Orthogonal Polynomials - 18.6 Symmetry, Special Values, and Limits to Monomials

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
18.6.E1 L n ( Ξ± ) ⁑ ( 0 ) = ( Ξ± + 1 ) n n ! Laguerre-polynomial-L 𝛼 𝑛 0 Pochhammer 𝛼 1 𝑛 𝑛 {\displaystyle{\displaystyle L^{(\alpha)}_{n}\left(0\right)=\frac{{\left(% \alpha+1\right)_{n}}}{n!}}}
\LaguerrepolyL[\alpha]{n}@{0} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}		 

LaguerreL(n, alpha, 0) = (pochhammer(alpha + 1, n))/(factorial(n))

LaguerreL[n, \[Alpha], 0] == Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]
Missing Macro Error Successful - Successful [Tested: 9]
18.6.E2 lim Ξ± β†’ ∞ ⁑ P n ( Ξ± , Ξ² ) ⁑ ( x ) P n ( Ξ± , Ξ² ) ⁑ ( 1 ) = ( 1 + x 2 ) n subscript β†’ 𝛼 Jacobi-polynomial-P 𝛼 𝛽 𝑛 π‘₯ Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 superscript 1 π‘₯ 2 𝑛 {\displaystyle{\displaystyle\lim_{\alpha\to\infty}\frac{P^{(\alpha,\beta)}_{n}% \left(x\right)}{P^{(\alpha,\beta)}_{n}\left(1\right)}=\left(\frac{1+x}{2}% \right)^{n}}}
\lim_{\alpha\to\infty}\frac{\JacobipolyP{\alpha}{\beta}{n}@{x}}{\JacobipolyP{\alpha}{\beta}{n}@{1}} = \left(\frac{1+x}{2}\right)^{n}		 

limit((JacobiP(n, alpha, beta, x))/(JacobiP(n, alpha, beta, 1)), alpha = infinity) = ((1 + x)/(2))^(n)

Limit[Divide[JacobiP[n, \[Alpha], \[Beta], x],JacobiP[n, \[Alpha], \[Beta], 1]], \[Alpha] -> Infinity, GenerateConditions->None] == (Divide[1 + x,2])^(n)
Failure Aborted Successful [Tested: 27] Skipped - Because timed out
18.6.E3 lim Ξ² β†’ ∞ ⁑ P n ( Ξ± , Ξ² ) ⁑ ( x ) P n ( Ξ± , Ξ² ) ⁑ ( - 1 ) = ( 1 - x 2 ) n subscript β†’ 𝛽 Jacobi-polynomial-P 𝛼 𝛽 𝑛 π‘₯ Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 superscript 1 π‘₯ 2 𝑛 {\displaystyle{\displaystyle\lim_{\beta\to\infty}\frac{P^{(\alpha,\beta)}_{n}% \left(x\right)}{P^{(\alpha,\beta)}_{n}\left(-1\right)}=\left(\frac{1-x}{2}% \right)^{n}}}
\lim_{\beta\to\infty}\frac{\JacobipolyP{\alpha}{\beta}{n}@{x}}{\JacobipolyP{\alpha}{\beta}{n}@{-1}} = \left(\frac{1-x}{2}\right)^{n}		 

limit((JacobiP(n, alpha, beta, x))/(JacobiP(n, alpha, beta, - 1)), beta = infinity) = ((1 - x)/(2))^(n)

Limit[Divide[JacobiP[n, \[Alpha], \[Beta], x],JacobiP[n, \[Alpha], \[Beta], - 1]], \[Beta] -> Infinity, GenerateConditions->None] == (Divide[1 - x,2])^(n)
Failure Failure Error Successful [Tested: 27]
18.6.E4 lim Ξ» β†’ ∞ ⁑ C n ( Ξ» ) ⁑ ( x ) C n ( Ξ» ) ⁑ ( 1 ) = x n subscript β†’ πœ† ultraspherical-Gegenbauer-polynomial πœ† 𝑛 π‘₯ ultraspherical-Gegenbauer-polynomial πœ† 𝑛 1 superscript π‘₯ 𝑛 {\displaystyle{\displaystyle\lim_{\lambda\to\infty}\frac{C^{(\lambda)}_{n}% \left(x\right)}{C^{(\lambda)}_{n}\left(1\right)}=x^{n}}}
\lim_{\lambda\to\infty}\frac{\ultrasphpoly{\lambda}{n}@{x}}{\ultrasphpoly{\lambda}{n}@{1}} = x^{n}		 

limit((GegenbauerC(n, lambda, x))/(GegenbauerC(n, lambda, 1)), lambda = infinity) = (x)^(n)

Limit[Divide[GegenbauerC[n, \[Lambda], x],GegenbauerC[n, \[Lambda], 1]], \[Lambda] -> Infinity, GenerateConditions->None] == (x)^(n)
Failure Aborted Successful [Tested: 9] Skipped - Because timed out
18.6.E5 lim Ξ± β†’ ∞ ⁑ L n ( Ξ± ) ⁑ ( Ξ± ⁒ x ) L n ( Ξ± ) ⁑ ( 0 ) = ( 1 - x ) n subscript β†’ 𝛼 Laguerre-polynomial-L 𝛼 𝑛 𝛼 π‘₯ Laguerre-polynomial-L 𝛼 𝑛 0 superscript 1 π‘₯ 𝑛 {\displaystyle{\displaystyle\lim_{\alpha\to\infty}\frac{L^{(\alpha)}_{n}\left(% \alpha x\right)}{L^{(\alpha)}_{n}\left(0\right)}=(1-x)^{n}}}
\lim_{\alpha\to\infty}\frac{\LaguerrepolyL[\alpha]{n}@{\alpha x}}{\LaguerrepolyL[\alpha]{n}@{0}} = (1-x)^{n}		 

limit((LaguerreL(n, alpha, alpha*x))/(LaguerreL(n, alpha, 0)), alpha = infinity) = (1 - x)^(n)

Limit[Divide[LaguerreL[n, \[Alpha], \[Alpha]*x],LaguerreL[n, \[Alpha], 0]], \[Alpha] -> Infinity, GenerateConditions->None] == (1 - x)^(n)
Missing Macro Error Aborted - Skipped - Because timed out
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