# 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)=(2*factorial(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, 2*x - 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+1.280284738*I <- {N = 2^(1/2)+I*2^(1/2), x[N+1,n] = 2^(1/2)+I*2^(1/2), n = 1}
1.251822237+.4629104109*I <- {N = 2^(1/2)+I*2^(1/2), x[N+1,n] = 2^(1/2)+I*2^(1/2), n = 2}
3.059241197+.132349918*I <- {N = 2^(1/2)+I*2^(1/2), x[N+1,n] = 2^(1/2)+I*2^(1/2), n = 3}
.4933988023-1.548142386*I <- {N = 2^(1/2)+I*2^(1/2), x[N+1,n] = 2^(1/2)-I*2^(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+1.911393109*I <- {theta = 2^(1/2)+I*2^(1/2), n = 1, x = 1}
1.660326008+1.911393109*I <- {theta = 2^(1/2)+I*2^(1/2), n = 1, x = 2}
2.660326008+1.911393109*I <- {theta = 2^(1/2)+I*2^(1/2), n = 1, x = 3}
9.076090394+2.597002114*I <- {theta = 2^(1/2)+I*2^(1/2), n = 2, x = 1}
... skip entries to safe data
Fail
Complex[0.6603260083052754, 1.9113931101642103] <- {Rule[n, 1], Rule[x, 1], Rule[Îž, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.6603260083052755, 1.9113931101642103] <- {Rule[n, 1], Rule[x, 2], Rule[Îž, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[2.6603260083052755, 1.9113931101642103] <- {Rule[n, 1], Rule[x, 3], Rule[Îž, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[9.076090401898771, 2.5970021097090865] <- {Rule[n, 2], Rule[x, 1], Rule[Îž, 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+3.822786219*I <- {theta = 2^(1/2)+I*2^(1/2), n = 1, x = 1}
3.320652015+3.822786219*I <- {theta = 2^(1/2)+I*2^(1/2), n = 1, x = 2}
5.320652015+3.822786219*I <- {theta = 2^(1/2)+I*2^(1/2), n = 1, x = 3}
18.15218079+5.194004229*I <- {theta = 2^(1/2)+I*2^(1/2), n = 2, x = 1}
... skip entries to safe data
Fail
Complex[1.3206520166105502, 3.82278622032842] <- {Rule[n, 1], Rule[x, 1], Rule[Îž, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[3.32065201661055, 3.82278622032842] <- {Rule[n, 1], Rule[x, 2], Rule[Îž, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[5.32065201661055, 3.82278622032842] <- {Rule[n, 1], Rule[x, 3], Rule[Îž, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[18.152180803797542, 5.194004219418172] <- {Rule[n, 2], Rule[x, 1], Rule[Îž, 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(2*lambda, n))/(factorial(n))*hypergeom([- n , n + 2*lambda], [lambda +(1)/(2)], (1 - x)/(2)) GegenbauerC[n, \[Lambda], x]=Divide[Pochhammer[2*\[Lambda], n],(n)!]*HypergeometricPFQ[{- n , n + 2*\[Lambda]}, {\[Lambda]+Divide[1,2]}, Divide[1 - x,2]] Successful Successful - -
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 - 2*ell))*(2*x)^(n - 2*ell), ell = 0..floor(n/ 2)) GegenbauerC[n, \[Lambda], x]= Sum[Divide[(- 1)^(\[ScriptL])* Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!*(n - 2*\[ScriptL])!]*(2*x)^(n - 2*\[ScriptL]), {\[ScriptL], 0, Floor[n/ 2]}] 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 - 2*ell))*(2*x)^(n - 2*ell), ell = 0..floor(n/ 2))=(2*x)^(n)*(pochhammer(lambda, n))/(factorial(n))*hypergeom([-(1)/(2)*n , -(1)/(2)*n +(1)/(2)], [1 - lambda - n], (1)/((x)^(2))) Sum[Divide[(- 1)^(\[ScriptL])* Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!*(n - 2*\[ScriptL])!]*(2*x)^(n - 2*\[ScriptL]), {\[ScriptL], 0, Floor[n/ 2]}]=(2*x)^(n)*Divide[Pochhammer[\[Lambda], n],(n)!]*HypergeometricPFQ[{-Divide[1,2]*n , -Divide[1,2]*n +Divide[1,2]}, {1 - \[Lambda]- n}, Divide[1,(x)^(2)]] Failure Failure 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 - 2*ell)* theta), ell = 0..n) GegenbauerC[n, \[Lambda], Cos[\[Theta]]]= Sum[Divide[Pochhammer[\[Lambda], \[ScriptL]]*Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!*(n - \[ScriptL])!]*Cos[(n - 2*\[ScriptL])* \[Theta]], {\[ScriptL], 0, n}] 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 - 2*ell)* theta), ell = 0..n)= exp(I*n*theta)*(pochhammer(lambda, n))/(factorial(n))*hypergeom([- n , lambda], [1 - lambda - n], exp(- 2*I*theta)) Sum[Divide[Pochhammer[\[Lambda], \[ScriptL]]*Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!*(n - \[ScriptL])!]*Cos[(n - 2*\[ScriptL])* \[Theta]], {\[ScriptL], 0, n}]= Exp[I*n*\[Theta]]*Divide[Pochhammer[\[Lambda], n],(n)!]*HypergeometricPFQ[{- n , \[Lambda]}, {1 - \[Lambda]- n}, Exp[- 2*I*\[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)*(2*x)^(n - 2*ell))/(factorial(ell)*factorial(n - 2*ell)), ell = 0..floor(n/ 2)) HermiteH[n, x]= (n)!*Sum[Divide[(- 1)^(\[ScriptL])*(2*x)^(n - 2*\[ScriptL]),(\[ScriptL])!*(n - 2*\[ScriptL])!], {\[ScriptL], 0, Floor[n/ 2]}] 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)*(2*x)^(n - 2*ell))/(factorial(ell)*factorial(n - 2*ell)), ell = 0..floor(n/ 2))=(2*x)^(n)* hypergeom([-(1)/(2)*n , -(1)/(2)*n +(1)/(2)], [-], -(1)/((x)^(2))) (n)!*Sum[Divide[(- 1)^(\[ScriptL])*(2*x)^(n - 2*\[ScriptL]),(\[ScriptL])!*(n - 2*\[ScriptL])!], {\[ScriptL], 0, Floor[n/ 2]}]=(2*x)^(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)- 3*x ChebyshevT[3, x]= 4*(x)^(3)- 3*x 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)- 4*x ChebyshevU[3, x]= 8*(x)^(3)- 4*x 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)- 2*x + 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)- 4*x + 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)- 6*x + 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)- 12*x HermiteH[3, x]= 8*(x)^(3)- 12*x 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(2*lambda, 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(2*alpha + 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, 2*x - 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(2*n, alpha, alpha, x))/(JacobiP(2*n, alpha, alpha, 1))=(JacobiP(n, alpha, -(1)/(2), 2*(x)^(2)- 1))/(JacobiP(n, alpha, -(1)/(2), 1)) Divide[JacobiP[2*n, \[Alpha], \[Alpha], x],JacobiP[2*n, \[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(2*n + 1, alpha, alpha, x))/(JacobiP(2*n + 1, alpha, alpha, 1))=(x*JacobiP(n, alpha, (1)/(2), 2*(x)^(2)- 1))/(JacobiP(n, alpha, (1)/(2), 1)) Divide[JacobiP[2*n + 1, \[Alpha], \[Alpha], x],JacobiP[2*n + 1, \[Alpha], \[Alpha], 1]]=Divide[x*JacobiP[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(2*n, lambda, x)=(pochhammer(lambda, n))/(pochhammer((1)/(2), n))*JacobiP(n, lambda -(1)/(2), -(1)/(2), 2*(x)^(2)- 1) GegenbauerC[2*n, \[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(2*n + 1, lambda, x)=(pochhammer(lambda, n + 1))/(pochhammer((1)/(2), n + 1))*x*JacobiP(n, lambda -(1)/(2), (1)/(2), 2*(x)^(2)- 1) GegenbauerC[2*n + 1, \[Lambda], x]=Divide[Pochhammer[\[Lambda], n + 1],Pochhammer[Divide[1,2], n + 1]]*x*JacobiP[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 ${\displaystyle{\displaystyle(2n+\alpha+\beta+1)P^{(\alpha,\beta)}_{n}\left(x% \right)=(n+\alpha+\beta+1)P^{(\alpha,\beta+1)}_{n}\left(x\right)+(n+\alpha)P^{% (\alpha,\beta+1)}_{n-1}\left(x\right)}}$ (2*n + 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) (2*n + \[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 ${\displaystyle{\displaystyle 4\lambda(n+\lambda+1)(1-x^{2})C^{(\lambda+1)}_{n}% \left(x\right)=-(n+1)(n+2)C^{(\lambda)}_{n+2}\left(x\right)+(n+2\lambda)(n+2% \lambda+1)C^{(\lambda)}_{n}\left(x\right)}}$ 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) 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)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), n = 1, x = 1}
Float(infinity)+Float(infinity)*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), n = 2, x = 1}
Float(infinity)+Float(infinity)*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), n = 3, x = 1}
Float(infinity)+Float(infinity)*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)-I*2^(1/2), n = 1, x = 1}
... skip entries to safe data
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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)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), n = 1, x = 1}
Float(undefined)+Float(undefined)*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), n = 2, x = 1}
Float(undefined)+Float(undefined)*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), n = 3, x = 1}
Float(undefined)+Float(undefined)*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)-I*2^(1/2), n = 1, x = 1}
... skip entries to safe data
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18.9.E17 ${\displaystyle{\displaystyle(2n+\alpha+\beta)(1-x^{2})\frac{\mathrm{d}}{% \mathrm{d}x}P^{(\alpha,\beta)}_{n}\left(x\right)=n\left(\alpha-\beta-(2n+% \alpha+\beta)x\right)P^{(\alpha,\beta)}_{n}\left(x\right)+2(n+\alpha)(n+\beta)% P^{(\alpha,\beta)}_{n-1}\left(x\right)}}$ (2*n + alpha + beta)*(1 - (x)^(2))* diff(JacobiP(n, alpha, beta, x), x)= n*(alpha - beta -(2*n + alpha + beta)*x)* JacobiP(n, alpha, beta, x)+ 2*(n + alpha)*(n + beta)* JacobiP(n - 1, alpha, beta, x) (2*n + \[Alpha]+ \[Beta])*(1 - (x)^(2))* D[JacobiP[n, \[Alpha], \[Beta], x], x]= n*(\[Alpha]- \[Beta]-(2*n + \[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 ${\displaystyle{\displaystyle(2n+\alpha+\beta+2)(1-x^{2})\frac{\mathrm{d}}{% \mathrm{d}x}P^{(\alpha,\beta)}_{n}\left(x\right)=(n+\alpha+\beta+1)\left(% \alpha-\beta+(2n+\alpha+\beta+2)x\right)P^{(\alpha,\beta)}_{n}\left(x\right)-2% (n+1)(n+\alpha+\beta+1)P^{(\alpha,\beta)}_{n+1}\left(x\right)}}$ (2*n + alpha + beta + 2)*(1 - (x)^(2))* diff(JacobiP(n, alpha, beta, x), x)=(n + alpha + beta + 1)*(alpha - beta +(2*n + alpha + beta + 2)*x)* JacobiP(n, alpha, beta, x)- 2*(n + 1)*(n + alpha + beta + 1)* JacobiP(n + 1, alpha, beta, x) (2*n + \[Alpha]+ \[Beta]+ 2)*(1 - (x)^(2))* D[JacobiP[n, \[Alpha], \[Beta], x], x]=(n + \[Alpha]+ \[Beta]+ 1)*(\[Alpha]- \[Beta]+(2*n + \[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)= 2*lambda*GegenbauerC(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 + 2*lambda - 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)+I*2^(1/2), n = 1, x = 1}
Float(infinity)+Float(infinity)*I <- {lambda = 2^(1/2)+I*2^(1/2), n = 2, x = 1}
Float(infinity)+Float(infinity)*I <- {lambda = 2^(1/2)+I*2^(1/2), n = 3, x = 1}
Float(infinity)+Float(infinity)*I <- {lambda = 2^(1/2)-I*2^(1/2), n = 1, x = 1}
... skip entries to safe data
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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)= 2*n*HermiteH(n - 1, x) D[HermiteH[n, x], x]= 2*n*HermiteH[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))^(- 2*alpha)* 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)+ I*sin(theta)*cos(phi))^(n), phi = 0..Pi) LegendreP[n, Cos[\[Theta]]]=Divide[1,Pi]*Integrate[(Cos[\[Theta]]+ I*Sin[\[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 + I*t)^(n)* exp(- (t)^(2)), t = - infinity..infinity) HermiteH[n, x]=Divide[(2)^(n),(Pi)^(Divide[1,2])]*Integrate[(x + I*t)^(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)=((- 2*I)^(n)* exp((x)^(2)))/((Pi)^((1)/(2)))*int(exp(- (t)^(2))*(t)^(n)* exp(2*I*x*t), t = - infinity..infinity) HermiteH[n, x]=Divide[(- 2*I)^(n)* Exp[(x)^(2)],(Pi)^(Divide[1,2])]*Integrate[Exp[- (t)^(2)]*(t)^(n)* Exp[2*I*x*t], {t, - Infinity, Infinity}] Failure Failure Skip Successful
18.10.E10 ${\displaystyle{\displaystyle\frac{(-2i)^{n}e^{x^{2}}}{\pi^{\frac{1}{2}}}\int_{% -\infty}^{\infty}e^{-t^{2}}t^{n}e^{2ixt}\mathrm{d}t=\frac{2^{n+1}}{\pi^{\frac{% 1}{2}}}e^{x^{2}}\int_{0}^{\infty}e^{-t^{2}}t^{n}\cos\left(2xt-\tfrac{1}{2}n\pi% \right)\mathrm{d}t}}$ ((- 2*I)^(n)* exp((x)^(2)))/((Pi)^((1)/(2)))*int(exp(- (t)^(2))*(t)^(n)* exp(2*I*x*t), t = - infinity..infinity)=((2)^(n + 1))/((Pi)^((1)/(2)))*exp((x)^(2))*int(exp(- (t)^(2))*(t)^(n)* cos(2*x*t -(1)/(2)*n*Pi), t = 0..infinity) Divide[(- 2*I)^(n)* Exp[(x)^(2)],(Pi)^(Divide[1,2])]*Integrate[Exp[- (t)^(2)]*(t)^(n)* Exp[2*I*x*t], {t, - Infinity, Infinity}]=Divide[(2)^(n + 1),(Pi)^(Divide[1,2])]*Exp[(x)^(2)]*Integrate[Exp[- (t)^(2)]*(t)^(n)* Cos[2*x*t -Divide[1,2]*n*Pi], {t, 0, Infinity}] Successful Failure - Error
18.11.E1