Orthogonal Polynomials - 18.18 Sums

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
18.18.E8 C n ( λ ) ⁡ ( cos ⁡ θ 1 ⁢ cos ⁡ θ 2 + sin ⁡ θ 1 ⁢ sin ⁡ θ 2 ⁢ cos ⁡ ϕ ) = ∑ ℓ = 0 n 2 2 ⁢ ℓ ⁢ ( n - ℓ ) ! ⁢ 2 ⁢ λ + 2 ⁢ ℓ - 1 2 ⁢ λ - 1 ⁢ ( ( λ ) ℓ ) 2 ( 2 ⁢ λ ) n + ℓ ⁢ ( sin ⁡ θ 1 ) ℓ ⁢ C n - ℓ ( λ + ℓ ) ⁡ ( cos ⁡ θ 1 ) ⁢ ( sin ⁡ θ 2 ) ℓ ⁢ C n - ℓ ( λ + ℓ ) ⁡ ( cos ⁡ θ 2 ) ⁢ C ℓ ( λ - 1 2 ) ⁡ ( cos ⁡ ϕ ) ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 subscript 𝜃 1 subscript 𝜃 2 subscript 𝜃 1 subscript 𝜃 2 italic-ϕ superscript subscript ℓ 0 𝑛 superscript 2 2 ℓ 𝑛 ℓ 2 𝜆 2 ℓ 1 2 𝜆 1 superscript Pochhammer 𝜆 ℓ 2 Pochhammer 2 𝜆 𝑛 ℓ superscript subscript 𝜃 1 ℓ ultraspherical-Gegenbauer-polynomial 𝜆 ℓ 𝑛 ℓ subscript 𝜃 1 superscript subscript 𝜃 2 ℓ ultraspherical-Gegenbauer-polynomial 𝜆 ℓ 𝑛 ℓ subscript 𝜃 2 ultraspherical-Gegenbauer-polynomial 𝜆 1 2 ℓ italic-ϕ {\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)}}
\ultrasphpoly{\lambda}{n}@{\cos@@{\theta_{1}}\cos@@{\theta_{2}}+\sin@@{\theta_{1}}\sin@@{\theta_{2}}\cos@@{\phi}} = \sum_{\ell=0}^{n}2^{2\ell}(n-\ell)!\frac{2\lambda+2\ell-1}{2\lambda-1}\frac{(\Pochhammersym{\lambda}{\ell})^{2}}{\Pochhammersym{2\lambda}{n+\ell}}(\sin@@{\theta_{1}})^{\ell}\ultrasphpoly{\lambda+\ell}{n-\ell}@{\cos@@{\theta_{1}}}(\sin@@{\theta_{2}})^{\ell}\ultrasphpoly{\lambda+\ell}{n-\ell}@{\cos@@{\theta_{2}}}\ultrasphpoly{\lambda-\frac{1}{2}}{\ell}@{\cos@@{\phi}}		 λ > 0 , λ ≠ 1 2 formulae-sequence 𝜆 0 𝜆 1 2 {\displaystyle{\displaystyle\lambda>0,\lambda\neq\frac{1}{2}}} 
GegenbauerC(n, lambda, cos(theta[1])*cos(theta[2])+ sin(theta[1])*sin(theta[2])*cos(phi)) = sum((2)^(2*ell)*factorial(n - ell)*(2*lambda + 2*ell - 1)/(2*lambda - 1)*((pochhammer(lambda, ell))^(2))/(pochhammer(2*lambda, 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}, GenerateConditions->None]
Failure Aborted Successful [Tested: 300] Skipped - Because timed out
18.18.E9 P n ⁡ ( cos ⁡ θ 1 ⁢ cos ⁡ θ 2 + sin ⁡ θ 1 ⁢ sin ⁡ θ 2 ⁢ cos ⁡ ϕ ) = P n ⁡ ( cos ⁡ θ 1 ) ⁢ P n ⁡ ( cos ⁡ θ 2 ) + 2 ⁢ ∑ ℓ = 1 n ( n - ℓ ) ! ⁢ ( n + ℓ ) ! 2 2 ⁢ ℓ ⁢ ( n ! ) 2 ⁢ ( sin ⁡ θ 1 ) ℓ ⁢ P n - ℓ ( ℓ , ℓ ) ⁡ ( cos ⁡ θ 1 ) ⁢ ( sin ⁡ θ 2 ) ℓ ⁢ P n - ℓ ( ℓ , ℓ ) ⁡ ( cos ⁡ θ 2 ) ⁢ cos ⁡ ( ℓ ⁢ ϕ ) Legendre-spherical-polynomial 𝑛 subscript 𝜃 1 subscript 𝜃 2 subscript 𝜃 1 subscript 𝜃 2 italic-ϕ Legendre-spherical-polynomial 𝑛 subscript 𝜃 1 Legendre-spherical-polynomial 𝑛 subscript 𝜃 2 2 superscript subscript ℓ 1 𝑛 𝑛 ℓ 𝑛 ℓ superscript 2 2 ℓ superscript 𝑛 2 superscript subscript 𝜃 1 ℓ Jacobi-polynomial-P ℓ ℓ 𝑛 ℓ subscript 𝜃 1 superscript subscript 𝜃 2 ℓ Jacobi-polynomial-P ℓ ℓ 𝑛 ℓ subscript 𝜃 2 ℓ italic-ϕ {\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)}}}
\LegendrepolyP{n}@{\cos@@{\theta_{1}}\cos@@{\theta_{2}}+\sin@@{\theta_{1}}\sin@@{\theta_{2}}\cos@@{\phi}} = {\LegendrepolyP{n}@{\cos@@{\theta_{1}}}\LegendrepolyP{n}@{\cos@@{\theta_{2}}}+2\sum_{\ell=1}^{n}\frac{(n-\ell)!\;(n+\ell)!}{2^{2\ell}(n!)^{2}}(\sin@@{\theta_{1}})^{\ell}\JacobipolyP{\ell}{\ell}{n-\ell}@{\cos@@{\theta_{1}}}(\sin@@{\theta_{2}})^{\ell}\JacobipolyP{\ell}{\ell}{n-\ell}@{\cos@@{\theta_{2}}}\cos@{\ell\phi}}		 

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]))+ 2*sum((factorial(n - ell)*factorial(n + ell))/((2)^(2*ell)*(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(ell*phi), 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]]]+ 2*Sum[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}, GenerateConditions->None]
Failure Aborted Successful [Tested: 300] Skipped - Because timed out
18.18.E12 L n ( α ) ⁡ ( λ ⁢ x ) L n ( α ) ⁡ ( 0 ) = ∑ ℓ = 0 n ( n ℓ ) ⁢ λ ℓ ⁢ ( 1 - λ ) n - ℓ ⁢ L ℓ ( α ) ⁡ ( x ) L ℓ ( α ) ⁡ ( 0 ) Laguerre-polynomial-L 𝛼 𝑛 𝜆 𝑥 Laguerre-polynomial-L 𝛼 𝑛 0 superscript subscript ℓ 0 𝑛 binomial 𝑛 ℓ superscript 𝜆 ℓ superscript 1 𝜆 𝑛 ℓ Laguerre-polynomial-L 𝛼 ℓ 𝑥 Laguerre-polynomial-L 𝛼 ℓ 0 {\displaystyle{\displaystyle\frac{L^{(\alpha)}_{n}\left(\lambda x\right)}{L^{(% \alpha)}_{n}\left(0\right)}=\sum_{\ell=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{\ell}% \lambda^{\ell}(1-\lambda)^{n-\ell}\frac{L^{(\alpha)}_{\ell}\left(x\right)}{L^{% (\alpha)}_{\ell}\left(0\right)}}}
\frac{\LaguerrepolyL[\alpha]{n}@{\lambda x}}{\LaguerrepolyL[\alpha]{n}@{0}} = \sum_{\ell=0}^{n}\binom{n}{\ell}\lambda^{\ell}(1-\lambda)^{n-\ell}\frac{\LaguerrepolyL[\alpha]{\ell}@{x}}{\LaguerrepolyL[\alpha]{\ell}@{0}}		 

(LaguerreL(n, alpha, lambda*x))/(LaguerreL(n, alpha, 0)) = sum(binomial(n,ell)*(lambda)^(ell)*(1 - lambda)^(n - ell)*(LaguerreL(ell, alpha, x))/(LaguerreL(ell, alpha, 0)), ell = 0..n)

Divide[LaguerreL[n, \[Alpha], \[Lambda]*x],LaguerreL[n, \[Alpha], 0]] == Sum[Binomial[n,\[ScriptL]]*\[Lambda]^\[ScriptL]*(1 - \[Lambda])^(n - \[ScriptL])*Divide[LaguerreL[\[ScriptL], \[Alpha], x],LaguerreL[\[ScriptL], \[Alpha], 0]], {\[ScriptL], 0, n}, GenerateConditions->None]
Missing Macro Error Failure - Skipped - Because timed out
18.18.E13 H n ⁡ ( λ ⁢ x ) = λ n ⁢ ∑ ℓ = 0 ⌊ n / 2 ⌋ ( - n ) 2 ⁢ ℓ ℓ ! ⁢ ( 1 - λ - 2 ) ℓ ⁢ H n - 2 ⁢ ℓ ⁡ ( x ) Hermite-polynomial-H 𝑛 𝜆 𝑥 superscript 𝜆 𝑛 superscript subscript ℓ 0 𝑛 2 Pochhammer 𝑛 2 ℓ ℓ superscript 1 superscript 𝜆 2 ℓ Hermite-polynomial-H 𝑛 2 ℓ 𝑥 {\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)}}
\HermitepolyH{n}@{\lambda x} = \lambda^{n}\sum_{\ell=0}^{\floor{n/2}}\frac{\Pochhammersym{-n}{2\ell}}{\ell!}(1-\lambda^{-2})^{\ell}\HermitepolyH{n-2\ell}@{x}		 

HermiteH(n, lambda*x) = (lambda)^(n)* sum((pochhammer(- n, 2*ell))/(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]}, GenerateConditions->None]
Failure Failure Successful [Tested: 90]
Failed [90 / 90]
Result: Complex[2.598076211353316, 1.4999999999999998]
Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[2.5, 7.794228634059947]
Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
18.18.E14 P n ( γ , β ) ⁡ ( x ) = ( β + 1 ) n ( α + β + 2 ) n ⁢ ∑ ℓ = 0 n α + β + 2 ⁢ ℓ + 1 α + β + 1 ⁢ ( α + β + 1 ) ℓ ⁢ ( n + β + γ + 1 ) ℓ ( β + 1 ) ℓ ⁢ ( n + α + β + 2 ) ℓ ⁢ ( γ - α ) n - ℓ ( n - ℓ ) ! ⁢ P ℓ ( α , β ) ⁡ ( x ) Jacobi-polynomial-P 𝛾 𝛽 𝑛 𝑥 Pochhammer 𝛽 1 𝑛 Pochhammer 𝛼 𝛽 2 𝑛 superscript subscript ℓ 0 𝑛 𝛼 𝛽 2 ℓ 1 𝛼 𝛽 1 Pochhammer 𝛼 𝛽 1 ℓ Pochhammer 𝑛 𝛽 𝛾 1 ℓ Pochhammer 𝛽 1 ℓ Pochhammer 𝑛 𝛼 𝛽 2 ℓ Pochhammer 𝛾 𝛼 𝑛 ℓ 𝑛 ℓ Jacobi-polynomial-P 𝛼 𝛽 ℓ 𝑥 {\displaystyle{\displaystyle P^{(\gamma,\beta)}_{n}\left(x\right)=\dfrac{{% \left(\beta+1\right)_{n}}}{{\left(\alpha+\beta+2\right)_{n}}}\sum_{\ell=0}^{n}% \dfrac{\alpha+\beta+2\ell+1}{\alpha+\beta+1}\dfrac{{\left(\alpha+\beta+1\right% )_{\ell}}{\left(n+\beta+\gamma+1\right)_{\ell}}}{{\left(\beta+1\right)_{\ell}}% {\left(n+\alpha+\beta+2\right)_{\ell}}}\dfrac{{\left(\gamma-\alpha\right)_{n-% \ell}}}{(n-\ell)!}P^{(\alpha,\beta)}_{\ell}\left(x\right)}}
\JacobipolyP{\gamma}{\beta}{n}@{x} = \dfrac{\Pochhammersym{\beta+1}{n}}{\Pochhammersym{\alpha+\beta+2}{n}}\sum_{\ell=0}^{n}\dfrac{\alpha+\beta+2\ell+1}{\alpha+\beta+1}\dfrac{\Pochhammersym{\alpha+\beta+1}{\ell}\Pochhammersym{n+\beta+\gamma+1}{\ell}}{\Pochhammersym{\beta+1}{\ell}\Pochhammersym{n+\alpha+\beta+2}{\ell}}\dfrac{\Pochhammersym{\gamma-\alpha}{n-\ell}}{(n-\ell)!}\JacobipolyP{\alpha}{\beta}{\ell}@{x}		 

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

JacobiP[n, \[Gamma], \[Beta], x] == Divide[Pochhammer[\[Beta]+ 1, n],Pochhammer[\[Alpha]+ \[Beta]+ 2, n]]*Sum[Divide[\[Alpha]+ \[Beta]+ 2*\[ScriptL]+ 1,\[Alpha]+ \[Beta]+ 1]*Divide[Pochhammer[\[Alpha]+ \[Beta]+ 1, \[ScriptL]]*Pochhammer[n + \[Beta]+ \[Gamma]+ 1, \[ScriptL]],Pochhammer[\[Beta]+ 1, \[ScriptL]]*Pochhammer[n + \[Alpha]+ \[Beta]+ 2, \[ScriptL]]]*Divide[Pochhammer[\[Gamma]- \[Alpha], n - \[ScriptL]],(n - \[ScriptL])!]*JacobiP[\[ScriptL], \[Alpha], \[Beta], x], {\[ScriptL], 0, n}, GenerateConditions->None]
Failure Aborted
Failed [299 / 300]
Result: -.361012173-.6250000000*I
Test Values: {alpha = 3/2, beta = 3/2, gamma = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 1}


Result: -1.123113229-2.395332347*I
Test Values: {alpha = 3/2, beta = 3/2, gamma = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 2}

... skip entries to safe data
 Skipped - Because timed out
18.18.E15 ( 1 + x 2 ) n = ( β + 1 ) n ( α + β + 2 ) n ⁢ ∑ ℓ = 0 n α + β + 2 ⁢ ℓ + 1 α + β + 1 ⁢ ( α + β + 1 ) ℓ ⁢ ( n - ℓ + 1 ) ℓ ( β + 1 ) ℓ ⁢ ( n + α + β + 2 ) ℓ ⁢ P ℓ ( α , β ) ⁡ ( x ) superscript 1 𝑥 2 𝑛 Pochhammer 𝛽 1 𝑛 Pochhammer 𝛼 𝛽 2 𝑛 superscript subscript ℓ 0 𝑛 𝛼 𝛽 2 ℓ 1 𝛼 𝛽 1 Pochhammer 𝛼 𝛽 1 ℓ Pochhammer 𝑛 ℓ 1 ℓ Pochhammer 𝛽 1 ℓ Pochhammer 𝑛 𝛼 𝛽 2 ℓ Jacobi-polynomial-P 𝛼 𝛽 ℓ 𝑥 {\displaystyle{\displaystyle\left(\frac{1+x}{2}\right)^{n}=\frac{{\left(\beta+% 1\right)_{n}}}{{\left(\alpha+\beta+2\right)_{n}}}\sum_{\ell=0}^{n}\frac{\alpha% +\beta+2\ell+1}{\alpha+\beta+1}\frac{{\left(\alpha+\beta+1\right)_{\ell}}{% \left(n-\ell+1\right)_{\ell}}}{{\left(\beta+1\right)_{\ell}}{\left(n+\alpha+% \beta+2\right)_{\ell}}}P^{(\alpha,\beta)}_{\ell}\left(x\right)}}
\left(\frac{1+x}{2}\right)^{n} = \frac{\Pochhammersym{\beta+1}{n}}{\Pochhammersym{\alpha+\beta+2}{n}}\sum_{\ell=0}^{n}\frac{\alpha+\beta+2\ell+1}{\alpha+\beta+1}\frac{\Pochhammersym{\alpha+\beta+1}{\ell}\Pochhammersym{n-\ell+1}{\ell}}{\Pochhammersym{\beta+1}{\ell}\Pochhammersym{n+\alpha+\beta+2}{\ell}}\JacobipolyP{\alpha}{\beta}{\ell}@{x}		 

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

(Divide[1 + x,2])^(n) == Divide[Pochhammer[\[Beta]+ 1, n],Pochhammer[\[Alpha]+ \[Beta]+ 2, n]]*Sum[Divide[\[Alpha]+ \[Beta]+ 2*\[ScriptL]+ 1,\[Alpha]+ \[Beta]+ 1]*Divide[Pochhammer[\[Alpha]+ \[Beta]+ 1, \[ScriptL]]*Pochhammer[n - \[ScriptL]+ 1, \[ScriptL]],Pochhammer[\[Beta]+ 1, \[ScriptL]]*Pochhammer[n + \[Alpha]+ \[Beta]+ 2, \[ScriptL]]]*JacobiP[\[ScriptL], \[Alpha], \[Beta], x], {\[ScriptL], 0, n}, GenerateConditions->None]
Failure Aborted Successful [Tested: 81]
Failed [78 / 81]
Result: Plus[1.25, Times[-0.125, Plus[Times[2.0, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-2, Plus[1, ], Plus[-1, Times[-1, ], 1], Plus[Times[-1, ], 1], Plus[1, , 1.5], Plus[1, , 1.5, 1.5], Plus[4, Times[2, ], 1.5, 1.5], []], Times[Plus[-1, Times[-1, ], 1], Plus[Times[-8, ], Times[-28, Power[, 2]], Times[-36, Power[, 3]], Times[-20, Power[, 4]], Times[-4, Power[, 5]], Times[8, 1], Times[28, , 1], Times[36, Power[, 2], 1], Times[20, Power[, 3], 1], Times[4, Power[, 4], 1], Times[48, , 1.5], Times[128, Power[, 2], 1.5], Times[124, Power[, 3], 1.5], Times[52, Power[, 4], 1.5], Times[8, Power[, 5], 1.5], Times[24, , 1, 1.5], Times[52, Power[, 2], 1, 1.5], Times[36, Power[, 3], 1, 1.5], Times[8, Power[, 4], 1, 1.5], Times[-18, , 1.5], Times[-46, Power[, 2], 1.5], Times[-38, Power[, 3], 1.5], Times[-10, Power[, 4], 1.5], Times[18, 1, 1.5], Times[46, , 1, 1.5], Times[38, Power[, 2], 1, 1.5], Times[10, Powe<syntaxhighlight lang=mathematica>Result: Plus[1.5625, Times[-0.07291666666666667, Plus[Times[2.0, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-2, Plus[1, ], Plus[-1, Times[-1, ], 2], Plus[Times[-1, ], 2], Plus[1, , 1.5], Plus[1, , 1.5, 1.5], Plus[4, Times[2, ], 1.5, 1.5], []], Times[Plus[-1, Times[-1, ], 2], Plus[Times[-8, ], Times[-28, Power[, 2]], Times[-36, Power[, 3]], Times[-20, Power[, 4]], Times[-4, Power[, 5]], Times[8, 2], Times[28, , 2], Times[36, Power[, 2], 2], Times[20, Power[, 3], 2], Times[4, Power[, 4], 2], Times[48, , 1.5], Times[128, Power[, 2], 1.5], Times[124, Power[, 3], 1.5], Times[52, Power[, 4], 1.5], Times[8, Power[, 5], 1.5], Times[24, , 2, 1.5], Times[52, Power[, 2], 2, 1.5], Times[36, Power[, 3], 2, 1.5], Times[8, Power[, 4], 2, 1.5], Times[-18, , 1.5], Times[-46, Power[, 2], 1.5], Times[-38, Power[, 3], 1.5], Times[-10, Power[, 4], 1.5], Times[18, 2, 1.5], Times[46, , 2, 1.5], Times[38, Power[, 2], 2, 1.5], Times[10, Power[, 3], 2, 1.5], Times[76, , 1.5, 1.5], Times[150, Power[, 2], 1.5, 1.5], Times[96, Power[, 3], 1.5, 1.5], Times[20, Power[, 4], 1.5, 1.5], Times[26, , 2, 1.5, 1.5], Times[36, Power[, 2], 2, 1.5, 1.5], Times[12, Power[, 3], 2, 1.5, 1.5], Times[-6, , Power[1.5, 2]], Times[-13, Power[, 2], Power[1.5, 2]], Times[-6, Power[, 3], Power[1.5, 2]], Times[12, 2, Power[1.5, 2]], Times[23, , 2, Power[1.5, 2]], Times[10, Power[, 2], 2, Power[1.5, 2]], Times[44, , 1.5, Power[1.5, 2]], Times[57, Power[, 2], 1.5, Power[1.5, 2]], Times[18, Power[, 3], 1.5, Power[1.5, 2]], Times[9, , 2, 1.5, Power[1.5, 2]], Times[6, Power[, 2], 2, 1.5, Power[1.5, 2]], Times[3, , Power[1.5, 3]], Times[Power[, 2], Power[1.5, 3]], Times[2, 2, Power[1.5, 3]], Times[3, , 2, Power[1.5, 3]], Times[11, , 1.5, Power[1.5, 3]], Times[7, Power[, 2], 1.5, Power[1.5, 3]], Times[, 2, 1.5, Power[1.5, 3]], Times[, Power[1.5, 4]], Times[, 1.5, Power[1.5, 4]], Times[-10, , 1.5], Times[-26, Power[, 2], 1.5], Times[-22, Power[, 3], 1.5], Times[-6, Power[, 4], 1.5], Times[10, 2, 1.5], Times[26, , 2, 1.5], Times[22, Power[, 2], 2, 1.5], Times[6, Power[, 3], 2, 1.5], Times[76, , 1.5, 1.5], Times[150, Power[, 2], 1.5, 1.5], Times[96, Power[, 3], 1.5, 1.5], Times[20, Power[, 4], 1.5, 1.5], Times[26, , 2, 1.5, 1.5], Times[36, Power[, 2], 2, 1.5, 1.5], Times[12, Power[, 3], 2, 1.5, 1.5], Times[-14, , 1.5, 1.5], Times[-24, Power[, 2], 1.5, 1.5], Times[-10, Power[, 3], 1.5, 1.5], Times[14, 2, 1.5, 1.5], Times[24, , 2, 1.5, 1.5], Times[10, Power[, 2], 2, 1.5, 1.5], Times[88, , 1.5, 1.5, 1.5], Times[114, Power[, 2], 1.5, 1.5, 1.5], Times[36, Power[, 3], 1.5, 1.5, 1.5], Times[18, , 2, 1.5, 1.5, 1.5], Times[12, Power[, 2], 2, 1.5, 1.5, 1.5], Times[, Power[1.5, 2], 1.5], Times[-1, Power[, 2], Power[1.5, 2], 1.5], Times[4, 2, Power[1.5, 2], 1.5], Times[5, , 2, Power[1.5, 2], 1.5], Times[33, , 1.5, Power[1.5, 2], 1.5], Times[21, Power[, 2], 1.5, Power[1.5, 2], 1.5], Times[3, , 2, 1.5, Power[1.5, 2], 1.5], Times[2, , Power[1.5, 3], 1.5], Times[4, , 1.5, Power[1.5, 3], 1.5], Times[-8, , Power[1.5, 2]], Times[-11, Power[, 2], Power[1.5, 2]], Times[-4, Power[, 3], Power[1.5, 2]], Times[2, 2, Power[1.5, 2]], Times[, 2, Power[1.5, 2]], Times[44, , 1.5, Power[1.5, 2]], Times[57, Power[, 2], 1.5, Power[1.5, 2]], Times[18, Power[, 3], 1.5, Power[1.5, 2]], Times[9, , 2, 1.5, Power[1.5, 2]], Times[6, Power[, 2], 2, 1.5, Power[1.5, 2]], Times[-7, , 1.5, Power[1.5, 2]], Times[-5, Power[, 2], 1.5, Power[1.5, 2]], Times[2, 2, 1.5, Power[1.5, 2]], Times[, 2, 1.5, Power[1.5, 2]], Times[33, , 1.5, 1.5, Power[1.5, 2]], Times[21, Power[, 2], 1.5, 1.5, Power[1.5, 2]], Times[3, , 2, 1.5, 1.5, Power[1.5, 2]], Times[6, , 1.5, Power[1.5, 2], Power[1.5, 2]], Times[-5, , Power[1.5, 3]], Times[-3, Power[, 2], Power[1.5, 3]], Times[-1, , 2, Power[1.5, 3]], Times[11, , 1.5, Power[1.5, 3]], Times[7, Power[, 2], 1.5, Power[1.5, 3]], Times[, 2, 1.5, Power[1.5, 3]], Times[-2, , 1.5, Power[1.5, 3]], Times[4, , 1.5, 1.5, Power[1.5, 3]], Times[-1, , Power[1.5, 4]], Times[, 1.5, Power[1.5, 4]]], [Plus[1, ]]], Times[, Plus[2, , 2, 1.5, 1.5], Plus[-24, Times[-68, ], Times[-68, Power[, 2]], Times[-28, Power[, 3]], Times[-4, Power[, 4]], Times[-8, 2], Times[-20, , 2], Times[-16, Power[, 2], 2], Times[-4, Power[, 3], 2], Times[24, 1.5], Times[76, , 1.5], Times[88, Power[, 2], 1.5], Times[44, Power[, 3], 1.5], Times[8, Power[, 4], 1.5], Times[-24, 2, 1.5], Times[-52, , 2, 1.5], Times[-36, Power[, 2], 2, 1.5], Times[-8, Power[, 3], 2, 1.5], Times[-20, 1.5], Times[-42, , 1.5], Times[-28, Power[, 2], 1.5], Times[-6, Power[, 3], 1.5], Times[-4, 2, 1.5], Times[-6, , 2, 1.5], Times[-2, Power[, 2], 2, 1.5], Times[26, 1.5, 1.5], Times[62, , 1.5, 1.5], Times[48, Power[, 2], 1.5, 1.5], Times[12, Power[, 3], 1.5, 1.5], Times[-26, 2, 1.5, 1.5], Times[-36, , 2, 1.5, 1.5], Times[-12, Power[, 2], 2, 1.5, 1.5], Times[-1, Power[1.5, 2]], Times[-1, , Power[1.5, 2]], Times[-3, 2, Power[1.5, 2]], Times[-2, , 2, Power[1.5, 2]], Times[9, 1.5, Power[1.5, 2]], Times[15, , 1.5, Power[1.5, 2]], Times[6, Power[, 2], 1.5, Power[1.5, 2]], Times[-9, 2, 1.5, Power[1.5, 2]], Times[-6, , 2, 1.5, Power[1.5, 2]], Power[1.5, 3], Times[, Power[1.5, 3]], Times[-1, 2, Power[1.5, 3]], Times[1.5, Power[1.5, 3]], Times[, 1.5, Power[1.5, 3]], Times[-1, 2, 1.5, Power[1.5, 3]], Times[-32, 1.5], Times[-70, , 1.5], Times[-48, Power[, 2], 1.5], Times[-10, Power[, 3], 1.5], Times[-8, 2, 1.5], Times[-14, , 2, 1.5], Times[-6, Power[, 2], 2, 1.5], Times[26, 1.5, 1.5], Times[62, , 1.5, 1.5], Times[48, Power[, 2], 1.5, 1.5], Times[12, Power[, 3], 1.5, 1.5], Times[-26, 2, 1.5, 1.5], Times[-36, , 2, 1.5, 1.5], Times[-12, Power[, 2], 2, 1.5, 1.5], Times[-18, 1.5, 1.5], Times[-28, , 1.5, 1.5], Times[-10, Power[, 2], 1.5, 1.5], Times[-2, 2, 1.5, 1.5], Times[-2, , 2, 1.5, 1.5], Times[18, 1.5, 1.5, 1.5], Times[30, , 1.5, 1.5, 1.5], Times[12, Power[, 2], 1.5, 1.5, 1.5], Times[-18, 2, 1.5, 1.5, 1.5], Times[-12, , 2, 1.5, 1.5, 1.5], Times[-1, Power[1.5, 2], 1.5], Times[-1, , Power[1.5, 2], 1.5], Times[-1, 2, Power[1.5, 2], 1.5], Times[3, 1.5, Power[1.5, 2], 1.5], Times[3, , 1.5, Power[1.5, 2], 1.5], Times[-3, 2, 1.5, Power[1.5, 2], 1.5], Times[-17, Power[1.5, 2]], Times[-27, , Power[1.5, 2]], Times[-10, Power[, 2], Power[1.5, 2]], Times[2, Power[1.5, 2]], Times[9, 1.5, Power[1.5, 2]], Times[15, , 1.5, Power[1.5, 2]], Times[6, Power[, 2], 1.5, Power[1.5, 2]], Times[-9, 2, 1.5, Power[1.5, 2]], Times[-6, , 2, 1.5, Power[1.5, 2]], Times[-5, 1.5, Power[1.5, 2]], Times[-5, , 1.5, Power[1.5, 2]], Times[2, 1.5, Power[1.5, 2]], Times[3, 1.5, 1.5, Power[1.5, 2]], Times[3, , 1.5, 1.5, Power[1.5, 2]], Times[-3, 2, 1.5, 1.5, Power[1.5, 2]], Times[-3, Power[1.5, 3]], Times[-3, , Power[1.5, 3]], Times[2, Power[1.5, 3]], Times[1.5, Power[1.5, 3]], Times[, 1.5, Power[1.5, 3]], Times[-1, 2, 1.5, Power[1.5, 3]]], [Plus[2, ]]], Times[2, , Plus[1, ], Plus[2, , 1.5], Plus[2, Times[2, ], 1.5, 1.5], Plus[2, , 2, 1.5, 1.5], Plus[3, , 2, 1.5, 1.5], [Plus[3, ]]]], 0], Equal[[1], 0], Equal[[2], Times[Rational[1, 2], 2, Power[Plus[1, 1.5], -1], Plus[1, 1.5, 1.5], Power[Plus[2, 2, 1.5, 1.5], -1], Plus[1.5, Times[-1, 1.5], Times[1.5, Plus[2, 1.5, 1.5]]]]], Equal[[3], Plus[Times[Rational[1, 2], 2, Power[Plus[1, 1.5], -1], Plus[1, 1.5, 1.5], Power[Plus[2, 2, 1.5, 1.5], -1], Plus[1.5, Times[-1, 1.5], Times[1.5, Plus[2, 1.5, 1.5]]]], Times[Rational[1, 2], Plus[-1, 2], 2, Power[Plus[1, 1.5], -1], Power[Plus[2, 1.5], -1], Plus[1, 1.5, 1.5], Power[Plus[2, 1.5, 1.5], -1], Power[Plus[2, 2, 1.5, 1.5], -1], Power[Plus[3, 2, 1.5, 1.5], -1], Plus[Times[-2, Plus[1, 1.5], Plus[1, 1.5], Plus[4, 1.5, 1.5]], Times[Rational[1, 2], Plus[3, 1.5, 1.5], Plus[Times[8, 1.5], Times[6, 1.5, 1.5], Power[1.5, 2], Times[1.5, Power[1.5, 2]], Times[6, 1.5, 1.5], Times[2, 1.5, 1.5, 1.5], Times[-1, Power[1.5, 2]], Times[1.5, Power[1.5, 2]]], Plus[1.5, Times[-1, 1.5], Times[1.5, Plus[2, 1.5, 1.5]]]]]]]]}]][3.0]], Times[4.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[-2, Plus[-1, Times[-1, ], 2], Plus[Times[-1, ], 2], Plus[1, , 1.5], Plus[1, , 1.5, 1.5], Plus[4, Times[2, ], 1.5, 1.5], []], Times[Plus[-1, Times[-1, ], 2], Plus[Times[-8, ], Times[-20, Power[, 2]], Times[-16, Power[, 3]], Times[-4, Power[, 4]], Times[8, 2], Times[20, , 2], Times[16, Power[, 2], 2], Times[4, Power[, 3], 2], Times[48, 1.5], Times[128, , 1.5], Times[124, Power[, 2], 1.5], Times[52, Power[, 3], 1.5], Times[8, Power[, 4], 1.5], Times[24, 2, 1.5], Times[52, , 2, 1.5], Times[36, Power[, 2], 2, 1.5], Times[8, Power[, 3], 2, 1.5], Times[-18, , 1.5], Times[-28, Power[, 2], 1.5], Times[-10, Power[, 3], 1.5], Times[18, 2, 1.5], Times[28, , 2, 1.5], Times[10, Power[, 2], 2, 1.5], Times[76, 1.5, 1.5], Times[150, , 1.5, 1.5], Times[96, Power[, 2], 1.5, 1.5], Times[20, Power[, 3], 1.5, 1.5], Times[26, 2, 1.5, 1.5], Times[36, , 2, 1.5, 1.5], Times[12, Power[, 2], 2, 1.5, 1.5], Times[6, Power[1.5, 2]], Times[-5, , Power[1.5, 2]], Times[-6, Power[, 2], Power[1.5, 2]], Times[15, 2, Power[1.5, 2]], Times[10, , 2, Power[1.5, 2]], Times[44, 1.5, Power[1.5, 2]], Times[57, , 1.5, Power[1.5, 2]], Times[18, Power[, 2], 1.5, Power[1.5, 2]], Times[9, 2, 1.5, Power[1.5, 2]], Times[6, , 2, 1.5, Power[1.5, 2]], Times[5, Power[1.5, 3]], Times[, Power[1.5, 3]], Times[3, 2, Power[1.5, 3]], Times[11, 1.5, Power[1.5, 3]], Times[7, , 1.5, Power[1.5, 3]], Times[2, 1.5, Power[1.5, 3]], Power[1.5, 4], Times[1.5, Power[1.5, 4]], Times[-10, , 1.5], Times[-16, Power[, 2], 1.5], Times[-6, Power[, 3], 1.5], Times[10, 2, 1.5], Times[16, , 2, 1.5], Times[6, Power[, 2], 2, 1.5], Times[76, 1.5, 1.5], Times[150, , 1.5, 1.5], Times[96, Power[, 2], 1.5, 1.5], Times[20, Power[, 3], 1.5, 1.5], Times[26, 2, 1.5, 1.5], Times[36, , 2, 1.5, 1.5], Times[12, Power[, 2], 2, 1.5, 1.5], Times[-14, , 1.5, 1.5], Times[-10, Power[, 2], 1.5, 1.5], Times[14, 2, 1.5, 1.5], Times[10, , 2, 1.5, 1.5], Times[88, 1.5, 1.5, 1.5], Times[114, , 1.5, 1.5, 1.5], Times[36, Power[, 2], 1.5, 1.5, 1.5], Times[18, 2, 1.5, 1.5, 1.5], Times[12, , 2, 1.5, 1.5, 1.5], Times[5, Power[1.5, 2], 1.5], Times[-1, , Power[1.5, 2], 1.5], Times[5, 2, Power[1.5, 2], 1.5], Times[33, 1.5, Power[1.5, 2], 1.5], Times[21, , 1.5, Power[1.5, 2], 1.5], Times[3, 2, 1.5, Power[1.5, 2], 1.5], Times[2, Power[1.5, 3], 1.5], Times[4, 1.5, Power[1.5, 3], 1.5], Times[-6, Power[1.5, 2]], Times[-9, , Power[1.5, 2]], Times[-4, Power[, 2], Power[1.5, 2]], Times[-1, 2, Power[1.5, 2]], Times[44, 1.5, Power[1.5, 2]], Times[57, , 1.5, Power[1.5, 2]], Times[18, Power[, 2], 1.5, Power[1.5, 2]], Times[9, 2, 1.5, Power[1.5, 2]], Times[6, , 2, 1.5, Power[1.5, 2]], Times[-5, 1.5, Power[1.5, 2]], Times[-5, , 1.5, Power[1.5, 2]], Times[2, 1.5, Power[1.5, 2]], Times[33, 1.5, 1.5, Power[1.5, 2]], Times[21, , 1.5, 1.5, Power[1.5, 2]], Times[3, 2, 1.5, 1.5, Power[1.5, 2]], Times[6, 1.5, Power[1.5, 2], Power[1.5, 2]], Times[-5, Power[1.5, 3]], Times[-3, , Power[1.5, 3]], Times[-1, 2, Power[1.5, 3]], Times[11, 1.5, Power[1.5, 3]], Times[7, , 1.5, Power[1.5, 3]], Times[2, 1.5, Power[1.5, 3]], Times[-2, 1.5, Power[1.5, 3]], Times[4, 1.5, 1.5, Power[1.5, 3]], Times[-1, Power[1.5, 4]], Times[1.5, Power[1.5, 4]]], [Plus[1, ]]], Times[-1, Plus[2, , 2, 1.5, 1.5], Plus[48, Times[112, ], Times[92, Power[, 2]], Times[32, Power[, 3]], Times[4, Power[, 4]], Times[16, 2], Times[32, , 2], Times[20, Power[, 2], 2], Times[4, Power[, 3], 2], Times[-24, 1.5], Times[-76, , 1.5], Times[-88, Power[, 2], 1.5], Times[-44, Power[, 3], 1.5], Times[-8, Power[, 4], 1.5], Times[24, 2, 1.5], Times[52, , 2, 1.5], Times[36, Power[, 2], 2, 1.5], Times[8, Power[, 3], 2, 1.5], Times[40, 1.5], Times[64, , 1.5], Times[34, Power[, 2], 1.5], Times[6, Power[, 3], 1.5], Times[8, 2, 1.5], Times[8, , 2, 1.5], Times[2, Power[, 2], 2, 1.5], Times[-26, 1.5, 1.5], Times[-62, , 1.5, 1.5], Times[-48, Power[, 2], 1.5, 1.5], Times[-12, Power[, 3], 1.5, 1.5], Times[26, 2, 1.5, 1.5], Times[36, , 2, 1.5, 1.5], Times[12, Power[, 2], 2, 1.5, 1.5], Times[5, Power[1.5, 2]], Times[3, , Power[1.5, 2]], Times[3, 2, Power[1.5, 2]], Times[2, , 2, Power[1.5, 2]], Times[-9, 1.5, Power[1.5, 2]], Times[-15, , 1.5, Power[1.5, 2]], Times[-6, Power[, 2], 1.5, Power[1.5, 2]], Times[9, 2, 1.5, Power[1.5, 2]], Times[6, , 2, 1.5, Power[1.5, 2]], Times[-1, Power[1.5, 3]], Times[-1, , Power[1.5, 3]], Times[2, Power[1.5, 3]], Times[-1, 1.5, Power[1.5, 3]], Times[-1, , 1.5, Power[1.5, 3]], Times[2, 1.5, Power[1.5, 3]], Times[64, 1.5], Times[108, , 1.5], Times[58, Power[, 2], 1.5], Times[10, Power[, 3], 1.5], Times[16, 2, 1.5], Times[20, , 2, 1.5], Times[6, Power[, 2], 2, 1.5], Times[-26, 1.5, 1.5], Times[-62, , 1.5, 1.5], Times[-48, Power[, 2], 1.5, 1.5], Times[-12, Power[, 3], 1.5, 1.5], Times[26, 2, 1.5, 1.5], Times[36, , 2, 1.5, 1.5], Times[12, Power[, 2], 2, 1.5, 1.5], Times[36, 1.5, 1.5], Times[38, , 1.5, 1.5], Times[10, Power[, 2], 1.5, 1.5], Times[4, 2, 1.5, 1.5], Times[2, , 2, 1.5, 1.5], Times[-18, 1.5, 1.5, 1.5], Times[-30, , 1.5, 1.5, 1.5], Times[-12, Power[, 2], 1.5, 1.5, 1.5], Times[18, 2, 1.5, 1.5, 1.5], Times[12, , 2, 1.5, 1.5, 1.5], Times[3, Power[1.5, 2], 1.5], Times[, Power[1.5, 2], 1.5], Times[2, Power[1.5, 2], 1.5], Times[-3, 1.5, Power[1.5, 2], 1.5], Times[-3, , 1.5, Power[1.5, 2], 1.5], Times[3, 2, 1.5, Power[1.5, 2], 1.5], Times[31, Power[1.5, 2]], Times[35, , Power[1.5, 2]], Times[10, Power[, 2], Power[1.5, 2]], Times[2, Power[1.5, 2]], Times[-9, 1.5, Power[1.5, 2]], Times[-15, , 1.5, Power[1.5, 2]], Times[-6, Power[, 2], 1.5, Power[1.5, 2]], Times[9, 2, 1.5, Power[1.5, 2]], Times[6, , 2, 1.5, Power[1.5, 2]], Times[9, 1.5, Power[1.5, 2]], Times[5, , 1.5, Power[1.5, 2]], Times[-1, 2, 1.5, Power[1.5, 2]], Times[-3, 1.5, 1.5, Power[1.5, 2]], Times[-3, , 1.5, 1.5, Power[1.5, 2]], Times[3, 2, 1.5, 1.5, Power[1.5, 2]], Times[5, Power[1.5, 3]], Times[3, , Power[1.5, 3]], Times[-1, 2, Power[1.5, 3]], Times[-1, 1.5, Power[1.5, 3]], Times[-1, , 1.5, Power[1.5, 3]], Times[2, 1.5, Power[1.5, 3]]], [Plus[2, ]]], Times[2, Plus[2, ], Plus[2, , 1.5], Plus[2, Times[2, ], 1.5, 1.5], Plus[2, , 2, 1.5, 1.5], Plus[3, , 2, 1.5, 1.5], [Plus[3, ]]]], 0], Equal[[-1], 0], Equal[[0], Times[Rational[1, 2], Power[Plus[1, 2], -1], Power[1.5, -1], Power[Plus[1.5, 1.5], -1], Power[Plus[2, 1.5, 1.5], -1], Plus[1, 2, 1.5, 1.5], Plus[Times[Plus[1, 1.5, 1.5], Plus[Times[2, 1.5, 1.5], Power[1.5, 2], Times[1.5, Power[1.5, 2]], Times[2, 1.5, 1.5], Times[2, 1.5, 1.5, 1.5], Times[-1, Power[1.5, 2]], Times[1.5, Power[1.5, 2]]]], Times[-1, Plus[1.5, 1.5], Plus[1, 1.5, 1.5], Plus[1.5, Times[-1, 1.5], Times[1.5, Plus[2, 1.5, 1.5]]]]]]], Equal[[1], Plus[1, Times[Rational[1, 2], Power[Plus[1, 2], -1], Power[1.5, -1], Power[Plus[1.5, 1.5], -1], Power[Plus[2, 1.5, 1.5], -1], Plus[1, 2, 1.5, 1.5], Plus[Times[Plus[1, 1.5, 1.5], Plus[Times[2, 1.5, 1.5], Power[1.5, 2], Times[1.5, Power[1.5, 2]], Times[2, 1.5, 1.5], Times[2, 1.5, 1.5, 1.5], Times[-1, Power[1.5, 2]], Times[1.5, Power[1.5, 2]]]], Times[-1, Plus[1.5, 1.5], Plus[1, 1.5, 1.5], Plus[1.5, Times[-1, 1.5], Times[1.5, Plus[2, 1.5, 1.5]]]]]]]]}]][3.0]]]]], {Rule[n, 2], Rule[x, 1.5], Rule[α, 1.5], Rule[β, 1.5]}

... skip entries to safe data
18.18.E16 C n ( μ ) ⁡ ( x ) = ∑ ℓ = 0 ⌊ n / 2 ⌋ λ + n - 2 ⁢ ℓ λ ⁢ ( μ ) n - ℓ ( λ + 1 ) n - ℓ ⁢ ( μ - λ ) ℓ ℓ ! ⁢ C n - 2 ⁢ ℓ ( λ ) ⁡ ( x ) ultraspherical-Gegenbauer-polynomial 𝜇 𝑛 𝑥 superscript subscript ℓ 0 𝑛 2 𝜆 𝑛 2 ℓ 𝜆 Pochhammer 𝜇 𝑛 ℓ Pochhammer 𝜆 1 𝑛 ℓ Pochhammer 𝜇 𝜆 ℓ ℓ ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 2 ℓ 𝑥 {\displaystyle{\displaystyle C^{(\mu)}_{n}\left(x\right)=\sum_{\ell=0}^{\left% \lfloor n/2\right\rfloor}\frac{\lambda+n-2\ell}{\lambda}\frac{{\left(\mu\right% )_{n-\ell}}}{{\left(\lambda+1\right)_{n-\ell}}}\frac{{\left(\mu-\lambda\right)% _{\ell}}}{\ell!}C^{(\lambda)}_{n-2\ell}\left(x\right)}}
\ultrasphpoly{\mu}{n}@{x} = \sum_{\ell=0}^{\floor{n/2}}\frac{\lambda+n-2\ell}{\lambda}\frac{\Pochhammersym{\mu}{n-\ell}}{\Pochhammersym{\lambda+1}{n-\ell}}\frac{\Pochhammersym{\mu-\lambda}{\ell}}{\ell!}\ultrasphpoly{\lambda}{n-2\ell}@{x}		 

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

GegenbauerC[n, \[Mu], x] == Sum[Divide[\[Lambda]+ n - 2*\[ScriptL],\[Lambda]]*Divide[Pochhammer[\[Mu], n - \[ScriptL]],Pochhammer[\[Lambda]+ 1, n - \[ScriptL]]]*Divide[Pochhammer[\[Mu]- \[Lambda], \[ScriptL]],(\[ScriptL])!]*GegenbauerC[n - 2*\[ScriptL], \[Lambda], x], {\[ScriptL], 0, Floor[n/2]}, GenerateConditions->None]
Failure Aborted Successful [Tested: 300]
Failed [300 / 300]
Result: Plus[Complex[2.598076211353316, 1.4999999999999998], Times[Complex[-0.8660254037844387, 0.49999999999999994], Plus[Times[-2.0, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[-1, Times[-2, ], 1], Plus[Times[-2, ], 1], Plus[-3, Times[-2, ], 1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[-1, Times[-1, ], 1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[Times[-1, ], 1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[1, , Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], []], Times[Plus[1, , Times[-1, 1], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Plus[1, , Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[Times[-12, ], Times[-56, Power<syntaxhighlight lang=mathematica>Result: Plus[Complex[5.281088913245535, 5.647114317029973], Times[Complex[-0.8660254037844387, 0.49999999999999994], Plus[Times[-2.0, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[-1, Times[-2, ], 2], Plus[Times[-2, ], 2], Plus[-3, Times[-2, ], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[-1, Times[-1, ], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[Times[-1, ], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[1, , Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], []], Times[Plus[1, , Times[-1, 2], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Plus[1, , Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[Times[-12, ], Times[-56, Power[, 2]], Times[-86, Power[, 3]], Times[-48, Power[, 4]], Times[-8, Power[, 5]], Times[34, , 2], Times[105, Power[, 2], 2], Times[88, Power[, 3], 2], Times[20, Power[, 4], 2], Times[-31, , Power[2, 2]], Times[-52, Power[, 2], Power[2, 2]], Times[-18, Power[, 3], Power[2, 2]], Times[10, , Power[2, 3]], Times[7, Power[, 2], Power[2, 3]], Times[-1, , Power[2, 4]], Times[24, , Power[1.5, 2]], Times[112, Power[, 2], Power[1.5, 2]], Times[184, Power[, 3], Power[1.5, 2]], Times[128, Power[, 4], Power[1.5, 2]], Times[32, Power[, 5], Power[1.5, 2]], Times[-68, , 2, Power[1.5, 2]], Times[-228, Power[, 2], 2, Power[1.5, 2]], Times[-240, Power[, 3], 2, Power[1.5, 2]], Times[-80, Power[, 4], 2, Power[1.5, 2]], Times[68, , Power[2, 2], Power[1.5, 2]], Times[144, Power[, 2], Power[2, 2], Power[1.5, 2]], Times[72, Power[, 3], Power[2, 2], Power[1.5, 2]], Times[-28, , Power[2, 3], Power[1.5, 2]], Times[-28, Power[, 2], Power[2, 3], Power[1.5, 2]], Times[4, , Power[2, 4], Power[1.5, 2]], Times[18, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[50, Power[, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[34, Power[, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[4, Power[, 4], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-28, , 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-39, Power[, 2], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-8, Power[, 3], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-3, Power[2, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[3, , Power[2, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[Power[, 2], Power[2, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[4, Power[2, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[3, , Power[2, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, Power[2, 4], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-44, , Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-140, Power[, 2], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-144, Power[, 3], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-48, Power[, 4], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[92, , 2, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[192, Power[, 2], 2, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[96, Power[, 3], 2, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-60, , Power[2, 2], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-60, Power[, 2], Power[2, 2], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[12, , Power[2, 3], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[6, Power[, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[8, Power[, 3], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-3, 2, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-10, , 2, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-12, Power[, 2], 2, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[5, Power[2, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[8, , Power[2, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-2, Power[2, 3], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[24, , Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[48, Power[, 2], Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[24, Power[, 3], Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-36, , 2, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-36, Power[, 2], 2, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[12, , Power[2, 2], Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-2, , Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3]], Times[-4, Power[, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3]], Times[2, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3]], Times[4, , 2, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3]], Times[-1, Power[2, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3]], Times[-4, , Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3]], Times[-4, Power[, 2], Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3]], Times[4, , 2, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3]], Times[12, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[50, Power[, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[64, Power[, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[24, Power[, 4], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-31, , 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-80, Power[, 2], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-44, Power[, 3], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[3, Power[2, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[32, , Power[2, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[30, Power[, 2], Power[2, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-4, Power[2, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-9, , Power[2, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[Power[2, 4], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-24, , Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-88, Power[, 2], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-96, Power[, 3], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-32, Power[, 4], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[44, , 2, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[96, Power[, 2], 2, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[48, Power[, 3], 2, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-24, , Power[2, 2], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-24, Power[, 2], Power[2, 2], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[4, , Power[2, 3], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-24, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-62, Power[, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-36, Power[, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[3, 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[41, , 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[44, Power[, 2], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-5, Power[2, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-17, , Power[2, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, Power[2, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[44, , Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[96, Power[, 2], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[48, Power[, 3], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-48, , 2, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-48, Power[, 2], 2, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[12, , Power[2, 2], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[8, , Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[12, Power[, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, 2, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-8, , 2, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[Power[2, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-24, , Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-24, Power[, 2], Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[12, , 2, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[4, , Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[1, ]]], Times[-1, , Plus[1, , Times[-1, 2], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Plus[12, Times[44, ], Times[42, Power[, 2]], Times[-6, Power[, 3]], Times[-24, Power[, 4]], Times[-8, Power[, 5]], Times[-16, 2], Times[-23, , 2], Times[25, Power[, 2], 2], Times[52, Power[, 3], 2], Times[20, Power[, 4], 2], Power[2, 2], Times[-19, , Power[2, 2]], Times[-38, Power[, 2], Power[2, 2]], Times[-18, Power[, 3], Power[2, 2]], Times[4, Power[2, 3]], Times[11, , Power[2, 3]], Times[7, Power[, 2], Power[2, 3]], Times[-1, Power[2, 4]], Times[-1, , Power[2, 4]], Times[24, Power[1.5, 2]], Times[136, , Power[1.5, 2]], Times[296, Power[, 2], Power[1.5, 2]], Times[312, Power[, 3], Power[1.5, 2]], Times[160, Power[, 4], Power[1.5, 2]], Times[32, Power[, 5], Power[1.5, 2]], Times[-68, 2, Power[1.5, 2]], Times[-296, , 2, Power[1.5, 2]], Times[-468, Power[, 2], 2, Power[1.5, 2]], Times[-320, Power[, 3], 2, Power[1.5, 2]], Times[-80, Power[, 4], 2, Power[1.5, 2]], Times[68, Power[2, 2], Power[1.5, 2]], Times[212, , Power[2, 2], Power[1.5, 2]], Times[216, Power[, 2], Power[2, 2], Power[1.5, 2]], Times[72, Power[, 3], Power[2, 2], Power[1.5, 2]], Times[-28, Power[2, 3], Power[1.5, 2]], Times[-56, , Power[2, 3], Power[1.5, 2]], Times[-28, Power[, 2], Power[2, 3], Power[1.5, 2]], Times[4, Power[2, 4], Power[1.5, 2]], Times[4, , Power[2, 4], Power[1.5, 2]], Times[-10, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[14, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[98, Power[, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[110, Power[, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[36, Power[, 4], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-20, 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-123, , 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-175, Power[, 2], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-72, Power[, 3], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[39, Power[2, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[94, , Power[2, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[53, Power[, 2], Power[2, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-17, Power[2, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-17, , Power[2, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, Power[2, 4], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-92, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-408, , Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-652, Power[, 2], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-448, Power[, 3], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-112, Power[, 4], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[204, 2, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[652, , 2, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[672, Power[, 2], 2, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[224, Power[, 3], 2, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-152, Power[2, 2], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-312, , Power[2, 2], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-156, Power[, 2], Power[2, 2], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[44, Power[2, 3], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[44, , Power[2, 3], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-4, Power[2, 4], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-18, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-102, , Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-140, Power[, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-56, Power[, 3], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[60, 2, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[148, , 2, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[84, Power[, 2], 2, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-42, Power[2, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-44, , Power[2, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[8, Power[2, 3], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[136, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[440, , Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[456, Power[, 2], Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[152, Power[, 3], Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-220, 2, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-456, , 2, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-228, Power[, 2], 2, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[108, Power[2, 2], Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[108, , Power[2, 2], Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-16, Power[2, 3], Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[22, Power[Power[E, Times[Complex[0, 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Rational[1, 6]], Pi]], 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-10, Power[2, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[68, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[144, , Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[72, Power[, 2], Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-84, 2, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-84, , 2, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[24, Power[2, 2], Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[14, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[12, , Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-4, 2, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-28, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-28, , Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[16, 2, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[4, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 4], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[2, ]]], Times[-1, Plus[1, ], Plus[2, ], Plus[-1, Times[-2, ], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[-4, Times[-2, ], 2, Times[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Plus[-3, Times[-2, ], 2, Times[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Plus[-2, Times[-1, ], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[-1, Times[-1, ], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Times[GegenbauerC[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 1.5], Power[Pochhammer[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2], -1], Pochhammer[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], Equal[[2], Plus[Times[-1, Plus[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Plus[-1, 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], -1], GegenbauerC[Plus[-2, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 1.5], Power[Pochhammer[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2], -1], Pochhammer[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[GegenbauerC[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 1.5], Power[Pochhammer[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2], -1], Pochhammer[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]]]}]][2.0]]]]], {Rule[n, 2], Rule[x, 1.5], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
18.18.E17 ( 2 ⁢ x ) n = n ! ⁢ ∑ ℓ = 0 ⌊ n / 2 ⌋ λ + n - 2 ⁢ ℓ λ ⁢ 1 ( λ + 1 ) n - ℓ ⁢ ℓ ! ⁢ C n - 2 ⁢ ℓ ( λ ) ⁡ ( x ) superscript 2 𝑥 𝑛 𝑛 superscript subscript ℓ 0 𝑛 2 𝜆 𝑛 2 ℓ 𝜆 1 Pochhammer 𝜆 1 𝑛 ℓ ℓ ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 2 ℓ 𝑥 {\displaystyle{\displaystyle(2x)^{n}=n!\sum_{\ell=0}^{\left\lfloor n/2\right% \rfloor}\frac{\lambda+n-2\ell}{\lambda}\frac{1}{{\left(\lambda+1\right)_{n-% \ell}}\,\ell!}C^{(\lambda)}_{n-2\ell}\left(x\right)}}
(2x)^{n} = n!\sum_{\ell=0}^{\floor{n/2}}\frac{\lambda+n-2\ell}{\lambda}\frac{1}{\Pochhammersym{\lambda+1}{n-\ell}\,\ell!}\ultrasphpoly{\lambda}{n-2\ell}@{x}		 

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

(2*x)^(n) == (n)!*Sum[Divide[\[Lambda]+ n - 2*\[ScriptL],\[Lambda]]*Divide[1,Pochhammer[\[Lambda]+ 1, n - \[ScriptL]]*(\[ScriptL])!]*GegenbauerC[n - 2*\[ScriptL], \[Lambda], x], {\[ScriptL], 0, Floor[n/2]}, GenerateConditions->None]
Failure Aborted Error
Failed [74 / 90]
Result: Plus[3.0, Times[Complex[-0.8660254037844387, 0.49999999999999994], Plus[Times[-2.0, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[-1, Times[-2, ], 1], Plus[Times[-2, ], 1], Plus[-3, Times[-2, ], 1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[-1, Times[-1, ], 1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[Times[-1, ], 1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], []], Times[-1, Plus[-1, Times[-1, ], 1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[Times[12, ], Times[50, Power[, 2]], Times[64, Power[, 3]], Times[24, Power[, 4]], Times[-31, , 1], Times[-80, Power[, 2], 1], Times[-44, Power[, 3], 1], Times[3, Power[1, 2]], Times[32, , Power[1, 2]], Times[30, Power[, 2], Power[1, 2]], Times[-4, Power[1, 3]], Times[-9, , Power[1, 3]], Power[1, 4], Times[-24, , Power[1.5, 2]], Times[-88, Power[, 2], Power[1.5, 2]], Times[-96, Power[, 3], Power[1.5, 2]], Times[-32, Power[,<syntaxhighlight lang=mathematica>Result: Plus[9.0, Times[Complex[-1.7320508075688774, 0.9999999999999999], Plus[Times[-2.0, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[-1, Times[-2, ], 2], Plus[Times[-2, ], 2], Plus[-3, Times[-2, ], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[-1, Times[-1, ], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[Times[-1, ], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], []], Times[-1, Plus[-1, Times[-1, ], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[Times[12, ], Times[50, Power[, 2]], Times[64, Power[, 3]], Times[24, Power[, 4]], Times[-31, , 2], Times[-80, Power[, 2], 2], Times[-44, Power[, 3], 2], Times[3, Power[2, 2]], Times[32, , Power[2, 2]], Times[30, Power[, 2], Power[2, 2]], Times[-4, Power[2, 3]], Times[-9, , Power[2, 3]], Power[2, 4], Times[-24, , Power[1.5, 2]], Times[-88, Power[, 2], Power[1.5, 2]], Times[-96, Power[, 3], Power[1.5, 2]], Times[-32, Power[, 4], Power[1.5, 2]], Times[44, , 2, Power[1.5, 2]], Times[96, Power[, 2], 2, Power[1.5, 2]], Times[48, Power[, 3], 2, Power[1.5, 2]], Times[-24, , Power[2, 2], Power[1.5, 2]], Times[-24, Power[, 2], Power[2, 2], Power[1.5, 2]], Times[4, , Power[2, 3], Power[1.5, 2]], Times[-24, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-62, Power[, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-36, Power[, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[3, 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[41, , 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[44, Power[, 2], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-5, Power[2, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-17, , Power[2, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, Power[2, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[44, , Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[96, Power[, 2], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[48, Power[, 3], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-48, , 2, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-48, Power[, 2], 2, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[12, , Power[2, 2], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[8, , Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[12, Power[, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-1, 2, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-8, , 2, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[Power[2, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-24, , Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-24, Power[, 2], Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[12, , 2, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[4, , Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3]]], [Plus[1, ]]], Times[, Plus[-24, Times[-106, ], Times[-162, Power[, 2]], Times[-104, Power[, 3]], Times[-24, Power[, 4]], Times[53, 2], Times[165, , 2], Times[164, Power[, 2], 2], Times[52, Power[, 3], 2], Times[-42, Power[2, 2]], Times[-86, , Power[2, 2]], Times[-42, Power[, 2], Power[2, 2]], Times[15, Power[2, 3]], Times[15, , Power[2, 3]], Times[-2, Power[2, 4]], Times[24, Power[1.5, 2]], Times[112, , Power[1.5, 2]], Times[184, Power[, 2], Power[1.5, 2]], Times[128, Power[, 3], Power[1.5, 2]], Times[32, Power[, 4], Power[1.5, 2]], Times[-68, 2, Power[1.5, 2]], Times[-228, , 2, Power[1.5, 2]], Times[-240, Power[, 2], 2, Power[1.5, 2]], Times[-80, Power[, 3], 2, Power[1.5, 2]], Times[68, Power[2, 2], Power[1.5, 2]], Times[144, , Power[2, 2], Power[1.5, 2]], Times[72, Power[, 2], Power[2, 2], Power[1.5, 2]], Times[-28, Power[2, 3], Power[1.5, 2]], Times[-28, , Power[2, 3], Power[1.5, 2]], Times[4, Power[2, 4], Power[1.5, 2]], Times[56, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[182, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[186, Power[, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[60, Power[, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-87, 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-183, , 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-92, Power[, 2], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[45, Power[2, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[47, , Power[2, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-8, Power[2, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-68, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-228, , Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-240, Power[, 2], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-80, Power[, 3], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[136, 2, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[288, , 2, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[144, Power[, 2], 2, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-84, Power[2, 2], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-84, , Power[2, 2], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[16, Power[2, 3], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-42, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-92, , Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-48, Power[, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[40, 2, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[44, , 2, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-10, Power[2, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[68, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[144, , Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[72, Power[, 2], Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-84, 2, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-84, , 2, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[24, Power[2, 2], Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[10, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3]], Times[12, , Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3]], Times[-4, 2, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3]], Times[-28, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3]], Times[-28, , Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3]], Times[16, 2, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3]], Times[4, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 4]]], [Plus[2, ]]], Times[, Plus[1, ], Plus[3, Times[2, ], Times[-1, 2], Times[-2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Plus[4, Times[2, ], Times[-1, 2], Times[-2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Plus[1, Times[2, ], Times[-1, 2], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[3, ]]]], 0], Equal[[1], 0], Equal[[2], Times[-1, Plus[Times[-1, 2], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], GegenbauerC[Plus[-2, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 1.5], Power[Pochhammer[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2], -1]]], Equal[[3], Plus[Times[-1, Plus[Times[-1, 2], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], GegenbauerC[Plus[-2, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 1.5], Power[Pochhammer[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2], -1]], Times[Plus[Times[-1, 2], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Plus[-4, 2, Times[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], -1], Power[Plus[-3, 2, Times[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], -1], Plus[Times[-2, Plus[-2, 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[-3, Times[4, 2], Times[-1, Power[2, 2]], Times[6, Power[1.5, 2]], Times[-8, 2, Power[1.5, 2]], Times[2, Power[2, 2], Power[1.5, 2]], Times[3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-2, 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-8, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[4, 2, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], GegenbauerC[Plus[-2, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 1.5]], Times[Plus[-1, 2], 2, Plus[-3, 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], GegenbauerC[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 1.5]]], Power[Pochhammer[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2], -1]]]]}]][2.0]], Times[Complex[2.866025403784439, 0.49999999999999994], DifferenceRoot[Function[{, }, {Equal[Plus[Times[Plus[-1, Times[-2, ], 2], Plus[Times[-2, ], 2], Plus[-3, Times[-2, ], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[-1, Times[-1, ], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[Times[-1, ], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], []], Times[-1, Plus[-1, Times[-1, ], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[12, Times[56, ], Times[98, Power[, 2]], Times[80, Power[, 3]], Times[24, Power[, 4]], Times[-22, 2], Times[-79, , 2], Times[-104, Power[, 2], 2], Times[-44, Power[, 3], 2], Times[15, Power[2, 2]], Times[44, , Power[2, 2]], Times[30, Power[, 2], Power[2, 2]], Times[-6, Power[2, 3]], Times[-9, , Power[2, 3]], Power[2, 4], Times[-24, Power[1.5, 2]], Times[-112, , Power[1.5, 2]], Times[-184, Power[, 2], Power[1.5, 2]], Times[-128, Power[, 3], Power[1.5, 2]], Times[-32, Power[, 4], Power[1.5, 2]], Times[44, 2, Power[1.5, 2]], Times[140, , 2, Power[1.5, 2]], Times[144, Power[, 2], 2, Power[1.5, 2]], Times[48, Power[, 3], 2, Power[1.5, 2]], Times[-24, Power[2, 2], Power[1.5, 2]], Times[-48, , Power[2, 2], Power[1.5, 2]], Times[-24, Power[, 2], Power[2, 2], Power[1.5, 2]], Times[4, Power[2, 3], Power[1.5, 2]], Times[4, , Power[2, 3], Power[1.5, 2]], Times[-18, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-68, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-86, Power[, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-36, Power[, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[25, 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[65, , 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[44, Power[, 2], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-11, Power[2, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-17, , Power[2, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, Power[2, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[44, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[140, , Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[144, Power[, 2], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[48, Power[, 3], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-48, 2, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-96, , 2, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-48, Power[, 2], 2, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[12, Power[2, 2], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[12, , Power[2, 2], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[6, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[16, , Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[12, Power[, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-5, 2, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-8, , 2, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[Power[2, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-24, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-48, , Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-24, Power[, 2], Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[12, 2, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[12, , 2, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[4, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3]], Times[4, , Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3]]], [Plus[1, ]]], Times[Plus[1, ], Plus[-36, Times[-144, ], Times[-194, Power[, 2]], Times[-112, Power[, 3]], Times[-24, Power[, 4]], Times[72, 2], Times[197, , 2], Times[176, Power[, 2], 2], Times[52, Power[, 3], 2], Times[-50, Power[2, 2]], Times[-92, , Power[2, 2]], Times[-42, Power[, 2], Power[2, 2]], Times[16, Power[2, 3]], Times[15, , Power[2, 3]], Times[-2, Power[2, 4]], Times[24, Power[1.5, 2]], Times[112, , Power[1.5, 2]], Times[184, Power[, 2], Power[1.5, 2]], Times[128, Power[, 3], Power[1.5, 2]], Times[32, Power[, 4], Power[1.5, 2]], Times[-68, 2, Power[1.5, 2]], Times[-228, , 2, Power[1.5, 2]], Times[-240, Power[, 2], 2, Power[1.5, 2]], Times[-80, Power[, 3], 2, Power[1.5, 2]], Times[68, Power[2, 2], Power[1.5, 2]], Times[144, , Power[2, 2], Power[1.5, 2]], Times[72, Power[, 2], Power[2, 2], Power[1.5, 2]], Times[-28, Power[2, 3], Power[1.5, 2]], Times[-28, , Power[2, 3], Power[1.5, 2]], Times[4, Power[2, 4], Power[1.5, 2]], Times[82, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[232, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[206, Power[, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[60, Power[, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-112, 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-203, , 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-92, Power[, 2], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[50, Power[2, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[47, , Power[2, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-8, Power[2, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-68, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-228, , Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-240, Power[, 2], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-80, Power[, 3], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[136, 2, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[288, , 2, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[144, Power[, 2], 2, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-84, Power[2, 2], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-84, , Power[2, 2], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[16, Power[2, 3], Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-60, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-108, , Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-48, Power[, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[48, 2, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[44, , 2, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-10, Power[2, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[68, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[144, , Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[72, Power[, 2], Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-84, 2, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-84, , 2, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[24, Power[2, 2], Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[14, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3]], Times[12, , Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3]], Times[-4, 2, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3]], Times[-28, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3]], Times[-28, , Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3]], Times[16, 2, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3]], Times[4, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 4]]], [Plus[2, ]]], Times[-1, Plus[1, ], Plus[2, ], Plus[-1, Times[-2, ], 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[-4, Times[-2, ], 2, Times[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Plus[-3, Times[-2, ], 2, Times[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Times[GegenbauerC[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 1.5], Power[Pochhammer[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2], -1]]], Equal[[2], Plus[Times[-1, Plus[Times[-1, 2], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], GegenbauerC[Plus[-2, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 1.5], Power[Pochhammer[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2], -1]], Times[GegenbauerC[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 1.5], Power[Pochhammer[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2], -1]]]]}]][2.0]]]]], {Rule[n, 2], Rule[x, 1.5], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
18.18.E18 L n ( β ) ⁡ ( x ) = ∑ ℓ = 0 n ( β - α ) n - ℓ ( n - ℓ ) ! ⁢ L ℓ ( α ) ⁡ ( x ) Laguerre-polynomial-L 𝛽 𝑛 𝑥 superscript subscript ℓ 0 𝑛 Pochhammer 𝛽 𝛼 𝑛 ℓ 𝑛 ℓ Laguerre-polynomial-L 𝛼 ℓ 𝑥 {\displaystyle{\displaystyle L^{(\beta)}_{n}\left(x\right)=\sum_{\ell=0}^{n}% \frac{{\left(\beta-\alpha\right)_{n-\ell}}}{(n-\ell)!}L^{(\alpha)}_{\ell}\left% (x\right)}}
\LaguerrepolyL[\beta]{n}@{x} = \sum_{\ell=0}^{n}\frac{\Pochhammersym{\beta-\alpha}{n-\ell}}{(n-\ell)!}\LaguerrepolyL[\alpha]{\ell}@{x}		 

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

LaguerreL[n, \[Beta], x] == Sum[Divide[Pochhammer[\[Beta]- \[Alpha], n - \[ScriptL]],(n - \[ScriptL])!]*LaguerreL[\[ScriptL], \[Alpha], x], {\[ScriptL], 0, n}, GenerateConditions->None]
Missing Macro Error Failure -
Failed [78 / 81]
Result: Plus[1.0, Times[-1.0, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[Times[-1, ], 1], Plus[, Times[-1, 1.5], 1.5], Plus[1, , Times[-1, 1.5], 1.5], []], Times[-1, Plus[1, , Times[-1, 1.5], 1.5], Plus[-1, Times[-3, ], Times[-3, Power[, 2]], Times[2, 1], Times[3, , 1], Times[-1, 1.5], Times[-1, , 1.5], 1.5, Times[2, , 1.5], Times[-1, 1, 1.5], Times[-1, , 1.5], Times[1, 1.5]], [Plus[1, ]]], Times[Plus[1, ], Plus[-3, Times[-6, ], Times[-3, Power[, 2]], Times[4, 1], Times[3, , 1], Times[-1, 1.5], Times[-1, , 1.5], Times[4, 1.5], Times[4, , 1.5], Times[-2, 1, 1.5], Times[1.5, 1.5], Times[-1, Power[1.5, 2]], Times[-1, 1.5], Times[-2, , 1.5], Times[2, 1, 1.5], Times[-1, 1.5, 1.5], Times[1.5, 1.5]], [Plus[2, ]]], Times[-1, Plus[1, ], Plus[2, ], Plus[-1, Times[-1, ], 1, 1.5], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], LaguerreL[1, 1.5, 1.5]], Equal[[2], Plus[Times[Plus[Times[-1, 1.5], 1.5], LaguerreL[Pl<syntaxhighlight lang=mathematica>Result: Plus[0.25, Times[-1.0, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[Times[-1, ], 2], Plus[, Times[-1, 1.5], 1.5], Plus[1, , Times[-1, 1.5], 1.5], []], Times[-1, Plus[1, , Times[-1, 1.5], 1.5], Plus[-1, Times[-3, ], Times[-3, Power[, 2]], Times[2, 2], Times[3, , 2], Times[-1, 1.5], Times[-1, , 1.5], 1.5, Times[2, , 1.5], Times[-1, 2, 1.5], Times[-1, , 1.5], Times[2, 1.5]], [Plus[1, ]]], Times[Plus[1, ], Plus[-3, Times[-6, ], Times[-3, Power[, 2]], Times[4, 2], Times[3, , 2], Times[-1, 1.5], Times[-1, , 1.5], Times[4, 1.5], Times[4, , 1.5], Times[-2, 2, 1.5], Times[1.5, 1.5], Times[-1, Power[1.5, 2]], Times[-1, 1.5], Times[-2, , 1.5], Times[2, 2, 1.5], Times[-1, 1.5, 1.5], Times[1.5, 1.5]], [Plus[2, ]]], Times[-1, Plus[1, ], Plus[2, ], Plus[-1, Times[-1, ], 2, 1.5], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], LaguerreL[2, 1.5, 1.5]], Equal[[2], Plus[Times[Plus[Times[-1, 1.5], 1.5], LaguerreL[Plus[-1, 2], 1.5, 1.5]], LaguerreL[2, 1.5, 1.5]]]}]][3.0]]], {Rule[n, 2], Rule[x, 1.5], Rule[α, 1.5], Rule[β, 1.5]}

... skip entries to safe data
18.18.E19 x n = ( α + 1 ) n ⁢ ∑ ℓ = 0 n ( - n ) ℓ ( α + 1 ) ℓ ⁢ L ℓ ( α ) ⁡ ( x ) superscript 𝑥 𝑛 Pochhammer 𝛼 1 𝑛 superscript subscript ℓ 0 𝑛 Pochhammer 𝑛 ℓ Pochhammer 𝛼 1 ℓ Laguerre-polynomial-L 𝛼 ℓ 𝑥 {\displaystyle{\displaystyle x^{n}={\left(\alpha+1\right)_{n}}\sum_{\ell=0}^{n% }\frac{{\left(-n\right)_{\ell}}}{{\left(\alpha+1\right)_{\ell}}}L^{(\alpha)}_{% \ell}\left(x\right)}}
x^{n} = \Pochhammersym{\alpha+1}{n}\sum_{\ell=0}^{n}\frac{\Pochhammersym{-n}{\ell}}{\Pochhammersym{\alpha+1}{\ell}}\LaguerrepolyL[\alpha]{\ell}@{x}		 

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

(x)^(n) == Pochhammer[\[Alpha]+ 1, n]*Sum[Divide[Pochhammer[- n, \[ScriptL]],Pochhammer[\[Alpha]+ 1, \[ScriptL]]]*LaguerreL[\[ScriptL], \[Alpha], x], {\[ScriptL], 0, n}, GenerateConditions->None]
Missing Macro Error Failure -
Failed [24 / 27]
Result: Plus[1.5, Times[-2.5, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-1, Plus[-1, Times[-1, ], 1], Plus[Times[-1, ], 1], []], Times[Plus[-1, Times[-1, ], 1], Plus[-3, Times[-3, ], 1, 1.5, Times[-1, 1.5]], [Plus[1, ]]], Times[Plus[-7, Times[-9, ], Times[-3, Power[, 2]], Times[3, 1], Times[2, , 1], 1.5, Times[, 1.5], Times[-1, 1, 1.5], Times[-3, 1.5], Times[-2, , 1.5], Times[1, 1.5]], [Plus[2, ]]], Times[Plus[2, ], Plus[2, , 1.5], [Plus[3, ]]]], 0], Equal[[-1], 0], Equal[[0], 0], Equal[[1], 1]}]][2.0]]], {Rule[n, 1], Rule[x, 1.5], Rule[α, 1.5]}


Result: Plus[2.25, Times[-8.75, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-1, Plus[-1, Times[-1, ], 2], Plus[Times[-1, ], 2], []], Times[Plus[-1, Times[-1, ], 2], Plus[-3, Times[-3, ], 2, 1.5, Times[-1, 1.5]], [Plus[1, ]]], Times[Plus[-7, Times[-9, ], Times[-3, Power[, 2]], Times[3, 2], Times[2, , 2], 1.5, Times[, 1.5], Times[-1, 2, 1.5], Times[-3, 1.5], Times[-2, , 1.5], Times[2, 1.5]], [Plus[2, ]]], Times[Plus[2, ], Plus[2, , 1.5], [Plus[3, ]]]], 0], Equal[[-1], 0], Equal[[0], 0], Equal[[1], 1]}]][3.0]]], {Rule[n, 2], Rule[x, 1.5], Rule[α, 1.5]}

... skip entries to safe data
18.18.E20 ( 2 ⁢ x ) n = ∑ ℓ = 0 ⌊ n / 2 ⌋ ( - n ) 2 ⁢ ℓ ℓ ! ⁢ H n - 2 ⁢ ℓ ⁡ ( x ) superscript 2 𝑥 𝑛 superscript subscript ℓ 0 𝑛 2 Pochhammer 𝑛 2 ℓ ℓ Hermite-polynomial-H 𝑛 2 ℓ 𝑥 {\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_{\ell=0}^{\floor{n/2}}\frac{\Pochhammersym{-n}{2\ell}}{\ell!}\HermitepolyH{n-2\ell}@{x}		 

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

(2*x)^(n) == Sum[Divide[Pochhammer[- n, 2*\[ScriptL]],(\[ScriptL])!]*HermiteH[n - 2*\[ScriptL], x], {\[ScriptL], 0, Floor[n/2]}, GenerateConditions->None]
Failure Failure Successful [Tested: 9]
Failed [9 / 9]
Result: 3.0
Test Values: {Rule[n, 1], Rule[x, 1.5]}


Result: 9.0
Test Values: {Rule[n, 2], Rule[x, 1.5]}

... skip entries to safe data
18.18.E21 T m ⁡ ( x ) ⁢ T n ⁡ ( x ) = 1 2 ⁢ ( T m + n ⁡ ( x ) + T m - n ⁡ ( x ) ) Chebyshev-polynomial-first-kind-T 𝑚 𝑥 Chebyshev-polynomial-first-kind-T 𝑛 𝑥 1 2 Chebyshev-polynomial-first-kind-T 𝑚 𝑛 𝑥 Chebyshev-polynomial-first-kind-T 𝑚 𝑛 𝑥 {\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))}}
\ChebyshevpolyT{m}@{x}\ChebyshevpolyT{n}@{x} = \tfrac{1}{2}(\ChebyshevpolyT{m+n}@{x}+\ChebyshevpolyT{m-n}@{x})		 

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 [Tested: 27] Successful [Tested: 27]
18.18.E24 b n , ℓ = ( n ℓ ) ⁢ ( n + α + β + 1 ) ℓ ⁢ ( - β - n ) n - ℓ 2 ℓ ⁢ ( α + 1 ) n subscript 𝑏 𝑛 ℓ binomial 𝑛 ℓ Pochhammer 𝑛 𝛼 𝛽 1 ℓ Pochhammer 𝛽 𝑛 𝑛 ℓ superscript 2 ℓ Pochhammer 𝛼 1 𝑛 {\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} = \binom{n}{\ell}\frac{\Pochhammersym{n+\alpha+\beta+1}{\ell}\Pochhammersym{-\beta-n}{n-\ell}}{2^{\ell}\Pochhammersym{\alpha+1}{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 Error
Failed [270 / 270]
Result: Plus[Complex[0.8660254037844387, 0.49999999999999994], Times[-0.4, Power[2.0, Times[-1.0, ℓ]], Binomial[1.0, ℓ], Pochhammer[-2.5, Plus[1.0, Times[-1.0, ℓ]]], Pochhammer[5.0, ℓ]]]
Test Values: {Rule[n, 1], Rule[α, 1.5], Rule[β, 1.5], Rule[Subscript[b, n, ℓ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[0.8660254037844387, 0.49999999999999994], Times[-0.11428571428571428, Power[2.0, Times[-1.0, ℓ]], Binomial[2.0, ℓ], Pochhammer[-3.5, Plus[2.0, Times[-1.0, ℓ]]], Pochhammer[6.0, ℓ]]]
Test Values: {Rule[n, 2], Rule[α, 1.5], Rule[β, 1.5], Rule[Subscript[b, n, ℓ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
18.18.E25 P n ( α , β ) ⁡ ( x ) P n ( α , β ) ⁡ ( 1 ) ⁢ P n ( α , β ) ⁡ ( y ) P n ( α , β ) ⁡ ( 1 ) = ∑ ℓ = 0 n b n , ℓ ⁢ ( x + y ) ℓ ⁢ P ℓ ( α , β ) ⁡ ( ( 1 + x ⁢ y ) / ( x + y ) ) P ℓ ( α , β ) ⁡ ( 1 ) Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑦 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 superscript subscript ℓ 0 𝑛 subscript 𝑏 𝑛 ℓ superscript 𝑥 𝑦 ℓ Jacobi-polynomial-P 𝛼 𝛽 ℓ 1 𝑥 𝑦 𝑥 𝑦 Jacobi-polynomial-P 𝛼 𝛽 ℓ 1 {\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)}}}
\frac{\JacobipolyP{\alpha}{\beta}{n}@{x}}{\JacobipolyP{\alpha}{\beta}{n}@{1}}\frac{\JacobipolyP{\alpha}{\beta}{n}@{y}}{\JacobipolyP{\alpha}{\beta}{n}@{1}} = \sum_{\ell=0}^{n}b_{n,\ell}(x+y)^{\ell}\*\frac{\JacobipolyP{\alpha}{\beta}{\ell}@{\ifrac{(1+xy)}{(x+y)}}}{\JacobipolyP{\alpha}{\beta}{\ell}@{1}}		 

(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 + x*y)/(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 + x*y,x + y]],JacobiP[\[ScriptL], \[Alpha], \[Beta], 1]], {\[ScriptL], 0, n}, GenerateConditions->None]
Failure Failure Error Skipped - Because timed out
18.18.E26 P n ( α , β ) ⁡ ( x ) P n ( α , β ) ⁡ ( 1 ) = ∑ ℓ = 0 n b n , ℓ ⁢ ( x + 1 ) ℓ Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 superscript subscript ℓ 0 𝑛 subscript 𝑏 𝑛 ℓ superscript 𝑥 1 ℓ {\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}}}
\frac{\JacobipolyP{\alpha}{\beta}{n}@{x}}{\JacobipolyP{\alpha}{\beta}{n}@{1}} = \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}, GenerateConditions->None]
Failure Failure
Failed [299 / 300]
Result: -1.531088914-1.750000000*I
Test Values: {alpha = 3/2, beta = 3/2, x = 3/2, b[n,ell] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: -5.943747689-4.875000000*I
Test Values: {alpha = 3/2, beta = 3/2, x = 3/2, b[n,ell] = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [299 / 300]
Result: Complex[-1.5310889132455356, -1.7499999999999998]
Test Values: {Rule[n, 1], Rule[x, Rational[3, 2]], Rule[α, Rational[3, 2]], Rule[β, Rational[3, 2]], Rule[Subscript[b, n, ℓ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-5.943747686898277, -4.874999999999999]
Test Values: {Rule[n, 2], Rule[x, Rational[3, 2]], Rule[α, Rational[3, 2]], Rule[β, Rational[3, 2]], Rule[Subscript[b, n, ℓ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
18.18.E27 ∑ n = 0 ∞ n ! ⁢ L n ( α ) ⁡ ( x ) ⁢ L n ( α ) ⁡ ( y ) ( α + 1 ) n ⁢ z n = Γ ⁡ ( α + 1 ) ⁢ ( x ⁢ y ⁢ z ) - 1 2 ⁢ α 1 - z ⁢ exp ⁡ ( - ( x + y ) ⁢ z 1 - z ) ⁢ I α ⁡ ( 2 ⁢ ( x ⁢ y ⁢ z ) 1 2 1 - z ) superscript subscript 𝑛 0 𝑛 Laguerre-polynomial-L 𝛼 𝑛 𝑥 Laguerre-polynomial-L 𝛼 𝑛 𝑦 Pochhammer 𝛼 1 𝑛 superscript 𝑧 𝑛 Euler-Gamma 𝛼 1 superscript 𝑥 𝑦 𝑧 1 2 𝛼 1 𝑧 𝑥 𝑦 𝑧 1 𝑧 modified-Bessel-first-kind 𝛼 2 superscript 𝑥 𝑦 𝑧 1 2 1 𝑧 {\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{n!\,L^{(\alpha)}_{n}\left% (x\right)L^{(\alpha)}_{n}\left(y\right)}{{\left(\alpha+1\right)_{n}}}z^{n}=% \frac{\Gamma\left(\alpha+1\right)(xyz)^{-\frac{1}{2}\alpha}}{1-z}\*\exp\left(% \frac{-(x+y)z}{1-z}\right)I_{\alpha}\left(\frac{2(xyz)^{\frac{1}{2}}}{1-z}% \right)}}
\sum_{n=0}^{\infty}\frac{n!\,\LaguerrepolyL[\alpha]{n}@{x}\LaguerrepolyL[\alpha]{n}@{y}}{\Pochhammersym{\alpha+1}{n}}z^{n} = \frac{\EulerGamma@{\alpha+1}(xyz)^{-\frac{1}{2}\alpha}}{1-z}\*\exp@{\frac{-(x+y)z}{1-z}}\modBesselI{\alpha}@{\frac{2(xyz)^{\frac{1}{2}}}{1-z}}		 | z | < 1 , ℜ ⁡ ( α + 1 ) > 0 , ℜ ⁡ ( ( α ) + k + 1 ) > 0 formulae-sequence 𝑧 1 formulae-sequence 𝛼 1 0 𝛼 𝑘 1 0 {\displaystyle{\displaystyle|z|<1,\Re(\alpha+1)>0,\Re((\alpha)+k+1)>0}} 
sum((factorial(n)*LaguerreL(n, alpha, x)*LaguerreL(n, alpha, y))/(pochhammer(alpha + 1, n))*(x + y*I)^(n), n = 0..infinity) = (GAMMA(alpha + 1)*(x*y*(x + y*I))^(-(1)/(2)*alpha))/(1 -(x + y*I))* exp((-(x + y)*(x + y*I))/(1 -(x + y*I)))*BesselI(alpha, (2*(x*y*(x + y*I))^((1)/(2)))/(1 -(x + y*I)))

Sum[Divide[(n)!*LaguerreL[n, \[Alpha], x]*LaguerreL[n, \[Alpha], y],Pochhammer[\[Alpha]+ 1, n]]*(x + y*I)^(n), {n, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[\[Alpha]+ 1]*(x*y*(x + y*I))^(-Divide[1,2]*\[Alpha]),1 -(x + y*I)]* Exp[Divide[-(x + y)*(x + y*I),1 -(x + y*I)]]*BesselI[\[Alpha], Divide[2*(x*y*(x + y*I))^(Divide[1,2]),1 -(x + y*I)]]
Missing Macro Error Failure -
Failed [54 / 54]
Result: Plus[Complex[-0.2554853305235294, -0.2809050421578725], NSum[Times[Power[Complex[1.5, -1.5], n], Factorial[n], LaguerreL[n, 1.5, -1.5], LaguerreL[n, 1.5, 1.5], Power[Pochhammer[2.5, n], -1]]
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[x, 1.5], Rule[y, -1.5], Rule[α, 1.5]}


Result: Plus[Complex[0.5256093118420817, -0.5266734460651719], NSum[Times[Power[Complex[1.5, -1.5], n], Factorial[n], LaguerreL[n, 0.5, -1.5], LaguerreL[n, 0.5, 1.5], Power[Pochhammer[1.5, n], -1]]
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[x, 1.5], Rule[y, -1.5], Rule[α, 0.5]}

... skip entries to safe data
18.18.E28 ∑ n = 0 ∞ H n ⁡ ( x ) ⁢ H n ⁡ ( y ) 2 n ⁢ n ! ⁢ z n = ( 1 - z 2 ) - 1 2 ⁢ exp ⁡ ( 2 ⁢ x ⁢ y ⁢ z - ( x 2 + y 2 ) ⁢ z 2 1 - z 2 ) superscript subscript 𝑛 0 Hermite-polynomial-H 𝑛 𝑥 Hermite-polynomial-H 𝑛 𝑦 superscript 2 𝑛 𝑛 superscript 𝑧 𝑛 superscript 1 superscript 𝑧 2 1 2 2 𝑥 𝑦 𝑧 superscript 𝑥 2 superscript 𝑦 2 superscript 𝑧 2 1 superscript 𝑧 2 {\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_{n=0}^{\infty}\frac{\HermitepolyH{n}@{x}\HermitepolyH{n}@{y}}{2^{n}n!}z^{n} = (1-z^{2})^{-\frac{1}{2}}\exp@{\frac{2xyz-(x^{2}+y^{2})z^{2}}{1-z^{2}}}		 | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}} 
sum((HermiteH(n, x)*HermiteH(n, y))/((2)^(n)* factorial(n))*(x + y*I)^(n), n = 0..infinity) = (1 -(x + y*I)^(2))^(-(1)/(2))* exp((2*x*y*(x + y*I)-((x)^(2)+ (y)^(2))*(x + y*I)^(2))/(1 -(x + y*I)^(2)))

Sum[Divide[HermiteH[n, x]*HermiteH[n, y],(2)^(n)* (n)!]*(x + y*I)^(n), {n, 0, Infinity}, GenerateConditions->None] == (1 -(x + y*I)^(2))^(-Divide[1,2])* Exp[Divide[2*x*y*(x + y*I)-((x)^(2)+ (y)^(2))*(x + y*I)^(2),1 -(x + y*I)^(2)]]
Failure Failure Manual Skip!
Failed [18 / 18]
Result: Plus[Complex[45.14577089044274, -92.71442284704277], NSum[Times[Power[Complex[0.75, -0.75], n], Power[Factorial[n], -1], HermiteH[n, -1.5], HermiteH[n, 1.5]]
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[x, 1.5], Rule[y, -1.5]}


Result: Plus[Complex[-1.1210206126790663, -11.104063395584024], NSum[Times[Power[Complex[0.75, 0.75], n], Power[Factorial[n], -1], Power[HermiteH[n, 1.5], 2]]
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
18.18.E29 ∑ ℓ = 0 n C ℓ ( λ ) ⁡ ( x ) ⁢ C n - ℓ ( μ ) ⁡ ( x ) = C n ( λ + μ ) ⁡ ( x ) superscript subscript ℓ 0 𝑛 ultraspherical-Gegenbauer-polynomial 𝜆 ℓ 𝑥 ultraspherical-Gegenbauer-polynomial 𝜇 𝑛 ℓ 𝑥 ultraspherical-Gegenbauer-polynomial 𝜆 𝜇 𝑛 𝑥 {\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_{\ell=0}^{n}\ultrasphpoly{\lambda}{\ell}@{x}\ultrasphpoly{\mu}{n-\ell}@{x} = \ultrasphpoly{\lambda+\mu}{n}@{x}		 

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}, GenerateConditions->None] == GegenbauerC[n, \[Lambda]+ \[Mu], x]
Failure Successful
Failed [36 / 300]
Result: -3.000000000+0.*I
Test Values: {lambda = 1/2*3^(1/2)+1/2*I, mu = -1/2*3^(1/2)-1/2*I, x = 3/2, n = 1}


Result: -3.499999999+0.*I
Test Values: {lambda = 1/2*3^(1/2)+1/2*I, mu = -1/2*3^(1/2)-1/2*I, x = 3/2, n = 2}

... skip entries to safe data
 Successful [Tested: 300]
18.18.E30 ∑ ℓ = 0 n ℓ + 2 ⁢ λ 2 ⁢ λ ⁢ C ℓ ( λ ) ⁡ ( x ) ⁢ x n - ℓ = C n ( λ + 1 ) ⁡ ( x ) superscript subscript ℓ 0 𝑛 ℓ 2 𝜆 2 𝜆 ultraspherical-Gegenbauer-polynomial 𝜆 ℓ 𝑥 superscript 𝑥 𝑛 ℓ ultraspherical-Gegenbauer-polynomial 𝜆 1 𝑛 𝑥 {\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=0}^{n}\frac{\ell+2\lambda}{2\lambda}\ultrasphpoly{\lambda}{\ell}@{x}x^{n-\ell} = \ultrasphpoly{\lambda+1}{n}@{x}		 

sum((ell + 2*lambda)/(2*lambda)*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}, GenerateConditions->None] == GegenbauerC[n, \[Lambda]+ 1, x]
Failure Failure Successful [Tested: 90] Successful [Tested: 90]
18.18.E31 ∑ ℓ = 0 n T ℓ ⁡ ( x ) ⁢ x n - ℓ = U n ⁡ ( x ) superscript subscript ℓ 0 𝑛 Chebyshev-polynomial-first-kind-T ℓ 𝑥 superscript 𝑥 𝑛 ℓ Chebyshev-polynomial-second-kind-U 𝑛 𝑥 {\displaystyle{\displaystyle\sum_{\ell=0}^{n}T_{\ell}\left(x\right)x^{n-\ell}=% U_{n}\left(x\right)}}
\sum_{\ell=0}^{n}\ChebyshevpolyT{\ell}@{x}x^{n-\ell} = \ChebyshevpolyU{n}@{x}		 

sum(ChebyshevT(ell, x)*(x)^(n - ell), ell = 0..n) = ChebyshevU(n, x)

Sum[ChebyshevT[\[ScriptL], x]*(x)^(n - \[ScriptL]), {\[ScriptL], 0, n}, GenerateConditions->None] == ChebyshevU[n, x]
Failure Aborted Successful [Tested: 9] Successful [Tested: 9]
18.18.E32 2 ⁢ ∑ ℓ = 0 n T 2 ⁢ ℓ ⁡ ( x ) = 1 + U 2 ⁢ n ⁡ ( x ) 2 superscript subscript ℓ 0 𝑛 Chebyshev-polynomial-first-kind-T 2 ℓ 𝑥 1 Chebyshev-polynomial-second-kind-U 2 𝑛 𝑥 {\displaystyle{\displaystyle 2\sum_{\ell=0}^{n}T_{2\ell}\left(x\right)=1+U_{2n% }\left(x\right)}}
2\sum_{\ell=0}^{n}\ChebyshevpolyT{2\ell}@{x} = 1+\ChebyshevpolyU{2n}@{x}		 

2*sum(ChebyshevT(2*ell, x), ell = 0..n) = 1 + ChebyshevU(2*n, x)

2*Sum[ChebyshevT[2*\[ScriptL], x], {\[ScriptL], 0, n}, GenerateConditions->None] == 1 + ChebyshevU[2*n, x]
Failure Successful Successful [Tested: 9] Successful [Tested: 9]
18.18.E33 2 ⁢ ∑ ℓ = 0 n T 2 ⁢ ℓ + 1 ⁡ ( x ) = U 2 ⁢ n + 1 ⁡ ( x ) 2 superscript subscript ℓ 0 𝑛 Chebyshev-polynomial-first-kind-T 2 ℓ 1 𝑥 Chebyshev-polynomial-second-kind-U 2 𝑛 1 𝑥 {\displaystyle{\displaystyle 2\sum_{\ell=0}^{n}T_{2\ell+1}\left(x\right)=U_{2n% +1}\left(x\right)}}
2\sum_{\ell=0}^{n}\ChebyshevpolyT{2\ell+1}@{x} = \ChebyshevpolyU{2n+1}@{x}		 

2*sum(ChebyshevT(2*ell + 1, x), ell = 0..n) = ChebyshevU(2*n + 1, x)

2*Sum[ChebyshevT[2*\[ScriptL]+ 1, x], {\[ScriptL], 0, n}, GenerateConditions->None] == ChebyshevU[2*n + 1, x]
Failure Successful Successful [Tested: 9] Successful [Tested: 9]
18.18.E34 2 ⁢ ( 1 - x 2 ) ⁢ ∑ ℓ = 0 n U 2 ⁢ ℓ ⁡ ( x ) = 1 - T 2 ⁢ n + 2 ⁡ ( x ) 2 1 superscript 𝑥 2 superscript subscript ℓ 0 𝑛 Chebyshev-polynomial-second-kind-U 2 ℓ 𝑥 1 Chebyshev-polynomial-first-kind-T 2 𝑛 2 𝑥 {\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_{\ell=0}^{n}\ChebyshevpolyU{2\ell}@{x} = 1-\ChebyshevpolyT{2n+2}@{x}		 

2*(1 - (x)^(2))*sum(ChebyshevU(2*ell, x), ell = 0..n) = 1 - ChebyshevT(2*n + 2, x)

2*(1 - (x)^(2))*Sum[ChebyshevU[2*\[ScriptL], x], {\[ScriptL], 0, n}, GenerateConditions->None] == 1 - ChebyshevT[2*n + 2, x]
Failure Successful Successful [Tested: 9] Successful [Tested: 9]
18.18.E35 2 ⁢ ( 1 - x 2 ) ⁢ ∑ ℓ = 0 n U 2 ⁢ ℓ + 1 ⁡ ( x ) = x - T 2 ⁢ n + 3 ⁡ ( x ) 2 1 superscript 𝑥 2 superscript subscript ℓ 0 𝑛 Chebyshev-polynomial-second-kind-U 2 ℓ 1 𝑥 𝑥 Chebyshev-polynomial-first-kind-T 2 𝑛 3 𝑥 {\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_{\ell=0}^{n}\ChebyshevpolyU{2\ell+1}@{x} = x-\ChebyshevpolyT{2n+3}@{x}		 

2*(1 - (x)^(2))*sum(ChebyshevU(2*ell + 1, x), ell = 0..n) = x - ChebyshevT(2*n + 3, x)

2*(1 - (x)^(2))*Sum[ChebyshevU[2*\[ScriptL]+ 1, x], {\[ScriptL], 0, n}, GenerateConditions->None] == x - ChebyshevT[2*n + 3, x]
Failure Successful Successful [Tested: 9] Successful [Tested: 9]
18.18.E36 ∑ ℓ = 0 n P ℓ ⁡ ( x ) ⁢ P n - ℓ ⁡ ( x ) = U n ⁡ ( x ) superscript subscript ℓ 0 𝑛 Legendre-spherical-polynomial ℓ 𝑥 Legendre-spherical-polynomial 𝑛 ℓ 𝑥 Chebyshev-polynomial-second-kind-U 𝑛 𝑥 {\displaystyle{\displaystyle\sum_{\ell=0}^{n}P_{\ell}\left(x\right)P_{n-\ell}% \left(x\right)=U_{n}\left(x\right)}}
\sum_{\ell=0}^{n}\LegendrepolyP{\ell}@{x}\LegendrepolyP{n-\ell}@{x} = \ChebyshevpolyU{n}@{x}		 

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}, GenerateConditions->None] == ChebyshevU[n, x]
Failure Successful Successful [Tested: 9] Successful [Tested: 9]
18.18.E37 ∑ ℓ = 0 n L ℓ ( α ) ⁡ ( x ) = L n ( α + 1 ) ⁡ ( x ) superscript subscript ℓ 0 𝑛 Laguerre-polynomial-L 𝛼 ℓ 𝑥 Laguerre-polynomial-L 𝛼 1 𝑛 𝑥 {\displaystyle{\displaystyle\sum_{\ell=0}^{n}L^{(\alpha)}_{\ell}\left(x\right)% =L^{(\alpha+1)}_{n}\left(x\right)}}
\sum_{\ell=0}^{n}\LaguerrepolyL[\alpha]{\ell}@{x} = \LaguerrepolyL[\alpha+1]{n}@{x}		 

sum(LaguerreL(ell, alpha, x), ell = 0..n) = LaguerreL(n, alpha + 1, x)
 
Sum[LaguerreL[\[ScriptL], \[Alpha], x], {\[ScriptL], 0, n}, GenerateConditions->None] == LaguerreL[n, \[Alpha]+ 1, x]
 Missing Macro Error Successful - Successful [Tested: 27]
18.18.E38 ∑ ℓ = 0 n L ℓ ( α ) ⁡ ( x ) ⁢ L n - ℓ ( β ) ⁡ ( y ) = L n ( α + β + 1 ) ⁡ ( x + y ) superscript subscript ℓ 0 𝑛 Laguerre-polynomial-L 𝛼 ℓ 𝑥 Laguerre-polynomial-L 𝛽 𝑛 ℓ 𝑦 Laguerre-polynomial-L 𝛼 𝛽 1 𝑛 𝑥 𝑦 {\displaystyle{\displaystyle\sum_{\ell=0}^{n}L^{(\alpha)}_{\ell}\left(x\right)% L^{(\beta)}_{n-\ell}\left(y\right)=L^{(\alpha+\beta+1)}_{n}\left(x+y\right)}}
\sum_{\ell=0}^{n}\LaguerrepolyL[\alpha]{\ell}@{x}\LaguerrepolyL[\beta]{n-\ell}@{y} = \LaguerrepolyL[\alpha+\beta+1]{n}@{x+y}		 

sum(LaguerreL(ell, alpha, x)*LaguerreL(n - ell, beta, y), ell = 0..n) = LaguerreL(n, alpha + beta + 1, x + y)
 
Sum[LaguerreL[\[ScriptL], \[Alpha], x]*LaguerreL[n - \[ScriptL], \[Beta], y], {\[ScriptL], 0, n}, GenerateConditions->None] == LaguerreL[n, \[Alpha]+ \[Beta]+ 1, x + y]
 Missing Macro Error Successful - Successful [Tested: 300]
18.18.E39 ∑ ℓ = 0 n ( n ℓ ) ⁢ H ℓ ⁡ ( 2 1 2 ⁢ x ) ⁢ H n - ℓ ⁡ ( 2 1 2 ⁢ y ) = 2 1 2 ⁢ n ⁢ H n ⁡ ( x + y ) superscript subscript ℓ 0 𝑛 binomial 𝑛 ℓ Hermite-polynomial-H ℓ superscript 2 1 2 𝑥 Hermite-polynomial-H 𝑛 ℓ superscript 2 1 2 𝑦 superscript 2 1 2 𝑛 Hermite-polynomial-H 𝑛 𝑥 𝑦 {\displaystyle{\displaystyle\sum_{\ell=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{\ell}H% _{\ell}\left(2^{\frac{1}{2}}x\right)H_{n-\ell}\left(2^{\frac{1}{2}}y\right)=2^% {\frac{1}{2}n}H_{n}\left(x+y\right)}}
\sum_{\ell=0}^{n}\binom{n}{\ell}\HermitepolyH{\ell}@{2^{\frac{1}{2}}x}\HermitepolyH{n-\ell}@{2^{\frac{1}{2}}y} = 2^{\frac{1}{2}n}\HermitepolyH{n}@{x+y}		 

sum(binomial(n,ell)*HermiteH(ell, (2)^((1)/(2))* x)*HermiteH(n - ell, (2)^((1)/(2))* y), ell = 0..n) = (2)^((1)/(2)*n)* HermiteH(n, x + y)
 
Sum[Binomial[n,\[ScriptL]]*HermiteH[\[ScriptL], (2)^(Divide[1,2])* x]*HermiteH[n - \[ScriptL], (2)^(Divide[1,2])* y], {\[ScriptL], 0, n}, GenerateConditions->None] == (2)^(Divide[1,2]*n)* HermiteH[n, x + y]
 Failure Successful Successful [Tested: 54] Successful [Tested: 54]
18.18.E40 ∑ ℓ = 0 n ( n ℓ ) ⁢ H 2 ⁢ ℓ ⁡ ( x ) ⁢ H 2 ⁢ n - 2 ⁢ ℓ ⁡ ( y ) = ( - 1 ) n ⁢ 2 2 ⁢ n ⁢ n ! ⁢ L n ⁡ ( x 2 + y 2 ) superscript subscript ℓ 0 𝑛 binomial 𝑛 ℓ Hermite-polynomial-H 2 ℓ 𝑥 Hermite-polynomial-H 2 𝑛 2 ℓ 𝑦 superscript 1 𝑛 superscript 2 2 𝑛 𝑛 shorthand-Laguerre-polynomial-L 𝑛 superscript 𝑥 2 superscript 𝑦 2 {\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_{\ell=0}^{n}\binom{n}{\ell}\HermitepolyH{2\ell}@{x}\HermitepolyH{2n-2\ell}@{y} = (-1)^{n}2^{2n}n!\LaguerrepolyL[]{n}@{x^{2}+y^{2}}		 

sum(binomial(n,ell)*HermiteH(2*ell, x)*HermiteH(2*n - 2*ell, y), ell = 0..n) = (- 1)^(n)* (2)^(2*n)* factorial(n)*LaguerreL(n, (x)^(2)+ (y)^(2))
 
Sum[Binomial[n,\[ScriptL]]*HermiteH[2*\[ScriptL], x]*HermiteH[2*n - 2*\[ScriptL], y], {\[ScriptL], 0, n}, GenerateConditions->None] == (- 1)^(n)* (2)^(2*n)* (n)!*LaguerreL[n, (x)^(2)+ (y)^(2)]
 Failure Successful Successful [Tested: 54] Successful [Tested: 54]
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