ Iy], {j, - Infinity, Infinity}],Gamma[2Iy]]])^(2)`	Failure	Failure	Skip	Skip
18.25.E8	${\displaystyle{\displaystyle h_{n}=n!\,2\pi\prod_{j<\ell}\Gamma\left(n+a_{j}+a% _{\ell}\right)}}$	`h[n]= factorial(n)2Pi*product(GAMMA(n + a[j]+ a[ell]), j = - infinity..ell - 1)`	`Subscript[h, n]= (n)!2Pi*Product[Gamma[n + Subscript[a, j]+ Subscript[a, \[ScriptL]]], {j, - Infinity, \[ScriptL] - 1}]`	Error	Error	-	-
18.25.E9	${\displaystyle{\displaystyle\sum_{y=0}^{N}p_{n}(y(y+\gamma+\delta+1))p_{m}(y(y% +\gamma+\delta+1))\*\frac{\gamma+\delta+1+2y}{\gamma+\delta+1+y}\omega_{y}=h_{% n}\delta_{n,m}}}$	`sum(p[n](y(y + gamma + delta + 1))* p[m](y(y + gamma + delta + 1))(gamma + delta + 1 + 2y)/(gamma + delta + 1 + y)omega[y], y = 0..N)= h[n]KroneckerDelta[n, m]`	`Sum[Subscript[p, n](y(y + \[Gamma]+ \[Delta]+ 1))* Subscript[p, m](y(y + \[Gamma]+ \[Delta]+ 1))Divide[\[Gamma]+ \[Delta]+ 1 + 2y,\[Gamma]+ \[Delta]+ 1 + y]Subscript[\[Omega], y], {y, 0, N}]= Subscript[h, n]KroneckerDelta[n, m]`	Failure	Failure	Skip	Skip
18.25.E11	${\displaystyle{\displaystyle\omega_{y}=\frac{{\left(\alpha+1\right)_{y}}{\left% (\beta+\delta+1\right)_{y}}{\left(\gamma+1\right)_{y}}{\left(\gamma+\delta+2% \right)_{y}}}{{\left(-\alpha+\gamma+\delta+1\right)_{y}}{\left(-\beta+\gamma+1% \right)_{y}}{\left(\delta+1\right)_{y}}y!}}}$	`omega[y]=(pochhammer(alpha + 1, y)pochhammer(beta + delta + 1, y)pochhammer(gamma + 1, y)pochhammer(gamma + delta + 2, y))/(pochhammer(- alpha + gamma + delta + 1, y)pochhammer(- beta + gamma + 1, y)pochhammer(delta + 1, y)factorial(y))`	`Subscript[\[Omega], y]=Divide[Pochhammer[\[Alpha]+ 1, y]Pochhammer[\[Beta]+ \[Delta]+ 1, y]Pochhammer[\[Gamma]+ 1, y]Pochhammer[\[Gamma]+ \[Delta]+ 2, y],Pochhammer[- \[Alpha]+ \[Gamma]+ \[Delta]+ 1, y]Pochhammer[- \[Beta]+ \[Gamma]+ 1, y]Pochhammer[\[Delta]+ 1, y](y)!]`	Failure	Failure	Fail `12.16329441-7.801522345I <- {alpha = 2^(1/2)+I2^(1/2), beta = 2^(1/2)+I2^(1/2), delta = 2^(1/2)+I2^(1/2), omega[y] = 2^(1/2)+I2^(1/2), y = 1}` `63.95461927+94.62189224I <- {alpha = 2^(1/2)+I2^(1/2), beta = 2^(1/2)+I2^(1/2), delta = 2^(1/2)+I2^(1/2), omega[y] = 2^(1/2)+I2^(1/2), y = 2}` `-357.9031707+453.3958173I <- {alpha = 2^(1/2)+I2^(1/2), beta = 2^(1/2)+I2^(1/2), delta = 2^(1/2)+I2^(1/2), omega[y] = 2^(1/2)+I2^(1/2), y = 3}` `12.16329441-10.62994947I <- {alpha = 2^(1/2)+I2^(1/2), beta = 2^(1/2)+I2^(1/2), delta = 2^(1/2)+I2^(1/2), omega[y] = 2^(1/2)-I2^(1/2), y = 1}` ... skip entries to safe data	Skip
18.25.E12	${\displaystyle{\displaystyle h_{n}=\frac{{\left(-\beta\right)_{N}}{\left(% \gamma+\delta+2\right)_{N}}}{{\left(-\beta+\gamma+1\right)_{N}}{\left(\delta+1% \right)_{N}}}\frac{{\left(n+\alpha+\beta+1\right)_{n}}n!}{{\left(\alpha+\beta+% 2\right)_{2n}}}\*\frac{{\left(\alpha+\beta-\gamma+1\right)_{n}}{\left(\alpha-% \delta+1\right)_{n}}{\left(\beta+1\right)_{n}}}{{\left(\alpha+1\right)_{n}}{% \left(\beta+\delta+1\right)_{n}}{\left(\gamma+1\right)_{n}}}}}$	`h[n]=(pochhammer(- beta, N)pochhammer(gamma + delta + 2, N))/(pochhammer(- beta + gamma + 1, N)pochhammer(delta + 1, N))(pochhammer(n + alpha + beta + 1, n)factorial(n))/(pochhammer(alpha + beta + 2, 2n))(pochhammer(alpha + beta - gamma + 1, n)pochhammer(alpha - delta + 1, n)pochhammer(beta + 1, n))/(pochhammer(alpha + 1, n)pochhammer(beta + delta + 1, n)pochhammer(gamma + 1, n))`	`Subscript[h, n]=Divide[Pochhammer[- \[Beta], N]Pochhammer[\[Gamma]+ \[Delta]+ 2, N],Pochhammer[- \[Beta]+ \[Gamma]+ 1, N]Pochhammer[\[Delta]+ 1, N]]Divide[Pochhammer[n + \[Alpha]+ \[Beta]+ 1, n](n)!,Pochhammer[\[Alpha]+ \[Beta]+ 2, 2n]]Divide[Pochhammer[\[Alpha]+ \[Beta]- \[Gamma]+ 1, n]Pochhammer[\[Alpha]- \[Delta]+ 1, n]Pochhammer[\[Beta]+ 1, n],Pochhammer[\[Alpha]+ 1, n]Pochhammer[\[Beta]+ \[Delta]+ 1, n]Pochhammer[\[Gamma]+ 1, n]]`	Failure	Failure	Skip	Skip
18.25.E14	${\displaystyle{\displaystyle\omega_{y}=\frac{(-1)^{y}{\left(-N\right)_{y}}{% \left(\gamma+1\right)_{y}}{\left(\gamma+\delta+1\right)_{2}}}{{\left(N+\gamma+% \delta+2\right)_{y}}{\left(\delta+1\right)_{y}}y!}}}$	`omega[y]=((- 1)^(y)* pochhammer(- N, y)pochhammer(gamma + 1, y)pochhammer(gamma + delta + 1, 2))/(pochhammer(N + gamma + delta + 2, y)pochhammer(delta + 1, y)factorial(y))`	`Subscript[\[Omega], y]=Divide[(- 1)^(y)* Pochhammer[- N, y]Pochhammer[\[Gamma]+ 1, y]Pochhammer[\[Gamma]+ \[Delta]+ 1, 2],Pochhammer[N + \[Gamma]+ \[Delta]+ 2, y]Pochhammer[\[Delta]+ 1, y](y)!]`	Failure	Failure	Fail `-.785651684+.48581315e-1I <- {N = 2^(1/2)+I2^(1/2), delta = 2^(1/2)+I2^(1/2), omega[y] = 2^(1/2)+I2^(1/2), y = 1}` `1.316871147+1.251035041I <- {N = 2^(1/2)+I2^(1/2), delta = 2^(1/2)+I2^(1/2), omega[y] = 2^(1/2)+I2^(1/2), y = 2}` `1.420652225+1.407312490I <- {N = 2^(1/2)+I2^(1/2), delta = 2^(1/2)+I2^(1/2), omega[y] = 2^(1/2)+I2^(1/2), y = 3}` `-.785651684-2.779845809I <- {N = 2^(1/2)+I2^(1/2), delta = 2^(1/2)+I2^(1/2), omega[y] = 2^(1/2)-I2^(1/2), y = 1}` ... skip entries to safe data	Skip
18.25.E15	${\displaystyle{\displaystyle h_{n}=\frac{n!\,(N-n)!\,{\left(\gamma+\delta+2% \right)_{N}}}{N!\,{\left(\gamma+1\right)_{n}}{\left(\delta+1\right)_{N-n}}}}}$	`h[n]=(factorial(n)factorial(N - n)pochhammer(gamma + delta + 2, N))/(factorial(N)pochhammer(gamma + 1, n)pochhammer(delta + 1, N - n))`	`Subscript[h, n]=Divide[(n)!(N - n)!Pochhammer[\[Gamma]+ \[Delta]+ 2, N],(N)!Pochhammer[\[Gamma]+ 1, n]Pochhammer[\[Delta]+ 1, N - n]]`	Failure	Failure	Fail `-1.311749265+.9066027273I <- {N = 2^(1/2)+I2^(1/2), delta = 2^(1/2)+I2^(1/2), h[n] = 2^(1/2)+I2^(1/2), n = 1}` `-3.476693885+1.928884423I <- {N = 2^(1/2)+I2^(1/2), delta = 2^(1/2)+I2^(1/2), h[n] = 2^(1/2)+I2^(1/2), n = 2}` `-4.217343867+7.012858818I <- {N = 2^(1/2)+I2^(1/2), delta = 2^(1/2)+I2^(1/2), h[n] = 2^(1/2)+I2^(1/2), n = 3}` `-1.311749265-1.921824397I <- {N = 2^(1/2)+I2^(1/2), delta = 2^(1/2)+I2^(1/2), h[n] = 2^(1/2)-I2^(1/2), n = 1}` ... skip entries to safe data	Skip
18.28.E2	${\displaystyle{\displaystyle\int_{-1}^{1}p_{n}(x)p_{m}(x)w(x)\mathrm{d}x=h_{n}% \delta_{n,m}}}$	`int(p[n](x) p[m](x) w(x), x = - 1..1)= h[n]KroneckerDelta[n, m]`	`Integrate[Subscript[p, n](x) Subscript[p, m](x) w(x), {x, - 1, 1}]= Subscript[h, n]KroneckerDelta[n, m]`	Failure	Failure	Skip	Successful
18.28.E6	${\displaystyle{\displaystyle\int_{-1}^{1}p_{n}(x)p_{m}(x)w(x)\mathrm{d}x+\sum_% {\ell}p_{n}(x_{\ell})p_{m}(x_{\ell})\omega_{\ell}=h_{n}\delta_{n,m}}}$	`int(p[n](x) p[m](x) w(x), x = - 1..1)+ sum(p[n](x[ell])* p[m](x[ell]) omega[ell], ell = - infinity..infinity)= h[n]*KroneckerDelta[n, m]`	`Integrate[Subscript[p, n](x) Subscript[p, m](x) w(x), {x, - 1, 1}]+ Sum[Subscript[p, n](Subscript[x, \[ScriptL]])* Subscript[p, m](Subscript[x, \[ScriptL]]) Subscript[\[Omega], \[ScriptL]], {\[ScriptL], - Infinity, Infinity}]= Subscript[h, n]*KroneckerDelta[n, m]`	Failure	Failure	Skip	Error
18.28.E11	${\displaystyle{\displaystyle 0<q}}$	`0 < q`	`0 < q`	Failure	Failure	Successful	Successful
18.28.E11	${\displaystyle{\displaystyle q<1,a,b\in\mathbb{R},ab}}$	`q < 1 , a , b in real , a*b`	`q < 1 , a , b \[Element]Reals , ab`	Failure	Failure	Successful	Error
18.28.E11	${\displaystyle{\displaystyle 1,a,b\in\mathbb{R},ab>1,a^{-1}b}}$	`1 , a , b in real , ab > 1 , (a)^(- 1) b`	`1 , a , b \[Element]Reals , ab > 1 , (a)^(- 1)* b`	Error	Failure	-	Error
18.28.E11	${\displaystyle{\displaystyle 1,a^{-1}b<q^{-1}}}$	`1 , (a)^(- 1)* b < (q)^(- 1)`	`1 , (a)^(- 1)* b < (q)^(- 1)`	Failure	Failure	Skip	-
18.28.E12	${\displaystyle{\displaystyle 0<q}}$	`0 < q`	`0 < q`	Failure	Failure	Successful	Successful
18.28.E12	${\displaystyle{\displaystyle q<1,\ifrac{a}{\mathrm{i}},\ifrac{b}{\mathrm{i}}% \in\mathbb{R},(\Im a)(\Im b)}}$	`q < 1 ,(a)/(I),(b)/(I)in real ,(Im(a))*(Im(b))`	`q < 1 ,Divide[a,I],Divide[b,I]\[Element]Reals ,(Im[a])(Im[b])`	Failure	Failure	Successful	Error
18.28.E12	${\displaystyle{\displaystyle 1,\ifrac{a}{\mathrm{i}},\ifrac{b}{\mathrm{i}}\in% \mathbb{R},(\Im a)(\Im b)>0,a^{-1}b}}$	`1 ,(a)/(I),(b)/(I)in real ,(Im(a))(Im(b))> 0 , (a)^(- 1) b`	`1 ,Divide[a,I],Divide[b,I]\[Element]Reals ,(Im[a])(Im[b])> 0 , (a)^(- 1)* b`	Error	Failure	-	Error
18.28.E12	${\displaystyle{\displaystyle 0,a^{-1}b<q^{-1}}}$	`0 , (a)^(- 1)* b < (q)^(- 1)`	`0 , (a)^(- 1)* b < (q)^(- 1)`	Failure	Failure	Skip	-
18.28.E20	${\displaystyle{\displaystyle\sum_{y=0}^{N}R_{n}(q^{-y}+\gamma\delta q^{y+1})R_% {m}(q^{-y}+\gamma\delta q^{y+1})\omega_{y}=h_{n}\delta_{n,m}}}$	`sum(R[n]((q)^(- y)+ gammadelta(q)^(y + 1)) R[m]((q)^(- y)+ gammadelta(q)^(y + 1)) omega[y], y = 0..N)= h[n]*KroneckerDelta[n, m]`	`Sum[Subscript[R, n]((q)^(- y)+ \[Gamma]\[Delta](q)^(y + 1)) Subscript[R, m]((q)^(- y)+ \[Gamma]\[Delta](q)^(y + 1)) Subscript[\[Omega], y], {y, 0, N}]= Subscript[h, n]*KroneckerDelta[n, m]`	Failure	Failure	Skip	Skip
18.33.E3	${\displaystyle{\displaystyle\phi_{n}^{*}(z)=z^{n}\overline{\phi_{n}({\overline% {z}^{-1}})}}}$	`(phi[n])^()(z)= (z)^(n)* conjugate(phi[n]*((conjugate(z))^(- 1)))`	`(Subscript[\[Phi], n])^()(z)= (z)^(n)* Conjugate[Subscript[\[Phi], n]*((Conjugate[z])^(- 1))]`	Error	Failure	-	Error
18.33.E3	${\displaystyle{\displaystyle z^{n}\overline{\phi_{n}({\overline{z}^{-1}})}={% \kappa_{n}}+\sum_{\ell=1}^{n}\overline{\kappa}_{n,n-\ell}z^{\ell}}}$	`(z)^(n)* conjugate(phi[n]((conjugate(z))^(- 1)))=kappa[n]+ sum(conjugate(kappa)[n , n - ell](z)^(ell), ell = 1..n)`	`(z)^(n)* Conjugate[Subscript[\[Phi], n]((Conjugate[z])^(- 1))]=Subscript[\[Kappa], n]+ Sum[Subscript[Conjugate[\[Kappa]], n , n - \[ScriptL]](z)^(\[ScriptL]), {\[ScriptL], 1, n}]`	Failure	Failure	Skip	Skip
18.33#Ex1	${\displaystyle{\displaystyle w_{1}(x)=(1-x^{2})^{-\frac{1}{2}}w\left(x+\mathrm% {i}(1-x^{2})^{\frac{1}{2}}\right)}}$	`w[1](x)=(1 - (x)^(2))^(-(1)/(2)) w(x + I(1 - (x)^(2))^((1)/(2)))`	`Subscript[w, 1](x)=(1 - (x)^(2))^(-Divide[1,2]) w(x + I(1 - (x)^(2))^(Divide[1,2]))`	Failure	Failure	Fail `Float(infinity)+Float(infinity)I <- {w = 2^(1/2)+I2^(1/2), w[1] = 2^(1/2)+I2^(1/2), x = 1}` `2.609647525+3.047206723I <- {w = 2^(1/2)+I2^(1/2), w[1] = 2^(1/2)+I2^(1/2), x = 2}` `4.156854249+4.328427123I <- {w = 2^(1/2)+I2^(1/2), w[1] = 2^(1/2)+I2^(1/2), x = 3}` `Float(infinity)+Float(infinity)I <- {w = 2^(1/2)+I2^(1/2), w[1] = 2^(1/2)-I2^(1/2), x = 1}` ... skip entries to safe data	Fail `DirectedInfinity[] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1]}` `DirectedInfinity[] <- {Rule[w, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 1]}` `DirectedInfinity[] <- {Rule[w, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[x, 1]}` `DirectedInfinity[] <- {Rule[w, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1]}`
18.33#Ex2	${\displaystyle{\displaystyle w_{2}(x)=(1-x^{2})^{\frac{1}{2}}w\left(x+\mathrm{% i}(1-x^{2})^{\frac{1}{2}}\right)}}$	`w[2](x)=(1 - (x)^(2))^((1)/(2)) w(x + I(1 - (x)^(2))^((1)/(2)))`	`Subscript[w, 2](x)=(1 - (x)^(2))^(Divide[1,2]) w(x + I(1 - (x)^(2))^(Divide[1,2]))`	Failure	Failure	Fail `1.414213562+1.414213562I <- {w = 2^(1/2)+I2^(1/2), w[2] = 2^(1/2)+I2^(1/2), x = 1}` `3.484765921+2.172088327I <- {w = 2^(1/2)+I2^(1/2), w[2] = 2^(1/2)+I2^(1/2), x = 2}` `4.928932186+3.556349186I <- {w = 2^(1/2)+I2^(1/2), w[2] = 2^(1/2)+I2^(1/2), x = 3}` `1.414213562-1.414213562I <- {w = 2^(1/2)+I2^(1/2), w[2] = 2^(1/2)-I2^(1/2), x = 1}` ... skip entries to safe data	Successful
18.33.E13	${\displaystyle{\displaystyle\phi_{n}(z)=\sum_{\ell=0}^{n}\frac{{\left(\lambda+% 1\right)_{\ell}}{\left(\lambda\right)_{n-\ell}}}{\ell!\,(n-\ell)!}\,z^{\ell}}}$	`phi[n](z)= sum((pochhammer(lambda + 1, ell)pochhammer(lambda, n - ell))/(factorial(ell)factorial(n - ell))(z)^(ell), ell = 0..n)`	`Subscript[\[Phi], n](z)= Sum[Divide[Pochhammer[\[Lambda]+ 1, \[ScriptL]]Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!(n - \[ScriptL])!](z)^(\[ScriptL]), {\[ScriptL], 0, n}]`	Failure	Failure	Skip	Skip
18.33.E13	${\displaystyle{\displaystyle\sum_{\ell=0}^{n}\frac{{\left(\lambda+1\right)_{% \ell}}{\left(\lambda\right)_{n-\ell}}}{\ell!\,(n-\ell)!}\,z^{\ell}=\frac{{% \left(\lambda\right)_{n}}}{n!}{{}_{2}F_{1}}\left({-n,\lambda+1\atop-\lambda-n+% 1};z\right)}}$	`sum((pochhammer(lambda + 1, ell)pochhammer(lambda, n - ell))/(factorial(ell)factorial(n - ell))(z)^(ell), ell = 0..n)=(pochhammer(lambda, n))/(factorial(n))hypergeom([- n , lambda + 1], [- lambda - n + 1], z)`	`Sum[Divide[Pochhammer[\[Lambda]+ 1, \[ScriptL]]Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!(n - \[ScriptL])!](z)^(\[ScriptL]), {\[ScriptL], 0, n}]=Divide[Pochhammer[\[Lambda], n],(n)!]HypergeometricPFQ[{- n , \[Lambda]+ 1}, {- \[Lambda]- n + 1}, z]`	Successful	Successful	-	-
18.34.E1	${\displaystyle{\displaystyle y_{n}\left(x;a\right)={{}_{2}F_{0}}\left({-n,n+a-% 1\atop-};-\frac{x}{2}\right)}}$	`Error`	`Pochhammer[n + a - 1, n] (x/2)^n Hypergeometric1F1[-n, -2 n - a + 2, 2/x]= HypergeometricPFQ[{- n , n + a - 1}, {-}, -Divide[x,2]]`	Error	Failure	-	Error
18.34.E1	${\displaystyle{\displaystyle{{}_{2}F_{0}}\left({-n,n+a-1\atop-};-\frac{x}{2}% \right)={\left(n+a-1\right)_{n}}\left(\frac{x}{2}\right)^{n}{{}_{1}F_{1}}\left% ({-n\atop-2n-a+2};\frac{2}{x}\right)}}$	`hypergeom([- n , n + a - 1], [-], -(x)/(2))= pochhammer(n + a - 1, n)((x)/(2))^(n) hypergeom([- n], [- 2*n - a + 2], (2)/(x))`	`HypergeometricPFQ[{- n , n + a - 1}, {-}, -Divide[x,2]]= Pochhammer[n + a - 1, n](Divide[x,2])^(n) HypergeometricPFQ[{- n}, {- 2*n - a + 2}, Divide[2,x]]`	Error	Failure	-	Error
18.34#Ex1	${\displaystyle{\displaystyle y_{n}(x)=y_{n}\left(x;2\right)}}$	`Error`	`Subscript[y, n]*(x)= Pochhammer[n + 2 - 1, n] (x/2)^n Hypergeometric1F1[-n, -2 n - 2 + 2, 2/x]`	Error	Failure	-	Successful
18.34.E4	${\displaystyle{\displaystyle y_{n+1}\left(x;a\right)=(A_{n}x+B_{n})y_{n}\left(% x;a\right)-C_{n}y_{n-1}\left(x;a\right)}}$	`Error`	`Pochhammer[n + 1 + a - 1, n + 1] (x/2)^n + 1 Hypergeometric1F1[-n + 1, -2 n + 1 - a + 2, 2/x]=(Subscript[A, n]x + Subscript[B, n]) Pochhammer[n + a - 1, n] (x/2)^n Hypergeometric1F1[-n, -2 n - a + 2, 2/x]- Subscript[C, n]*Pochhammer[n - 1 + a - 1, n - 1] (x/2)^n - 1 Hypergeometric1F1[-n - 1, -2 n - 1 - a + 2, 2/x]`	Error	Failure	-	Skip
18.34.E7	${\displaystyle{\displaystyle x^{2}y_{n}''\left(x;a\right)+(ax+2)y_{n}'\left(x;% a\right)-n(n+a-1)y_{n}\left(x;a\right)=0}}$	`Error`	`(x)^(2)* (D[Pochhammer[n + a - 1, n] (temp/2)^n Hypergeometric1F1[-n, -2 n - a + 2, 2/temp], {temp, 2}]/.temp-> x)+(ax + 2) (D[Pochhammer[n + a - 1, n] (temp/2)^n Hypergeometric1F1[-n, -2 n - a + 2, 2/temp], {temp, 1}]/.temp-> x)- n(n + a - 1) Pochhammer[n + a - 1, n] (x/2)^n Hypergeometric1F1[-n, -2 n - a + 2, 2/x]= 0`	Error	Successful	-	-
18.34.E8	${\displaystyle{\displaystyle\lim_{\alpha\to\infty}\frac{P^{(\alpha,a-\alpha-2)% }_{n}\left(1+\alpha x\right)}{P^{(\alpha,a-\alpha-2)}_{n}\left(1\right)}=y_{n}% \left(x;a\right)}}$	`Error`	`Limit[Divide[JacobiP[n, \[Alpha], a - \[Alpha]- 2, 1 + \[Alpha]*x],JacobiP[n, \[Alpha], a - \[Alpha]- 2, 1]], \[Alpha] -> Infinity]= Pochhammer[n + a - 1, n] (x/2)^n Hypergeometric1F1[-n, -2 n - a + 2, 2/x]`	Error	Failure	-	Error
18.35.E4	${\displaystyle{\displaystyle\frac{{\left(\lambda-\mathrm{i}\tau_{a,b}(\theta)% \right)_{n}}}{n!}e^{\mathrm{i}n\theta}\*{{}_{2}F_{1}}\left({-n,\lambda+\mathrm% {i}\tau_{a,b}(\theta)\atop-n-\lambda+1+\mathrm{i}\tau_{a,b}(\theta)};e^{-2% \mathrm{i}\theta}\right)=\sum_{\ell=0}^{n}\frac{{\left(\lambda+\mathrm{i}\tau_% {a,b}(\theta)\right)_{\ell}}}{\ell!}\frac{{\left(\lambda-\mathrm{i}\tau_{a,b}(% \theta)\right)_{n-\ell}}}{(n-\ell)!}e^{\mathrm{i}(n-2\ell)\theta}}}$	`(pochhammer(lambda - Itau[a , b](theta), n))/(factorial(n))exp(Intheta) hypergeom([- n , lambda + Itau[a , b](theta)], [- n - lambda + 1 + Itau[a , b](theta)], exp(- 2Itheta))= sum((pochhammer(lambda + Itau[a , b](theta), ell))/(factorial(ell))(pochhammer(lambda - Itau[a , b](theta), n - ell))/(factorial(n - ell))exp(I(n - 2ell)* theta), ell = 0..n)`	Divide[Pochhammer[\[Lambda]- ISubscript[\[Tau], a , b](\[Theta]), n],(n)!]Exp[In\[Theta]] HypergeometricPFQ[{- n , \[Lambda]+ ISubscript[\[Tau], a , b](\[Theta])}, {- n - \[Lambda]+ 1 + ISubscript[\[Tau], a , b](\[Theta])}, Exp[- 2I\[Theta]]]= Sum[Divide[Pochhammer[\[Lambda]+ ISubscript[\[Tau], a , b](\[Theta]), \[ScriptL]],(\[ScriptL])!]Divide[Pochhammer[\[Lambda]- ISubscript[\[Tau], a , b](\[Theta]), n - \[ScriptL]],(n - \[ScriptL])!]Exp[I(n - 2\[ScriptL])* \[Theta]], {\[ScriptL], 0, n}]	Successful	Successful	-	-
18.38.E1	${\displaystyle{\displaystyle V_{n}(x)=\ifrac{2nH_{n+1}\left(x\right)H_{n-1}% \left(x\right)}{(H_{n}\left(x\right))^{2}}}}$	`V[n](x)=(2nHermiteH(n + 1, x)HermiteH(n - 1, x))/((HermiteH(n, x))^(2))`	`Subscript[V, n](x)=Divide[2nHermiteH[n + 1, x]HermiteH[n - 1, x],(HermiteH[n, x])^(2)]`	Failure	Failure	Fail `.414213562+1.414213562I <- {V[n] = 2^(1/2)+I2^(1/2), n = 1, x = 1}` `1.078427124+2.828427124I <- {V[n] = 2^(1/2)+I2^(1/2), n = 1, x = 2}` `2.353751797+4.242640686I <- {V[n] = 2^(1/2)+I2^(1/2), n = 1, x = 3}` `9.414213562+1.414213562I <- {V[n] = 2^(1/2)+I2^(1/2), n = 2, x = 1}` ... skip entries to safe data	Skip
18.38.E3	${\displaystyle{\displaystyle\sum_{m=0}^{n}P^{(\alpha,0)}_{m}\left(x\right)>=0}}$	`sum(JacobiP(m, alpha, 0, x), m = 0..n)> = 0`	`Sum[JacobiP[m, \[Alpha], 0, x], {m, 0, n}]> = 0`	Failure	Failure	Skip	Successful
18.39.E5	${\displaystyle{\displaystyle\eta_{n}(x)=\pi^{-\frac{1}{4}}2^{-\frac{1}{2}n}(n!% \,b)^{-\frac{1}{2}}H_{n}\left(x/b\right)e^{-x^{2}/2b^{2}}}}$	`eta[n](x)= (Pi)^(-(1)/(4)) (2)^(-(1)/(2)n)(factorial(n)b)^(-(1)/(2)) HermiteH(n, x/ b)exp(- (x)^(2)/ 2(b)^(2))`	`Subscript[\[Eta], n](x)= (Pi)^(-Divide[1,4]) (2)^(-Divide[1,2]n)((n)!b)^(-Divide[1,2]) HermiteH[n, x/ b]Exp[- (x)^(2)/ 2(b)^(2)]`	Failure	Failure	Fail `1.789526128+1.400506842I <- {b = 2^(1/2)+I2^(1/2), eta[n] = 2^(1/2)+I2^(1/2), n = 1, x = 1}` `3.556814811+3.011841849I <- {b = 2^(1/2)+I2^(1/2), eta[n] = 2^(1/2)+I2^(1/2), n = 1, x = 2}` `3.176214405+4.606180986I <- {b = 2^(1/2)+I2^(1/2), eta[n] = 2^(1/2)+I2^(1/2), n = 1, x = 3}` `1.266982360+1.020980529I <- {b = 2^(1/2)+I2^(1/2), eta[n] = 2^(1/2)+I2^(1/2), n = 2, x = 1}` ... skip entries to safe data	Successful

Results of Orthogonal Polynomials

Navigation menu

Search