Orthogonal Polynomials - 18.5 Explicit Representations

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
18.5.E1 T n ⁡ ( x ) = cos ⁡ ( n ⁢ θ ) Chebyshev-polynomial-first-kind-T 𝑛 𝑥 𝑛 𝜃 {\displaystyle{\displaystyle T_{n}\left(x\right)=\cos\left(n\theta\right)}}
\ChebyshevpolyT{n}@{x} = \cos@{n\theta}		 

ChebyshevT(n, x) = cos(n*theta)

ChebyshevT[n, x] == Cos[n*\[Theta]]
Failure Failure
Failed [90 / 90]
Result: .7694569811+.3969495503*I
Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 1}


Result: 3.747751686+1.159954891*I
Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 2}

... skip entries to safe data
Failed [90 / 90]
Result: Complex[0.7694569809427748, 0.3969495502290325]
Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[3.747751685467572, 1.1599548913509004]
Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
18.5.E2 U n ⁡ ( x ) = ( sin ⁡ ( n + 1 ) ⁢ θ ) / sin ⁡ θ Chebyshev-polynomial-second-kind-U 𝑛 𝑥 𝑛 1 𝜃 𝜃 {\displaystyle{\displaystyle U_{n}\left(x\right)=\ifrac{(\sin(n+1)\theta)}{% \sin\theta}}}
\ChebyshevpolyU{n}@{x} = \ifrac{(\sin@@{(n+1)\theta})}{\sin@@{\theta}}		 

ChebyshevU(n, x) = (sin((n + 1)*theta))/(sin(theta))

ChebyshevU[n, x] == Divide[Sin[(n + 1)*\[Theta]],Sin[\[Theta]]]
Failure Failure
Failed [90 / 90]
Result: 1.538913962+.7938991006*I
Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 1}


Result: 7.495503373+2.319909783*I
Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 2}

... skip entries to safe data
Failed [90 / 90]
Result: Complex[1.5389139618855496, 0.7938991004580651]
Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[7.495503370935143, 2.3199097827018003]
Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
18.5.E6 L n ( α ) ⁡ ( 1 x ) = ( - 1 ) n n ! ⁢ x n + α + 1 ⁢ e 1 / x ⁢ d n d x n ⁡ ( x - α - 1 ⁢ e - 1 / x ) Laguerre-polynomial-L 𝛼 𝑛 1 𝑥 superscript 1 𝑛 𝑛 superscript 𝑥 𝑛 𝛼 1 superscript 𝑒 1 𝑥 derivative 𝑥 𝑛 superscript 𝑥 𝛼 1 superscript 𝑒 1 𝑥 {\displaystyle{\displaystyle L^{(\alpha)}_{n}\left(\frac{1}{x}\right)=\frac{(-% 1)^{n}}{n!}x^{n+\alpha+1}e^{\ifrac{1}{x}}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}% ^{n}}\left(x^{-\alpha-1}e^{-\ifrac{1}{x}}\right)}}
\LaguerrepolyL[\alpha]{n}@{\frac{1}{x}} = \frac{(-1)^{n}}{n!}x^{n+\alpha+1}e^{\ifrac{1}{x}}\deriv[n]{}{x}\left(x^{-\alpha-1}e^{-\ifrac{1}{x}}\right)		 

LaguerreL(n, alpha, (1)/(x)) = ((- 1)^(n))/(factorial(n))*(x)^(n + alpha + 1)* exp((1)/(x))*diff((x)^(- alpha - 1)* exp(-(1)/(x)), [x$(n)])

LaguerreL[n, \[Alpha], Divide[1,x]] == Divide[(- 1)^(n),(n)!]*(x)^(n + \[Alpha]+ 1)* Exp[Divide[1,x]]*D[(x)^(- \[Alpha]- 1)* Exp[-Divide[1,x]], {x, n}]
Missing Macro Error Failure -
Failed [24 / 27]
Result: Plus[1.8333333333333335, Times[1.9477340410546757, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-1, , Plus[-1, Times[-1, ], 1], Plus[Times[-1, ], 1], 1.5, []], Times[Plus[-1, Times[-1, ], 1], Plus[, Times[-1, 1], Times[-2, , 1.5], Times[-3, Power[, 2], 1.5], Times[2, 1, 1.5], Times[3, , 1, 1.5], Times[-1, 1.5], Times[2, 1.5, 1.5], Times[2, , 1.5, 1.5]], [Plus[1, ]]], Times[-1, Plus[Times[-1, ], 1, 1.5], Plus[1, , Times[-1, 1], Times[-4, 1.5], Times[-7, , 1.5], Times[-3, Power[, 2], 1.5], Times[4, 1, 1.5], Times[3, , 1, 1.5], Times[2, 1.5, 1.5], Times[, 1.5, 1.5]], [Plus[2, ]]], Times[Plus[2, ], 1.5, Plus[-1, Times[-1, ], 1, 1.5], Plus[Times[-1, ], 1, 1.5], [Plus[3, ]]]], 0], Equal[[-1], 0], Equal[[0], 0], Equal[[1], Times[Power[E, Times[-1, Power[1.5, -1]]], Binomial[Plus[-1, Times[-1, 1.5]], 1]]]}]][2.0]]], {Rule[n, 1], Rule[x, 1.5], Rule[α, 1.5]}


Result: Plus[2.2638888888888893, Times[-1.9477340410546757, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-1, , Plus[-1, Times[-1, ], 2], Plus[Times[-1, ], 2], 1.5, []], Times[Plus[-1, Times[-1, ], 2], Plus[, Times[-1, 2], Times[-2, , 1.5], Times[-3, Power[, 2], 1.5], Times[2, 2, 1.5], Times[3, , 2, 1.5], Times[-1, 1.5], Times[2, 1.5, 1.5], Times[2, , 1.5, 1.5]], [Plus[1, ]]], Times[-1, Plus[Times[-1, ], 2, 1.5], Plus[1, , Times[-1, 2], Times[-4, 1.5], Times[-7, , 1.5], Times[-3, Power[, 2], 1.5], Times[4, 2, 1.5], Times[3, , 2, 1.5], Times[2, 1.5, 1.5], Times[, 1.5, 1.5]], [Plus[2, ]]], Times[Plus[2, ], 1.5, Plus[-1, Times[-1, ], 2, 1.5], Plus[Times[-1, ], 2, 1.5], [Plus[3, ]]]], 0], Equal[[-1], 0], Equal[[0], 0], Equal[[1], Times[Power[E, Times[-1, Power[1.5, -1]]], Binomial[Plus[-1, Times[-1, 1.5]], 2]]]}]][3.0]]], {Rule[n, 2], Rule[x, 1.5], Rule[α, 1.5]}

... skip entries to safe data
18.5.E7 P n ( α , β ) ⁡ ( x ) = ∑ ℓ = 0 n ( n + α + β + 1 ) ℓ ⁢ ( α + ℓ + 1 ) n - ℓ ℓ ! ⁢ ( n - ℓ ) ! ⁢ ( x - 1 2 ) ℓ Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 superscript subscript ℓ 0 𝑛 Pochhammer 𝑛 𝛼 𝛽 1 ℓ Pochhammer 𝛼 ℓ 1 𝑛 ℓ ℓ 𝑛 ℓ superscript 𝑥 1 2 ℓ {\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(x\right)=\sum_{\ell=0% }^{n}\frac{{\left(n+\alpha+\beta+1\right)_{\ell}}{\left(\alpha+\ell+1\right)_{% n-\ell}}}{\ell!\;(n-\ell)!}\left(\frac{x-1}{2}\right)^{\ell}}}
\JacobipolyP{\alpha}{\beta}{n}@{x} = \sum_{\ell=0}^{n}\frac{\Pochhammersym{n+\alpha+\beta+1}{\ell}\Pochhammersym{\alpha+\ell+1}{n-\ell}}{\ell!\;(n-\ell)!}\left(\frac{x-1}{2}\right)^{\ell}		 

JacobiP(n, alpha, beta, x) = sum((pochhammer(n + alpha + beta + 1, ell)*pochhammer(alpha + ell + 1, n - ell))/(factorial(ell)*factorial(n - ell))*((x - 1)/(2))^(ell), ell = 0..n)

JacobiP[n, \[Alpha], \[Beta], x] == Sum[Divide[Pochhammer[n + \[Alpha]+ \[Beta]+ 1, \[ScriptL]]*Pochhammer[\[Alpha]+ \[ScriptL]+ 1, n - \[ScriptL]],(\[ScriptL])!*(n - \[ScriptL])!]*(Divide[x - 1,2])^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None]
Successful Successful - Successful [Tested: 81]
18.5.E7 ∑ ℓ = 0 n ( n + α + β + 1 ) ℓ ⁢ ( α + ℓ + 1 ) n - ℓ ℓ ! ⁢ ( n - ℓ ) ! ⁢ ( x - 1 2 ) ℓ = ( α + 1 ) n n ! ⁢ F 1 2 ⁡ ( - n , n + α + β + 1 α + 1 ; 1 - x 2 ) superscript subscript ℓ 0 𝑛 Pochhammer 𝑛 𝛼 𝛽 1 ℓ Pochhammer 𝛼 ℓ 1 𝑛 ℓ ℓ 𝑛 ℓ superscript 𝑥 1 2 ℓ Pochhammer 𝛼 1 𝑛 𝑛 Gauss-hypergeometric-F-as-2F1 𝑛 𝑛 𝛼 𝛽 1 𝛼 1 1 𝑥 2 {\displaystyle{\displaystyle\sum_{\ell=0}^{n}\frac{{\left(n+\alpha+\beta+1% \right)_{\ell}}{\left(\alpha+\ell+1\right)_{n-\ell}}}{\ell!\;(n-\ell)!}\left(% \frac{x-1}{2}\right)^{\ell}=\frac{{\left(\alpha+1\right)_{n}}}{n!}{{}_{2}F_{1}% }\left({-n,n+\alpha+\beta+1\atop\alpha+1};\frac{1-x}{2}\right)}}
\sum_{\ell=0}^{n}\frac{\Pochhammersym{n+\alpha+\beta+1}{\ell}\Pochhammersym{\alpha+\ell+1}{n-\ell}}{\ell!\;(n-\ell)!}\left(\frac{x-1}{2}\right)^{\ell} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}\genhyperF{2}{1}@@{-n,n+\alpha+\beta+1}{\alpha+1}{\frac{1-x}{2}}		 

sum((pochhammer(n + alpha + beta + 1, ell)*pochhammer(alpha + ell + 1, n - ell))/(factorial(ell)*factorial(n - ell))*((x - 1)/(2))^(ell), ell = 0..n) = (pochhammer(alpha + 1, n))/(factorial(n))*hypergeom([- n , n + alpha + beta + 1], [alpha + 1], (1 - x)/(2))

Sum[Divide[Pochhammer[n + \[Alpha]+ \[Beta]+ 1, \[ScriptL]]*Pochhammer[\[Alpha]+ \[ScriptL]+ 1, n - \[ScriptL]],(\[ScriptL])!*(n - \[ScriptL])!]*(Divide[x - 1,2])^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None] == Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*HypergeometricPFQ[{- n , n + \[Alpha]+ \[Beta]+ 1}, {\[Alpha]+ 1}, Divide[1 - x,2]]
Successful Successful - Successful [Tested: 81]
18.5.E8 P n ( α , β ) ⁡ ( x ) = 2 - n ⁢ ∑ ℓ = 0 n ( n + α ℓ ) ⁢ ( n + β n - ℓ ) ⁢ ( x - 1 ) n - ℓ ⁢ ( x + 1 ) ℓ Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 superscript 2 𝑛 superscript subscript ℓ 0 𝑛 binomial 𝑛 𝛼 ℓ binomial 𝑛 𝛽 𝑛 ℓ superscript 𝑥 1 𝑛 ℓ superscript 𝑥 1 ℓ {\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(x\right)=2^{-n}\sum_{% \ell=0}^{n}\genfrac{(}{)}{0.0pt}{}{n+\alpha}{\ell}\genfrac{(}{)}{0.0pt}{}{n+% \beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell}}}
\JacobipolyP{\alpha}{\beta}{n}@{x} = 2^{-n}\sum_{\ell=0}^{n}\binom{n+\alpha}{\ell}\binom{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell}		 

JacobiP(n, alpha, beta, x) = (2)^(- n)* sum(binomial(n + alpha,ell)*binomial(n + beta,n - ell)*(x - 1)^(n - ell)*(x + 1)^(ell), ell = 0..n)

JacobiP[n, \[Alpha], \[Beta], x] == (2)^(- n)* Sum[Binomial[n + \[Alpha],\[ScriptL]]*Binomial[n + \[Beta],n - \[ScriptL]]*(x - 1)^(n - \[ScriptL])*(x + 1)^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None]
Failure Failure Successful [Tested: 81] Successful [Tested: 81]
18.5.E8 2 - n ⁢ ∑ ℓ = 0 n ( n + α ℓ ) ⁢ ( n + β n - ℓ ) ⁢ ( x - 1 ) n - ℓ ⁢ ( x + 1 ) ℓ = ( α + 1 ) n n ! ⁢ ( x + 1 2 ) n ⁢ F 1 2 ⁡ ( - n , - n - β α + 1 ; x - 1 x + 1 ) superscript 2 𝑛 superscript subscript ℓ 0 𝑛 binomial 𝑛 𝛼 ℓ binomial 𝑛 𝛽 𝑛 ℓ superscript 𝑥 1 𝑛 ℓ superscript 𝑥 1 ℓ Pochhammer 𝛼 1 𝑛 𝑛 superscript 𝑥 1 2 𝑛 Gauss-hypergeometric-F-as-2F1 𝑛 𝑛 𝛽 𝛼 1 𝑥 1 𝑥 1 {\displaystyle{\displaystyle 2^{-n}\sum_{\ell=0}^{n}\genfrac{(}{)}{0.0pt}{}{n+% \alpha}{\ell}\genfrac{(}{)}{0.0pt}{}{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell% }=\frac{{\left(\alpha+1\right)_{n}}}{n!}\left(\frac{x+1}{2}\right)^{n}{{}_{2}F% _{1}}\left({-n,-n-\beta\atop\alpha+1};\frac{x-1}{x+1}\right)}}
2^{-n}\sum_{\ell=0}^{n}\binom{n+\alpha}{\ell}\binom{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}\left(\frac{x+1}{2}\right)^{n}\genhyperF{2}{1}@@{-n,-n-\beta}{\alpha+1}{\frac{x-1}{x+1}}		 

(2)^(- n)* sum(binomial(n + alpha,ell)*binomial(n + beta,n - ell)*(x - 1)^(n - ell)*(x + 1)^(ell), ell = 0..n) = (pochhammer(alpha + 1, n))/(factorial(n))*((x + 1)/(2))^(n)* hypergeom([- n , - n - beta], [alpha + 1], (x - 1)/(x + 1))

(2)^(- n)* Sum[Binomial[n + \[Alpha],\[ScriptL]]*Binomial[n + \[Beta],n - \[ScriptL]]*(x - 1)^(n - \[ScriptL])*(x + 1)^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None] == Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*(Divide[x + 1,2])^(n)* HypergeometricPFQ[{- n , - n - \[Beta]}, {\[Alpha]+ 1}, Divide[x - 1,x + 1]]
Failure Failure Successful [Tested: 81] Successful [Tested: 81]
18.5.E9 C n ( λ ) ⁡ ( x ) = ( 2 ⁢ λ ) n n ! ⁢ F 1 2 ⁡ ( - n , n + 2 ⁢ λ λ + 1 2 ; 1 - x 2 ) ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 Pochhammer 2 𝜆 𝑛 𝑛 Gauss-hypergeometric-F-as-2F1 𝑛 𝑛 2 𝜆 𝜆 1 2 1 𝑥 2 {\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(x\right)=\frac{{\left(2% \lambda\right)_{n}}}{n!}{{}_{2}F_{1}}\left({-n,n+2\lambda\atop\lambda+\tfrac{1% }{2}};\frac{1-x}{2}\right)}}
\ultrasphpoly{\lambda}{n}@{x} = \frac{\Pochhammersym{2\lambda}{n}}{n!}\genhyperF{2}{1}@@{-n,n+2\lambda}{\lambda+\tfrac{1}{2}}{\frac{1-x}{2}}		 

GegenbauerC(n, lambda, x) = (pochhammer(2*lambda, n))/(factorial(n))*hypergeom([- n , n + 2*lambda], [lambda +(1)/(2)], (1 - x)/(2))

GegenbauerC[n, \[Lambda], x] == Divide[Pochhammer[2*\[Lambda], n],(n)!]*HypergeometricPFQ[{- n , n + 2*\[Lambda]}, {\[Lambda]+Divide[1,2]}, Divide[1 - x,2]]
Successful Successful -
Failed [15 / 90]
Result: 0.375
Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[λ, -1.5]}


Result: 0.4375
Test Values: {Rule[n, 3], Rule[x, 1.5], Rule[λ, -1.5]}

... skip entries to safe data
18.5.E10 C n ( λ ) ⁡ ( x ) = ∑ ℓ = 0 ⌊ n / 2 ⌋ ( - 1 ) ℓ ⁢ ( λ ) n - ℓ ℓ ! ⁢ ( n - 2 ⁢ ℓ ) ! ⁢ ( 2 ⁢ x ) n - 2 ⁢ ℓ ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 superscript subscript ℓ 0 𝑛 2 superscript 1 ℓ Pochhammer 𝜆 𝑛 ℓ ℓ 𝑛 2 ℓ superscript 2 𝑥 𝑛 2 ℓ {\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(x\right)=\sum_{\ell=0}^{% \left\lfloor n/2\right\rfloor}\frac{(-1)^{\ell}{\left(\lambda\right)_{n-\ell}}% }{\ell!\;(n-2\ell)!}(2x)^{n-2\ell}}}
\ultrasphpoly{\lambda}{n}@{x} = \sum_{\ell=0}^{\floor{n/2}}\frac{(-1)^{\ell}\Pochhammersym{\lambda}{n-\ell}}{\ell!\;(n-2\ell)!}(2x)^{n-2\ell}		 

GegenbauerC(n, lambda, x) = sum(((- 1)^(ell)* pochhammer(lambda, n - ell))/(factorial(ell)*factorial(n - 2*ell))*(2*x)^(n - 2*ell), ell = 0..floor(n/2))

GegenbauerC[n, \[Lambda], x] == Sum[Divide[(- 1)^\[ScriptL]* Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!*(n - 2*\[ScriptL])!]*(2*x)^(n - 2*\[ScriptL]), {\[ScriptL], 0, Floor[n/2]}, GenerateConditions->None]
Failure Successful Manual Skip! Successful [Tested: 90]
18.5.E10 ∑ ℓ = 0 ⌊ n / 2 ⌋ ( - 1 ) ℓ ⁢ ( λ ) n - ℓ ℓ ! ⁢ ( n - 2 ⁢ ℓ ) ! ⁢ ( 2 ⁢ x ) n - 2 ⁢ ℓ = ( 2 ⁢ x ) n ⁢ ( λ ) n n ! ⁢ F 1 2 ⁡ ( - 1 2 ⁢ n , - 1 2 ⁢ n + 1 2 1 - λ - n ; 1 x 2 ) superscript subscript ℓ 0 𝑛 2 superscript 1 ℓ Pochhammer 𝜆 𝑛 ℓ ℓ 𝑛 2 ℓ superscript 2 𝑥 𝑛 2 ℓ superscript 2 𝑥 𝑛 Pochhammer 𝜆 𝑛 𝑛 Gauss-hypergeometric-F-as-2F1 1 2 𝑛 1 2 𝑛 1 2 1 𝜆 𝑛 1 superscript 𝑥 2 {\displaystyle{\displaystyle\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac% {(-1)^{\ell}{\left(\lambda\right)_{n-\ell}}}{\ell!\;(n-2\ell)!}(2x)^{n-2\ell}=% (2x)^{n}\frac{{\left(\lambda\right)_{n}}}{n!}{{}_{2}F_{1}}\left({-\tfrac{1}{2}% n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop 1-\lambda-n};\frac{1}{x^{2}}\right)}}
\sum_{\ell=0}^{\floor{n/2}}\frac{(-1)^{\ell}\Pochhammersym{\lambda}{n-\ell}}{\ell!\;(n-2\ell)!}(2x)^{n-2\ell} = (2x)^{n}\frac{\Pochhammersym{\lambda}{n}}{n!}\genhyperF{2}{1}@@{-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}}{1-\lambda-n}{\frac{1}{x^{2}}}		 

sum(((- 1)^(ell)* pochhammer(lambda, n - ell))/(factorial(ell)*factorial(n - 2*ell))*(2*x)^(n - 2*ell), ell = 0..floor(n/2)) = (2*x)^(n)*(pochhammer(lambda, n))/(factorial(n))*hypergeom([-(1)/(2)*n , -(1)/(2)*n +(1)/(2)], [1 - lambda - n], (1)/((x)^(2)))

Sum[Divide[(- 1)^\[ScriptL]* Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!*(n - 2*\[ScriptL])!]*(2*x)^(n - 2*\[ScriptL]), {\[ScriptL], 0, Floor[n/2]}, GenerateConditions->None] == (2*x)^(n)*Divide[Pochhammer[\[Lambda], n],(n)!]*HypergeometricPFQ[{-Divide[1,2]*n , -Divide[1,2]*n +Divide[1,2]}, {1 - \[Lambda]- n}, Divide[1,(x)^(2)]]
Failure Failure Manual Skip!
Failed [3 / 90]
Result: Indeterminate
Test Values: {Rule[n, 3], Rule[x, 1.5], Rule[λ, -2]}


Result: Indeterminate
Test Values: {Rule[n, 3], Rule[x, 0.5], Rule[λ, -2]}

... skip entries to safe data
18.5.E11 C n ( λ ) ⁡ ( cos ⁡ θ ) = ∑ ℓ = 0 n ( λ ) ℓ ⁢ ( λ ) n - ℓ ℓ ! ⁢ ( n - ℓ ) ! ⁢ cos ⁡ ( ( n - 2 ⁢ ℓ ) ⁢ θ ) ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝜃 superscript subscript ℓ 0 𝑛 Pochhammer 𝜆 ℓ Pochhammer 𝜆 𝑛 ℓ ℓ 𝑛 ℓ 𝑛 2 ℓ 𝜃 {\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(\cos\theta\right)=\sum_{% \ell=0}^{n}\frac{{\left(\lambda\right)_{\ell}}{\left(\lambda\right)_{n-\ell}}}% {\ell!\;(n-\ell)!}\cos\left((n-2\ell)\theta\right)}}
\ultrasphpoly{\lambda}{n}@{\cos@@{\theta}} = \sum_{\ell=0}^{n}\frac{\Pochhammersym{\lambda}{\ell}\Pochhammersym{\lambda}{n-\ell}}{\ell!\;(n-\ell)!}\cos@{(n-2\ell)\theta}		 

GegenbauerC(n, lambda, cos(theta)) = sum((pochhammer(lambda, ell)*pochhammer(lambda, n - ell))/(factorial(ell)*factorial(n - ell))*cos((n - 2*ell)*theta), ell = 0..n)

GegenbauerC[n, \[Lambda], Cos[\[Theta]]] == Sum[Divide[Pochhammer[\[Lambda], \[ScriptL]]*Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!*(n - \[ScriptL])!]*Cos[(n - 2*\[ScriptL])*\[Theta]], {\[ScriptL], 0, n}, GenerateConditions->None]
Failure Failure Error
Failed [30 / 300]
Result: Indeterminate
Test Values: {Rule[n, 1], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, -2]}


Result: Indeterminate
Test Values: {Rule[n, 2], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, -2]}

... skip entries to safe data
18.5.E11 ∑ ℓ = 0 n ( λ ) ℓ ⁢ ( λ ) n - ℓ ℓ ! ⁢ ( n - ℓ ) ! ⁢ cos ⁡ ( ( n - 2 ⁢ ℓ ) ⁢ θ ) = e i ⁢ n ⁢ θ ⁢ ( λ ) n n ! ⁢ F 1 2 ⁡ ( - n , λ 1 - λ - n ; e - 2 ⁢ i ⁢ θ ) superscript subscript ℓ 0 𝑛 Pochhammer 𝜆 ℓ Pochhammer 𝜆 𝑛 ℓ ℓ 𝑛 ℓ 𝑛 2 ℓ 𝜃 superscript 𝑒 imaginary-unit 𝑛 𝜃 Pochhammer 𝜆 𝑛 𝑛 Gauss-hypergeometric-F-as-2F1 𝑛 𝜆 1 𝜆 𝑛 superscript 𝑒 2 imaginary-unit 𝜃 {\displaystyle{\displaystyle\sum_{\ell=0}^{n}\frac{{\left(\lambda\right)_{\ell% }}{\left(\lambda\right)_{n-\ell}}}{\ell!\;(n-\ell)!}\cos\left((n-2\ell)\theta% \right)=e^{\mathrm{i}n\theta}\frac{{\left(\lambda\right)_{n}}}{n!}{{}_{2}F_{1}% }\left({-n,\lambda\atop 1-\lambda-n};e^{-2\mathrm{i}\theta}\right)}}
\sum_{\ell=0}^{n}\frac{\Pochhammersym{\lambda}{\ell}\Pochhammersym{\lambda}{n-\ell}}{\ell!\;(n-\ell)!}\cos@{(n-2\ell)\theta} = e^{\iunit n\theta}\frac{\Pochhammersym{\lambda}{n}}{n!}\genhyperF{2}{1}@@{-n,\lambda}{1-\lambda-n}{e^{-2\iunit\theta}}		 

sum((pochhammer(lambda, ell)*pochhammer(lambda, n - ell))/(factorial(ell)*factorial(n - ell))*cos((n - 2*ell)*theta), ell = 0..n) = exp(I*n*theta)*(pochhammer(lambda, n))/(factorial(n))*hypergeom([- n , lambda], [1 - lambda - n], exp(- 2*I*theta))

Sum[Divide[Pochhammer[\[Lambda], \[ScriptL]]*Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!*(n - \[ScriptL])!]*Cos[(n - 2*\[ScriptL])*\[Theta]], {\[ScriptL], 0, n}, GenerateConditions->None] == Exp[I*n*\[Theta]]*Divide[Pochhammer[\[Lambda], n],(n)!]*HypergeometricPFQ[{- n , \[Lambda]}, {1 - \[Lambda]- n}, Exp[- 2*I*\[Theta]]]
Failure Failure Error
Failed [30 / 300]
Result: Indeterminate
Test Values: {Rule[n, 1], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, -2]}


Result: Indeterminate
Test Values: {Rule[n, 2], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, -2]}

... skip entries to safe data
18.5.E12 L n ( α ) ⁡ ( x ) = ∑ ℓ = 0 n ( α + ℓ + 1 ) n - ℓ ( n - ℓ ) ! ⁢ ℓ ! ⁢ ( - x ) ℓ Laguerre-polynomial-L 𝛼 𝑛 𝑥 superscript subscript ℓ 0 𝑛 Pochhammer 𝛼 ℓ 1 𝑛 ℓ 𝑛 ℓ ℓ superscript 𝑥 ℓ {\displaystyle{\displaystyle L^{(\alpha)}_{n}\left(x\right)=\sum_{\ell=0}^{n}% \frac{{\left(\alpha+\ell+1\right)_{n-\ell}}}{(n-\ell)!\;\ell!}(-x)^{\ell}}}
\LaguerrepolyL[\alpha]{n}@{x} = \sum_{\ell=0}^{n}\frac{\Pochhammersym{\alpha+\ell+1}{n-\ell}}{(n-\ell)!\;\ell!}(-x)^{\ell}		 

LaguerreL(n, alpha, x) = sum((pochhammer(alpha + ell + 1, n - ell))/(factorial(n - ell)*factorial(ell))*(- x)^(ell), ell = 0..n)

LaguerreL[n, \[Alpha], x] == Sum[Divide[Pochhammer[\[Alpha]+ \[ScriptL]+ 1, n - \[ScriptL]],(n - \[ScriptL])!*(\[ScriptL])!]*(- x)^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None]
Missing Macro Error Successful - Successful [Tested: 27]
18.5.E12 ∑ ℓ = 0 n ( α + ℓ + 1 ) n - ℓ ( n - ℓ ) ! ⁢ ℓ ! ⁢ ( - x ) ℓ = ( α + 1 ) n n ! ⁢ F 1 1 ⁡ ( - n α + 1 ; x ) superscript subscript ℓ 0 𝑛 Pochhammer 𝛼 ℓ 1 𝑛 ℓ 𝑛 ℓ ℓ superscript 𝑥 ℓ Pochhammer 𝛼 1 𝑛 𝑛 Kummer-confluent-hypergeometric-M-as-1F1 𝑛 𝛼 1 𝑥 {\displaystyle{\displaystyle\sum_{\ell=0}^{n}\frac{{\left(\alpha+\ell+1\right)% _{n-\ell}}}{(n-\ell)!\;\ell!}(-x)^{\ell}=\frac{{\left(\alpha+1\right)_{n}}}{n!% }{{}_{1}F_{1}}\left({-n\atop\alpha+1};x\right)}}
\sum_{\ell=0}^{n}\frac{\Pochhammersym{\alpha+\ell+1}{n-\ell}}{(n-\ell)!\;\ell!}(-x)^{\ell} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}\genhyperF{1}{1}@@{-n}{\alpha+1}{x}		 

sum((pochhammer(alpha + ell + 1, n - ell))/(factorial(n - ell)*factorial(ell))*(- x)^(ell), ell = 0..n) = (pochhammer(alpha + 1, n))/(factorial(n))*hypergeom([- n], [alpha + 1], x)

Sum[Divide[Pochhammer[\[Alpha]+ \[ScriptL]+ 1, n - \[ScriptL]],(n - \[ScriptL])!*(\[ScriptL])!]*(- x)^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None] == Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*HypergeometricPFQ[{- n}, {\[Alpha]+ 1}, x]
Successful Successful - Successful [Tested: 27]
18.5.E13 H n ⁡ ( x ) = n ! ⁢ ∑ ℓ = 0 ⌊ n / 2 ⌋ ( - 1 ) ℓ ⁢ ( 2 ⁢ x ) n - 2 ⁢ ℓ ℓ ! ⁢ ( n - 2 ⁢ ℓ ) ! Hermite-polynomial-H 𝑛 𝑥 𝑛 superscript subscript ℓ 0 𝑛 2 superscript 1 ℓ superscript 2 𝑥 𝑛 2 ℓ ℓ 𝑛 2 ℓ {\displaystyle{\displaystyle H_{n}\left(x\right)=n!\sum_{\ell=0}^{\left\lfloor n% /2\right\rfloor}\frac{(-1)^{\ell}(2x)^{n-2\ell}}{\ell!\;(n-2\ell)!}}}
\HermitepolyH{n}@{x} = n!\sum_{\ell=0}^{\floor{n/2}}\frac{(-1)^{\ell}(2x)^{n-2\ell}}{\ell!\;(n-2\ell)!}		 

HermiteH(n, x) = factorial(n)*sum(((- 1)^(ell)*(2*x)^(n - 2*ell))/(factorial(ell)*factorial(n - 2*ell)), ell = 0..floor(n/2))

HermiteH[n, x] == (n)!*Sum[Divide[(- 1)^\[ScriptL]*(2*x)^(n - 2*\[ScriptL]),(\[ScriptL])!*(n - 2*\[ScriptL])!], {\[ScriptL], 0, Floor[n/2]}, GenerateConditions->None]
Failure Failure Successful [Tested: 9] Successful [Tested: 9]
18.5.E13 n ! ⁢ ∑ ℓ = 0 ⌊ n / 2 ⌋ ( - 1 ) ℓ ⁢ ( 2 ⁢ x ) n - 2 ⁢ ℓ ℓ ! ⁢ ( n - 2 ⁢ ℓ ) ! = ( 2 ⁢ x ) n ⁢ F 0 2 ⁡ ( - 1 2 ⁢ n , - 1 2 ⁢ n + 1 2 - ; - 1 x 2 ) 𝑛 superscript subscript ℓ 0 𝑛 2 superscript 1 ℓ superscript 2 𝑥 𝑛 2 ℓ ℓ 𝑛 2 ℓ superscript 2 𝑥 𝑛 Gauss-hypergeometric-pFq 2 0 1 2 𝑛 1 2 𝑛 1 2 1 superscript 𝑥 2 {\displaystyle{\displaystyle n!\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}% \frac{(-1)^{\ell}(2x)^{n-2\ell}}{\ell!\;(n-2\ell)!}=(2x)^{n}{{}_{2}F_{0}}\left% ({-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop-};-\frac{1}{x^{2}}\right)}}
n!\sum_{\ell=0}^{\floor{n/2}}\frac{(-1)^{\ell}(2x)^{n-2\ell}}{\ell!\;(n-2\ell)!} = (2x)^{n}\genhyperF{2}{0}@@{-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}}{-}{-\frac{1}{x^{2}}}		 

factorial(n)*sum(((- 1)^(ell)*(2*x)^(n - 2*ell))/(factorial(ell)*factorial(n - 2*ell)), ell = 0..floor(n/2)) = (2*x)^(n)* hypergeom([-(1)/(2)*n , -(1)/(2)*n +(1)/(2)], [-], -(1)/((x)^(2)))

(n)!*Sum[Divide[(- 1)^\[ScriptL]*(2*x)^(n - 2*\[ScriptL]),(\[ScriptL])!*(n - 2*\[ScriptL])!], {\[ScriptL], 0, Floor[n/2]}, GenerateConditions->None] == (2*x)^(n)* HypergeometricPFQ[{-Divide[1,2]*n , -Divide[1,2]*n +Divide[1,2]}, {-}, -Divide[1,(x)^(2)]]
Error Failure Skip - symbolical successful subtest Error
18.5#Ex1 T 0 ⁡ ( x ) = 1 Chebyshev-polynomial-first-kind-T 0 𝑥 1 {\displaystyle{\displaystyle T_{0}\left(x\right)=1}}
\ChebyshevpolyT{0}@{x} = 1		 

ChebyshevT(0, x) = 1

ChebyshevT[0, x] == 1
Successful Successful - Successful [Tested: 3]
18.5#Ex2 T 1 ⁡ ( x ) = x Chebyshev-polynomial-first-kind-T 1 𝑥 𝑥 {\displaystyle{\displaystyle T_{1}\left(x\right)=x}}
\ChebyshevpolyT{1}@{x} = x		 

ChebyshevT(1, x) = x

ChebyshevT[1, x] == x
Successful Successful - Successful [Tested: 3]
18.5#Ex3 T 2 ⁡ ( x ) = 2 ⁢ x 2 - 1 Chebyshev-polynomial-first-kind-T 2 𝑥 2 superscript 𝑥 2 1 {\displaystyle{\displaystyle T_{2}\left(x\right)=2x^{2}-1}}
\ChebyshevpolyT{2}@{x} = 2x^{2}-1		 

ChebyshevT(2, x) = 2*(x)^(2)- 1

ChebyshevT[2, x] == 2*(x)^(2)- 1
Successful Successful - Successful [Tested: 3]
18.5#Ex4 T 3 ⁡ ( x ) = 4 ⁢ x 3 - 3 ⁢ x Chebyshev-polynomial-first-kind-T 3 𝑥 4 superscript 𝑥 3 3 𝑥 {\displaystyle{\displaystyle T_{3}\left(x\right)=4x^{3}-3x}}
\ChebyshevpolyT{3}@{x} = 4x^{3}-3x		 

ChebyshevT(3, x) = 4*(x)^(3)- 3*x

ChebyshevT[3, x] == 4*(x)^(3)- 3*x
Successful Successful - Successful [Tested: 3]
18.5#Ex5 T 4 ⁡ ( x ) = 8 ⁢ x 4 - 8 ⁢ x 2 + 1 Chebyshev-polynomial-first-kind-T 4 𝑥 8 superscript 𝑥 4 8 superscript 𝑥 2 1 {\displaystyle{\displaystyle T_{4}\left(x\right)=8x^{4}-8x^{2}+1}}
\ChebyshevpolyT{4}@{x} = 8x^{4}-8x^{2}+1		 

ChebyshevT(4, x) = 8*(x)^(4)- 8*(x)^(2)+ 1

ChebyshevT[4, x] == 8*(x)^(4)- 8*(x)^(2)+ 1
Successful Successful - Successful [Tested: 3]
18.5#Ex6 T 5 ⁡ ( x ) = 16 ⁢ x 5 - 20 ⁢ x 3 + 5 ⁢ x Chebyshev-polynomial-first-kind-T 5 𝑥 16 superscript 𝑥 5 20 superscript 𝑥 3 5 𝑥 {\displaystyle{\displaystyle T_{5}\left(x\right)=16x^{5}-20x^{3}+5x}}
\ChebyshevpolyT{5}@{x} = 16x^{5}-20x^{3}+5x		 

ChebyshevT(5, x) = 16*(x)^(5)- 20*(x)^(3)+ 5*x

ChebyshevT[5, x] == 16*(x)^(5)- 20*(x)^(3)+ 5*x
Successful Successful - Successful [Tested: 3]
18.5#Ex7 T 6 ⁡ ( x ) = 32 ⁢ x 6 - 48 ⁢ x 4 + 18 ⁢ x 2 - 1 Chebyshev-polynomial-first-kind-T 6 𝑥 32 superscript 𝑥 6 48 superscript 𝑥 4 18 superscript 𝑥 2 1 {\displaystyle{\displaystyle T_{6}\left(x\right)=32x^{6}-48x^{4}+18x^{2}-1}}
\ChebyshevpolyT{6}@{x} = 32x^{6}-48x^{4}+18x^{2}-1		 

ChebyshevT(6, x) = 32*(x)^(6)- 48*(x)^(4)+ 18*(x)^(2)- 1

ChebyshevT[6, x] == 32*(x)^(6)- 48*(x)^(4)+ 18*(x)^(2)- 1
Successful Successful - Successful [Tested: 3]
18.5#Ex8 U 0 ⁡ ( x ) = 1 Chebyshev-polynomial-second-kind-U 0 𝑥 1 {\displaystyle{\displaystyle U_{0}\left(x\right)=1}}
\ChebyshevpolyU{0}@{x} = 1		 

ChebyshevU(0, x) = 1

ChebyshevU[0, x] == 1
Successful Successful - Successful [Tested: 3]
18.5#Ex9 U 1 ⁡ ( x ) = 2 ⁢ x Chebyshev-polynomial-second-kind-U 1 𝑥 2 𝑥 {\displaystyle{\displaystyle U_{1}\left(x\right)=2x}}
\ChebyshevpolyU{1}@{x} = 2x		 

ChebyshevU(1, x) = 2*x

ChebyshevU[1, x] == 2*x
Successful Successful - Successful [Tested: 3]
18.5#Ex10 U 2 ⁡ ( x ) = 4 ⁢ x 2 - 1 Chebyshev-polynomial-second-kind-U 2 𝑥 4 superscript 𝑥 2 1 {\displaystyle{\displaystyle U_{2}\left(x\right)=4x^{2}-1}}
\ChebyshevpolyU{2}@{x} = 4x^{2}-1		 

ChebyshevU(2, x) = 4*(x)^(2)- 1

ChebyshevU[2, x] == 4*(x)^(2)- 1
Successful Successful - Successful [Tested: 3]
18.5#Ex11 U 3 ⁡ ( x ) = 8 ⁢ x 3 - 4 ⁢ x Chebyshev-polynomial-second-kind-U 3 𝑥 8 superscript 𝑥 3 4 𝑥 {\displaystyle{\displaystyle U_{3}\left(x\right)=8x^{3}-4x}}
\ChebyshevpolyU{3}@{x} = 8x^{3}-4x		 

ChebyshevU(3, x) = 8*(x)^(3)- 4*x

ChebyshevU[3, x] == 8*(x)^(3)- 4*x
Successful Successful - Successful [Tested: 3]
18.5#Ex12 U 4 ⁡ ( x ) = 16 ⁢ x 4 - 12 ⁢ x 2 + 1 Chebyshev-polynomial-second-kind-U 4 𝑥 16 superscript 𝑥 4 12 superscript 𝑥 2 1 {\displaystyle{\displaystyle U_{4}\left(x\right)=16x^{4}-12x^{2}+1}}
\ChebyshevpolyU{4}@{x} = 16x^{4}-12x^{2}+1		 

ChebyshevU(4, x) = 16*(x)^(4)- 12*(x)^(2)+ 1
 
ChebyshevU[4, x] == 16*(x)^(4)- 12*(x)^(2)+ 1
 Successful Successful - Successful [Tested: 3]
18.5#Ex13 U 5 ⁡ ( x ) = 32 ⁢ x 5 - 32 ⁢ x 3 + 6 ⁢ x Chebyshev-polynomial-second-kind-U 5 𝑥 32 superscript 𝑥 5 32 superscript 𝑥 3 6 𝑥 {\displaystyle{\displaystyle U_{5}\left(x\right)=32x^{5}-32x^{3}+6x}}
\ChebyshevpolyU{5}@{x} = 32x^{5}-32x^{3}+6x		 

ChebyshevU(5, x) = 32*(x)^(5)- 32*(x)^(3)+ 6*x
 
ChebyshevU[5, x] == 32*(x)^(5)- 32*(x)^(3)+ 6*x
 Successful Successful - Successful [Tested: 3]
18.5#Ex14 U 6 ⁡ ( x ) = 64 ⁢ x 6 - 80 ⁢ x 4 + 24 ⁢ x 2 - 1 Chebyshev-polynomial-second-kind-U 6 𝑥 64 superscript 𝑥 6 80 superscript 𝑥 4 24 superscript 𝑥 2 1 {\displaystyle{\displaystyle U_{6}\left(x\right)=64x^{6}-80x^{4}+24x^{2}-1}}
\ChebyshevpolyU{6}@{x} = 64x^{6}-80x^{4}+24x^{2}-1		 

ChebyshevU(6, x) = 64*(x)^(6)- 80*(x)^(4)+ 24*(x)^(2)- 1
 
ChebyshevU[6, x] == 64*(x)^(6)- 80*(x)^(4)+ 24*(x)^(2)- 1
 Successful Successful - Successful [Tested: 3]
18.5#Ex15 P 0 ⁡ ( x ) = 1 Legendre-spherical-polynomial 0 𝑥 1 {\displaystyle{\displaystyle P_{0}\left(x\right)=1}}
\LegendrepolyP{0}@{x} = 1		 

LegendreP(0, x) = 1
 
LegendreP[0, x] == 1
 Successful Successful - Successful [Tested: 3]
18.5#Ex16 P 1 ⁡ ( x ) = x Legendre-spherical-polynomial 1 𝑥 𝑥 {\displaystyle{\displaystyle P_{1}\left(x\right)=x}}
\LegendrepolyP{1}@{x} = x		 

LegendreP(1, x) = x
 
LegendreP[1, x] == x
 Successful Successful - Successful [Tested: 3]
18.5#Ex17 P 2 ⁡ ( x ) = 3 2 ⁢ x 2 - 1 2 Legendre-spherical-polynomial 2 𝑥 3 2 superscript 𝑥 2 1 2 {\displaystyle{\displaystyle P_{2}\left(x\right)=\tfrac{3}{2}x^{2}-\tfrac{1}{2% }}}
\LegendrepolyP{2}@{x} = \tfrac{3}{2}x^{2}-\tfrac{1}{2}		 

LegendreP(2, x) = (3)/(2)*(x)^(2)-(1)/(2)
 
LegendreP[2, x] == Divide[3,2]*(x)^(2)-Divide[1,2]
 Successful Successful - Successful [Tested: 3]
18.5#Ex18 P 3 ⁡ ( x ) = 5 2 ⁢ x 3 - 3 2 ⁢ x Legendre-spherical-polynomial 3 𝑥 5 2 superscript 𝑥 3 3 2 𝑥 {\displaystyle{\displaystyle P_{3}\left(x\right)=\tfrac{5}{2}x^{3}-\tfrac{3}{2% }x}}
\LegendrepolyP{3}@{x} = \tfrac{5}{2}x^{3}-\tfrac{3}{2}x		 

LegendreP(3, x) = (5)/(2)*(x)^(3)-(3)/(2)*x
 
LegendreP[3, x] == Divide[5,2]*(x)^(3)-Divide[3,2]*x
 Successful Successful - Successful [Tested: 3]
18.5#Ex19 P 4 ⁡ ( x ) = 35 8 ⁢ x 4 - 15 4 ⁢ x 2 + 3 8 Legendre-spherical-polynomial 4 𝑥 35 8 superscript 𝑥 4 15 4 superscript 𝑥 2 3 8 {\displaystyle{\displaystyle P_{4}\left(x\right)=\tfrac{35}{8}x^{4}-\tfrac{15}% {4}x^{2}+\tfrac{3}{8}}}
\LegendrepolyP{4}@{x} = \tfrac{35}{8}x^{4}-\tfrac{15}{4}x^{2}+\tfrac{3}{8}		 

LegendreP(4, x) = (35)/(8)*(x)^(4)-(15)/(4)*(x)^(2)+(3)/(8)
 
LegendreP[4, x] == Divide[35,8]*(x)^(4)-Divide[15,4]*(x)^(2)+Divide[3,8]
 Successful Successful - Successful [Tested: 3]
18.5#Ex20 P 5 ⁡ ( x ) = 63 8 ⁢ x 5 - 35 4 ⁢ x 3 + 15 8 ⁢ x Legendre-spherical-polynomial 5 𝑥 63 8 superscript 𝑥 5 35 4 superscript 𝑥 3 15 8 𝑥 {\displaystyle{\displaystyle P_{5}\left(x\right)=\tfrac{63}{8}x^{5}-\tfrac{35}% {4}x^{3}+\tfrac{15}{8}x}}
\LegendrepolyP{5}@{x} = \tfrac{63}{8}x^{5}-\tfrac{35}{4}x^{3}+\tfrac{15}{8}x		 

LegendreP(5, x) = (63)/(8)*(x)^(5)-(35)/(4)*(x)^(3)+(15)/(8)*x
 
LegendreP[5, x] == Divide[63,8]*(x)^(5)-Divide[35,4]*(x)^(3)+Divide[15,8]*x
 Successful Successful - Successful [Tested: 3]
18.5#Ex21 P 6 ⁡ ( x ) = 231 16 ⁢ x 6 - 315 16 ⁢ x 4 + 105 16 ⁢ x 2 - 5 16 Legendre-spherical-polynomial 6 𝑥 231 16 superscript 𝑥 6 315 16 superscript 𝑥 4 105 16 superscript 𝑥 2 5 16 {\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}}}
\LegendrepolyP{6}@{x} = \tfrac{231}{16}x^{6}-\tfrac{315}{16}x^{4}+\tfrac{105}{16}x^{2}-\tfrac{5}{16}		 

LegendreP(6, x) = (231)/(16)*(x)^(6)-(315)/(16)*(x)^(4)+(105)/(16)*(x)^(2)-(5)/(16)
 
LegendreP[6, x] == Divide[231,16]*(x)^(6)-Divide[315,16]*(x)^(4)+Divide[105,16]*(x)^(2)-Divide[5,16]
 Successful Successful - Successful [Tested: 3]
18.5#Ex22 L 0 ⁡ ( x ) = 1 shorthand-Laguerre-polynomial-L 0 𝑥 1 {\displaystyle{\displaystyle L_{0}\left(x\right)=1}}
\LaguerrepolyL[]{0}@{x} = 1		 

LaguerreL(0, x) = 1
 
LaguerreL[0, x] == 1
 Successful Successful - Successful [Tested: 3]
18.5#Ex23 L 1 ⁡ ( x ) = - x + 1 shorthand-Laguerre-polynomial-L 1 𝑥 𝑥 1 {\displaystyle{\displaystyle L_{1}\left(x\right)=-x+1}}
\LaguerrepolyL[]{1}@{x} = -x+1		 

LaguerreL(1, x) = - x + 1
 
LaguerreL[1, x] == - x + 1
 Successful Successful - Successful [Tested: 3]
18.5#Ex24 L 2 ⁡ ( x ) = 1 2 ⁢ x 2 - 2 ⁢ x + 1 shorthand-Laguerre-polynomial-L 2 𝑥 1 2 superscript 𝑥 2 2 𝑥 1 {\displaystyle{\displaystyle L_{2}\left(x\right)=\tfrac{1}{2}x^{2}-2x+1}}
\LaguerrepolyL[]{2}@{x} = \tfrac{1}{2}x^{2}-2x+1		 

LaguerreL(2, x) = (1)/(2)*(x)^(2)- 2*x + 1
 
LaguerreL[2, x] == Divide[1,2]*(x)^(2)- 2*x + 1
 Successful Successful - Successful [Tested: 3]
18.5#Ex25 L 3 ⁡ ( x ) = - 1 6 ⁢ x 3 + 3 2 ⁢ x 2 - 3 ⁢ x + 1 shorthand-Laguerre-polynomial-L 3 𝑥 1 6 superscript 𝑥 3 3 2 superscript 𝑥 2 3 𝑥 1 {\displaystyle{\displaystyle L_{3}\left(x\right)=-\tfrac{1}{6}x^{3}+\tfrac{3}{% 2}x^{2}-3x+1}}
\LaguerrepolyL[]{3}@{x} = -\tfrac{1}{6}x^{3}+\tfrac{3}{2}x^{2}-3x+1		 

LaguerreL(3, x) = -(1)/(6)*(x)^(3)+(3)/(2)*(x)^(2)- 3*x + 1
 
LaguerreL[3, x] == -Divide[1,6]*(x)^(3)+Divide[3,2]*(x)^(2)- 3*x + 1
 Successful Successful - Successful [Tested: 3]
18.5#Ex26 L 4 ⁡ ( x ) = 1 24 ⁢ x 4 - 2 3 ⁢ x 3 + 3 ⁢ x 2 - 4 ⁢ x + 1 shorthand-Laguerre-polynomial-L 4 𝑥 1 24 superscript 𝑥 4 2 3 superscript 𝑥 3 3 superscript 𝑥 2 4 𝑥 1 {\displaystyle{\displaystyle L_{4}\left(x\right)=\tfrac{1}{24}x^{4}-\tfrac{2}{% 3}x^{3}+3x^{2}-4x+1}}
\LaguerrepolyL[]{4}@{x} = \tfrac{1}{24}x^{4}-\tfrac{2}{3}x^{3}+3x^{2}-4x+1		 

LaguerreL(4, x) = (1)/(24)*(x)^(4)-(2)/(3)*(x)^(3)+ 3*(x)^(2)- 4*x + 1
 
LaguerreL[4, x] == Divide[1,24]*(x)^(4)-Divide[2,3]*(x)^(3)+ 3*(x)^(2)- 4*x + 1
 Successful Successful - Successful [Tested: 3]
18.5#Ex27 L 5 ⁡ ( x ) = - 1 120 ⁢ x 5 + 5 24 ⁢ x 4 - 5 3 ⁢ x 3 + 5 ⁢ x 2 - 5 ⁢ x + 1 shorthand-Laguerre-polynomial-L 5 𝑥 1 120 superscript 𝑥 5 5 24 superscript 𝑥 4 5 3 superscript 𝑥 3 5 superscript 𝑥 2 5 𝑥 1 {\displaystyle{\displaystyle L_{5}\left(x\right)=-\tfrac{1}{120}x^{5}+\tfrac{5% }{24}x^{4}-\tfrac{5}{3}x^{3}+5x^{2}-5x+1}}
\LaguerrepolyL[]{5}@{x} = -\tfrac{1}{120}x^{5}+\tfrac{5}{24}x^{4}-\tfrac{5}{3}x^{3}+5x^{2}-5x+1		 

LaguerreL(5, x) = -(1)/(120)*(x)^(5)+(5)/(24)*(x)^(4)-(5)/(3)*(x)^(3)+ 5*(x)^(2)- 5*x + 1
 
LaguerreL[5, x] == -Divide[1,120]*(x)^(5)+Divide[5,24]*(x)^(4)-Divide[5,3]*(x)^(3)+ 5*(x)^(2)- 5*x + 1
 Successful Successful - Successful [Tested: 3]
18.5#Ex28 L 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 shorthand-Laguerre-polynomial-L 6 𝑥 1 720 superscript 𝑥 6 1 20 superscript 𝑥 5 5 8 superscript 𝑥 4 10 3 superscript 𝑥 3 15 2 superscript 𝑥 2 6 𝑥 1 {\displaystyle{\displaystyle L_{6}\left(x\right)=\tfrac{1}{720}x^{6}-\tfrac{1}% {20}x^{5}+\tfrac{5}{8}x^{4}-\tfrac{10}{3}x^{3}+\tfrac{15}{2}x^{2}-6x+1}}
\LaguerrepolyL[]{6}@{x} = \tfrac{1}{720}x^{6}-\tfrac{1}{20}x^{5}+\tfrac{5}{8}x^{4}-\tfrac{10}{3}x^{3}+\tfrac{15}{2}x^{2}-6x+1		 

LaguerreL(6, x) = (1)/(720)*(x)^(6)-(1)/(20)*(x)^(5)+(5)/(8)*(x)^(4)-(10)/(3)*(x)^(3)+(15)/(2)*(x)^(2)- 6*x + 1
 
LaguerreL[6, x] == Divide[1,720]*(x)^(6)-Divide[1,20]*(x)^(5)+Divide[5,8]*(x)^(4)-Divide[10,3]*(x)^(3)+Divide[15,2]*(x)^(2)- 6*x + 1
 Successful Successful - Successful [Tested: 3]
18.5#Ex29 H 0 ⁡ ( x ) = 1 Hermite-polynomial-H 0 𝑥 1 {\displaystyle{\displaystyle H_{0}\left(x\right)=1}}
\HermitepolyH{0}@{x} = 1		 

HermiteH(0, x) = 1
 
HermiteH[0, x] == 1
 Successful Successful - Successful [Tested: 3]
18.5#Ex30 H 1 ⁡ ( x ) = 2 ⁢ x Hermite-polynomial-H 1 𝑥 2 𝑥 {\displaystyle{\displaystyle H_{1}\left(x\right)=2x}}
\HermitepolyH{1}@{x} = 2x		 

HermiteH(1, x) = 2*x
 
HermiteH[1, x] == 2*x
 Successful Successful - Successful [Tested: 3]
18.5#Ex31 H 2 ⁡ ( x ) = 4 ⁢ x 2 - 2 Hermite-polynomial-H 2 𝑥 4 superscript 𝑥 2 2 {\displaystyle{\displaystyle H_{2}\left(x\right)=4x^{2}-2}}
\HermitepolyH{2}@{x} = 4x^{2}-2		 

HermiteH(2, x) = 4*(x)^(2)- 2
 
HermiteH[2, x] == 4*(x)^(2)- 2
 Successful Successful - Successful [Tested: 3]
18.5#Ex32 H 3 ⁡ ( x ) = 8 ⁢ x 3 - 12 ⁢ x Hermite-polynomial-H 3 𝑥 8 superscript 𝑥 3 12 𝑥 {\displaystyle{\displaystyle H_{3}\left(x\right)=8x^{3}-12x}}
\HermitepolyH{3}@{x} = 8x^{3}-12x		 

HermiteH(3, x) = 8*(x)^(3)- 12*x
 
HermiteH[3, x] == 8*(x)^(3)- 12*x
 Successful Successful - Successful [Tested: 3]
18.5#Ex33 H 4 ⁡ ( x ) = 16 ⁢ x 4 - 48 ⁢ x 2 + 12 Hermite-polynomial-H 4 𝑥 16 superscript 𝑥 4 48 superscript 𝑥 2 12 {\displaystyle{\displaystyle H_{4}\left(x\right)=16x^{4}-48x^{2}+12}}
\HermitepolyH{4}@{x} = 16x^{4}-48x^{2}+12		 

HermiteH(4, x) = 16*(x)^(4)- 48*(x)^(2)+ 12
 
HermiteH[4, x] == 16*(x)^(4)- 48*(x)^(2)+ 12
 Successful Successful - Successful [Tested: 3]
18.5#Ex34 H 5 ⁡ ( x ) = 32 ⁢ x 5 - 160 ⁢ x 3 + 120 ⁢ x Hermite-polynomial-H 5 𝑥 32 superscript 𝑥 5 160 superscript 𝑥 3 120 𝑥 {\displaystyle{\displaystyle H_{5}\left(x\right)=32x^{5}-160x^{3}+120x}}
\HermitepolyH{5}@{x} = 32x^{5}-160x^{3}+120x		 

HermiteH(5, x) = 32*(x)^(5)- 160*(x)^(3)+ 120*x
 
HermiteH[5, x] == 32*(x)^(5)- 160*(x)^(3)+ 120*x
 Successful Successful - Successful [Tested: 3]
18.5#Ex35 H 6 ⁡ ( x ) = 64 ⁢ x 6 - 480 ⁢ x 4 + 720 ⁢ x 2 - 120 Hermite-polynomial-H 6 𝑥 64 superscript 𝑥 6 480 superscript 𝑥 4 720 superscript 𝑥 2 120 {\displaystyle{\displaystyle H_{6}\left(x\right)=64x^{6}-480x^{4}+720x^{2}-120}}
\HermitepolyH{6}@{x} = 64x^{6}-480x^{4}+720x^{2}-120		 

HermiteH(6, x) = 64*(x)^(6)- 480*(x)^(4)+ 720*(x)^(2)- 120
 
HermiteH[6, x] == 64*(x)^(6)- 480*(x)^(4)+ 720*(x)^(2)- 120
 Successful Successful - Successful [Tested: 3]
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