Orthogonal Polynomials - 18.11 Relations to Other Functions

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DLMF Formula Constraints Maple Mathematica Symbolic
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18.11.E1 𝖯 n m ⁑ ( x ) = ( 1 2 ) m ⁒ ( - 2 ) m ⁒ ( 1 - x 2 ) 1 2 ⁒ m ⁒ C n - m ( m + 1 2 ) ⁑ ( x ) Ferrers-Legendre-P-first-kind π‘š 𝑛 π‘₯ Pochhammer 1 2 π‘š superscript 2 π‘š superscript 1 superscript π‘₯ 2 1 2 π‘š ultraspherical-Gegenbauer-polynomial π‘š 1 2 𝑛 π‘š π‘₯ {\displaystyle{\displaystyle\mathsf{P}^{m}_{n}\left(x\right)={\left(\tfrac{1}{% 2}\right)_{m}}(-2)^{m}(1-x^{2})^{\frac{1}{2}m}C^{(m+\frac{1}{2})}_{n-m}\left(x% \right)}}
\FerrersP[m]{n}@{x} = \Pochhammersym{\tfrac{1}{2}}{m}(-2)^{m}(1-x^{2})^{\frac{1}{2}m}\ultrasphpoly{m+\frac{1}{2}}{n-m}@{x}		 0 ≀ m , m ≀ n , | ( 1 2 - 1 2 ⁒ x ) | < 1 formulae-sequence 0 π‘š formulae-sequence π‘š 𝑛 1 2 1 2 π‘₯ 1 {\displaystyle{\displaystyle 0\leq m,m\leq n,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}} 
LegendreP(n, m, x) = pochhammer((1)/(2), m)*(- 2)^(m)*(1 - (x)^(2))^((1)/(2)*m)* GegenbauerC(n - m, m +(1)/(2), x)

LegendreP[n, m, x] == Pochhammer[Divide[1,2], m]*(- 2)^(m)*(1 - (x)^(2))^(Divide[1,2]*m)* GegenbauerC[n - m, m +Divide[1,2], x]
Failure Failure Successful [Tested: 18] Successful [Tested: 18]
18.11.E1 ( 1 2 ) m ⁒ ( - 2 ) m ⁒ ( 1 - x 2 ) 1 2 ⁒ m ⁒ C n - m ( m + 1 2 ) ⁑ ( x ) = ( n + 1 ) m ⁒ ( - 2 ) - m ⁒ ( 1 - x 2 ) 1 2 ⁒ m ⁒ P n - m ( m , m ) ⁑ ( x ) Pochhammer 1 2 π‘š superscript 2 π‘š superscript 1 superscript π‘₯ 2 1 2 π‘š ultraspherical-Gegenbauer-polynomial π‘š 1 2 𝑛 π‘š π‘₯ Pochhammer 𝑛 1 π‘š superscript 2 π‘š superscript 1 superscript π‘₯ 2 1 2 π‘š Jacobi-polynomial-P π‘š π‘š 𝑛 π‘š π‘₯ {\displaystyle{\displaystyle{\left(\tfrac{1}{2}\right)_{m}}(-2)^{m}(1-x^{2})^{% \frac{1}{2}m}C^{(m+\frac{1}{2})}_{n-m}\left(x\right)={\left(n+1\right)_{m}}(-2% )^{-m}(1-x^{2})^{\frac{1}{2}m}P^{(m,m)}_{n-m}\left(x\right)}}
\Pochhammersym{\tfrac{1}{2}}{m}(-2)^{m}(1-x^{2})^{\frac{1}{2}m}\ultrasphpoly{m+\frac{1}{2}}{n-m}@{x} = \Pochhammersym{n+1}{m}(-2)^{-m}(1-x^{2})^{\frac{1}{2}m}\JacobipolyP{m}{m}{n-m}@{x}		 0 ≀ m , m ≀ n , | ( 1 2 - 1 2 ⁒ x ) | < 1 formulae-sequence 0 π‘š formulae-sequence π‘š 𝑛 1 2 1 2 π‘₯ 1 {\displaystyle{\displaystyle 0\leq m,m\leq n,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}} 
pochhammer((1)/(2), m)*(- 2)^(m)*(1 - (x)^(2))^((1)/(2)*m)* GegenbauerC(n - m, m +(1)/(2), x) = pochhammer(n + 1, m)*(- 2)^(- m)*(1 - (x)^(2))^((1)/(2)*m)* JacobiP(n - m, m, m, x)

Pochhammer[Divide[1,2], m]*(- 2)^(m)*(1 - (x)^(2))^(Divide[1,2]*m)* GegenbauerC[n - m, m +Divide[1,2], x] == Pochhammer[n + 1, m]*(- 2)^(- m)*(1 - (x)^(2))^(Divide[1,2]*m)* JacobiP[n - m, m, m, x]
Successful Failure Skip - symbolical successful subtest Successful [Tested: 18]
18.11.E2 L n ( Ξ± ) ⁑ ( x ) = ( Ξ± + 1 ) n n ! ⁒ M ⁑ ( - n , Ξ± + 1 , x ) Laguerre-polynomial-L 𝛼 𝑛 π‘₯ Pochhammer 𝛼 1 𝑛 𝑛 Kummer-confluent-hypergeometric-M 𝑛 𝛼 1 π‘₯ {\displaystyle{\displaystyle L^{(\alpha)}_{n}\left(x\right)=\frac{{\left(% \alpha+1\right)_{n}}}{n!}M\left(-n,\alpha+1,x\right)}}
\LaguerrepolyL[\alpha]{n}@{x} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}\KummerconfhyperM@{-n}{\alpha+1}{x}		 

LaguerreL(n, alpha, x) = (pochhammer(alpha + 1, n))/(factorial(n))*KummerM(- n, alpha + 1, x)

LaguerreL[n, \[Alpha], x] == Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*Hypergeometric1F1[- n, \[Alpha]+ 1, x]
Missing Macro Error Successful Skip - symbolical successful subtest Successful [Tested: 27]
18.11.E2 ( Ξ± + 1 ) n n ! ⁒ M ⁑ ( - n , Ξ± + 1 , x ) = ( - 1 ) n n ! ⁒ U ⁑ ( - n , Ξ± + 1 , x ) Pochhammer 𝛼 1 𝑛 𝑛 Kummer-confluent-hypergeometric-M 𝑛 𝛼 1 π‘₯ superscript 1 𝑛 𝑛 Kummer-confluent-hypergeometric-U 𝑛 𝛼 1 π‘₯ {\displaystyle{\displaystyle\frac{{\left(\alpha+1\right)_{n}}}{n!}M\left(-n,% \alpha+1,x\right)=\frac{(-1)^{n}}{n!}U\left(-n,\alpha+1,x\right)}}
\frac{\Pochhammersym{\alpha+1}{n}}{n!}\KummerconfhyperM@{-n}{\alpha+1}{x} = \frac{(-1)^{n}}{n!}\KummerconfhyperU@{-n}{\alpha+1}{x}		 

(pochhammer(alpha + 1, n))/(factorial(n))*KummerM(- n, alpha + 1, x) = ((- 1)^(n))/(factorial(n))*KummerU(- n, alpha + 1, x)

Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*Hypergeometric1F1[- n, \[Alpha]+ 1, x] == Divide[(- 1)^(n),(n)!]*HypergeometricU[- n, \[Alpha]+ 1, x]
Failure Failure Successful [Tested: 27] Successful [Tested: 27]
18.11.E2 ( - 1 ) n n ! ⁒ U ⁑ ( - n , Ξ± + 1 , x ) = ( Ξ± + 1 ) n n ! ⁒ x - 1 2 ⁒ ( Ξ± + 1 ) ⁒ e 1 2 ⁒ x ⁒ M n + 1 2 ⁒ ( Ξ± + 1 ) , 1 2 ⁒ Ξ± ⁑ ( x ) superscript 1 𝑛 𝑛 Kummer-confluent-hypergeometric-U 𝑛 𝛼 1 π‘₯ Pochhammer 𝛼 1 𝑛 𝑛 superscript π‘₯ 1 2 𝛼 1 superscript 𝑒 1 2 π‘₯ Whittaker-confluent-hypergeometric-M 𝑛 1 2 𝛼 1 1 2 𝛼 π‘₯ {\displaystyle{\displaystyle\frac{(-1)^{n}}{n!}U\left(-n,\alpha+1,x\right)=% \frac{{\left(\alpha+1\right)_{n}}}{n!}x^{-\frac{1}{2}(\alpha+1)}e^{\frac{1}{2}% x}M_{n+\frac{1}{2}(\alpha+1),\frac{1}{2}\alpha}\left(x\right)}}
\frac{(-1)^{n}}{n!}\KummerconfhyperU@{-n}{\alpha+1}{x} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}x^{-\frac{1}{2}(\alpha+1)}e^{\frac{1}{2}x}\WhittakerconfhyperM{n+\frac{1}{2}(\alpha+1)}{\frac{1}{2}\alpha}@{x}		 

((- 1)^(n))/(factorial(n))*KummerU(- n, alpha + 1, x) = (pochhammer(alpha + 1, n))/(factorial(n))*(x)^(-(1)/(2)*(alpha + 1))* exp((1)/(2)*x)*WhittakerM(n +(1)/(2)*(alpha + 1), (1)/(2)*alpha, x)

Divide[(- 1)^(n),(n)!]*HypergeometricU[- n, \[Alpha]+ 1, x] == Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*(x)^(-Divide[1,2]*(\[Alpha]+ 1))* Exp[Divide[1,2]*x]*WhittakerM[n +Divide[1,2]*(\[Alpha]+ 1), Divide[1,2]*\[Alpha], x]
Failure Failure Successful [Tested: 27] Successful [Tested: 27]
18.11.E2 ( Ξ± + 1 ) n n ! ⁒ x - 1 2 ⁒ ( Ξ± + 1 ) ⁒ e 1 2 ⁒ x ⁒ M n + 1 2 ⁒ ( Ξ± + 1 ) , 1 2 ⁒ Ξ± ⁑ ( x ) = ( - 1 ) n n ! ⁒ x - 1 2 ⁒ ( Ξ± + 1 ) ⁒ e 1 2 ⁒ x ⁒ W n + 1 2 ⁒ ( Ξ± + 1 ) , 1 2 ⁒ Ξ± ⁑ ( x ) Pochhammer 𝛼 1 𝑛 𝑛 superscript π‘₯ 1 2 𝛼 1 superscript 𝑒 1 2 π‘₯ Whittaker-confluent-hypergeometric-M 𝑛 1 2 𝛼 1 1 2 𝛼 π‘₯ superscript 1 𝑛 𝑛 superscript π‘₯ 1 2 𝛼 1 superscript 𝑒 1 2 π‘₯ Whittaker-confluent-hypergeometric-W 𝑛 1 2 𝛼 1 1 2 𝛼 π‘₯ {\displaystyle{\displaystyle\frac{{\left(\alpha+1\right)_{n}}}{n!}x^{-\frac{1}% {2}(\alpha+1)}e^{\frac{1}{2}x}M_{n+\frac{1}{2}(\alpha+1),\frac{1}{2}\alpha}% \left(x\right)=\frac{(-1)^{n}}{n!}x^{-\frac{1}{2}(\alpha+1)}e^{\frac{1}{2}x}W_% {n+\frac{1}{2}(\alpha+1),\frac{1}{2}\alpha}\left(x\right)}}
\frac{\Pochhammersym{\alpha+1}{n}}{n!}x^{-\frac{1}{2}(\alpha+1)}e^{\frac{1}{2}x}\WhittakerconfhyperM{n+\frac{1}{2}(\alpha+1)}{\frac{1}{2}\alpha}@{x} = \frac{(-1)^{n}}{n!}x^{-\frac{1}{2}(\alpha+1)}e^{\frac{1}{2}x}\WhittakerconfhyperW{n+\frac{1}{2}(\alpha+1)}{\frac{1}{2}\alpha}@{x}		 

(pochhammer(alpha + 1, n))/(factorial(n))*(x)^(-(1)/(2)*(alpha + 1))* exp((1)/(2)*x)*WhittakerM(n +(1)/(2)*(alpha + 1), (1)/(2)*alpha, x) = ((- 1)^(n))/(factorial(n))*(x)^(-(1)/(2)*(alpha + 1))* exp((1)/(2)*x)*WhittakerW(n +(1)/(2)*(alpha + 1), (1)/(2)*alpha, x)

Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*(x)^(-Divide[1,2]*(\[Alpha]+ 1))* Exp[Divide[1,2]*x]*WhittakerM[n +Divide[1,2]*(\[Alpha]+ 1), Divide[1,2]*\[Alpha], x] == Divide[(- 1)^(n),(n)!]*(x)^(-Divide[1,2]*(\[Alpha]+ 1))* Exp[Divide[1,2]*x]*WhittakerW[n +Divide[1,2]*(\[Alpha]+ 1), Divide[1,2]*\[Alpha], x]
Failure Failure Successful [Tested: 27] Successful [Tested: 27]
18.11.E3 H n ⁑ ( x ) = 2 n ⁒ U ⁑ ( - 1 2 ⁒ n , 1 2 , x 2 ) Hermite-polynomial-H 𝑛 π‘₯ superscript 2 𝑛 Kummer-confluent-hypergeometric-U 1 2 𝑛 1 2 superscript π‘₯ 2 {\displaystyle{\displaystyle H_{n}\left(x\right)=2^{n}U\left(-\tfrac{1}{2}n,% \tfrac{1}{2},x^{2}\right)}}
\HermitepolyH{n}@{x} = 2^{n}\KummerconfhyperU@{-\tfrac{1}{2}n}{\tfrac{1}{2}}{x^{2}}		 

HermiteH(n, x) = (2)^(n)* KummerU(-(1)/(2)*n, (1)/(2), (x)^(2))

HermiteH[n, x] == (2)^(n)* HypergeometricU[-Divide[1,2]*n, Divide[1,2], (x)^(2)]
Failure Failure Successful [Tested: 9] Successful [Tested: 9]
18.11.E3 2 n ⁒ U ⁑ ( - 1 2 ⁒ n , 1 2 , x 2 ) = 2 n ⁒ x ⁒ U ⁑ ( - 1 2 ⁒ n + 1 2 , 3 2 , x 2 ) superscript 2 𝑛 Kummer-confluent-hypergeometric-U 1 2 𝑛 1 2 superscript π‘₯ 2 superscript 2 𝑛 π‘₯ Kummer-confluent-hypergeometric-U 1 2 𝑛 1 2 3 2 superscript π‘₯ 2 {\displaystyle{\displaystyle 2^{n}U\left(-\tfrac{1}{2}n,\tfrac{1}{2},x^{2}% \right)=2^{n}xU\left(-\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{3}{2},x^{2}\right)}}
2^{n}\KummerconfhyperU@{-\tfrac{1}{2}n}{\tfrac{1}{2}}{x^{2}} = 2^{n}x\KummerconfhyperU@{-\tfrac{1}{2}n+\tfrac{1}{2}}{\tfrac{3}{2}}{x^{2}}		 

(2)^(n)* KummerU(-(1)/(2)*n, (1)/(2), (x)^(2)) = (2)^(n)* x*KummerU(-(1)/(2)*n +(1)/(2), (3)/(2), (x)^(2))

(2)^(n)* HypergeometricU[-Divide[1,2]*n, Divide[1,2], (x)^(2)] == (2)^(n)* x*HypergeometricU[-Divide[1,2]*n +Divide[1,2], Divide[3,2], (x)^(2)]
Failure Failure Successful [Tested: 9] Successful [Tested: 9]
18.11.E3 2 n ⁒ x ⁒ U ⁑ ( - 1 2 ⁒ n + 1 2 , 3 2 , x 2 ) = 2 1 2 ⁒ n ⁒ e 1 2 ⁒ x 2 ⁒ U ⁑ ( - n - 1 2 , 2 1 2 ⁒ x ) superscript 2 𝑛 π‘₯ Kummer-confluent-hypergeometric-U 1 2 𝑛 1 2 3 2 superscript π‘₯ 2 superscript 2 1 2 𝑛 superscript 𝑒 1 2 superscript π‘₯ 2 parabolic-U 𝑛 1 2 superscript 2 1 2 π‘₯ {\displaystyle{\displaystyle 2^{n}xU\left(-\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{3% }{2},x^{2}\right)=2^{\frac{1}{2}n}e^{\frac{1}{2}x^{2}}U\left(-n-\tfrac{1}{2},2% ^{\frac{1}{2}}x\right)}}
2^{n}x\KummerconfhyperU@{-\tfrac{1}{2}n+\tfrac{1}{2}}{\tfrac{3}{2}}{x^{2}} = 2^{\frac{1}{2}n}e^{\frac{1}{2}x^{2}}\paraU@{-n-\tfrac{1}{2}}{2^{\frac{1}{2}}x}		 

(2)^(n)* x*KummerU(-(1)/(2)*n +(1)/(2), (3)/(2), (x)^(2)) = (2)^((1)/(2)*n)* exp((1)/(2)*(x)^(2))*CylinderU(- n -(1)/(2), (2)^((1)/(2))* x)

(2)^(n)* x*HypergeometricU[-Divide[1,2]*n +Divide[1,2], Divide[3,2], (x)^(2)] == (2)^(Divide[1,2]*n)* Exp[Divide[1,2]*(x)^(2)]*ParabolicCylinderD[- 1/2 -(- n -Divide[1,2]), (2)^(Divide[1,2])* x]
Failure Failure Successful [Tested: 9] Successful [Tested: 9]
18.11.E4 2 1 2 ⁒ n ⁒ U ⁑ ( - 1 2 ⁒ n , 1 2 , 1 2 ⁒ x 2 ) = 2 1 2 ⁒ ( n - 1 ) ⁒ x ⁒ U ⁑ ( - 1 2 ⁒ n + 1 2 , 3 2 , 1 2 ⁒ x 2 ) superscript 2 1 2 𝑛 Kummer-confluent-hypergeometric-U 1 2 𝑛 1 2 1 2 superscript π‘₯ 2 superscript 2 1 2 𝑛 1 π‘₯ Kummer-confluent-hypergeometric-U 1 2 𝑛 1 2 3 2 1 2 superscript π‘₯ 2 {\displaystyle{\displaystyle 2^{\frac{1}{2}n}U\left(-\tfrac{1}{2}n,\tfrac{1}{2% },\tfrac{1}{2}x^{2}\right)=2^{\frac{1}{2}(n-1)}xU\left(-\tfrac{1}{2}n+\tfrac{1% }{2},\tfrac{3}{2},\tfrac{1}{2}x^{2}\right)}}
2^{\frac{1}{2}n}\KummerconfhyperU@{-\tfrac{1}{2}n}{\tfrac{1}{2}}{\tfrac{1}{2}x^{2}} = 2^{\frac{1}{2}(n-1)}x\KummerconfhyperU@{-\tfrac{1}{2}n+\tfrac{1}{2}}{\tfrac{3}{2}}{\tfrac{1}{2}x^{2}}		 

(2)^((1)/(2)*n)* KummerU(-(1)/(2)*n, (1)/(2), (1)/(2)*(x)^(2)) = (2)^((1)/(2)*(n - 1))* x*KummerU(-(1)/(2)*n +(1)/(2), (3)/(2), (1)/(2)*(x)^(2))

(2)^(Divide[1,2]*n)* HypergeometricU[-Divide[1,2]*n, Divide[1,2], Divide[1,2]*(x)^(2)] == (2)^(Divide[1,2]*(n - 1))* x*HypergeometricU[-Divide[1,2]*n +Divide[1,2], Divide[3,2], Divide[1,2]*(x)^(2)]
Failure Failure Successful [Tested: 9] Successful [Tested: 9]
18.11.E4 2 1 2 ⁒ ( n - 1 ) ⁒ x ⁒ U ⁑ ( - 1 2 ⁒ n + 1 2 , 3 2 , 1 2 ⁒ x 2 ) = e 1 4 ⁒ x 2 ⁒ U ⁑ ( - n - 1 2 , x ) superscript 2 1 2 𝑛 1 π‘₯ Kummer-confluent-hypergeometric-U 1 2 𝑛 1 2 3 2 1 2 superscript π‘₯ 2 superscript 𝑒 1 4 superscript π‘₯ 2 parabolic-U 𝑛 1 2 π‘₯ {\displaystyle{\displaystyle 2^{\frac{1}{2}(n-1)}xU\left(-\tfrac{1}{2}n+\tfrac% {1}{2},\tfrac{3}{2},\tfrac{1}{2}x^{2}\right)=e^{\tfrac{1}{4}x^{2}}U\left(-n-% \tfrac{1}{2},x\right)}}
2^{\frac{1}{2}(n-1)}x\KummerconfhyperU@{-\tfrac{1}{2}n+\tfrac{1}{2}}{\tfrac{3}{2}}{\tfrac{1}{2}x^{2}} = e^{\tfrac{1}{4}x^{2}}\paraU@{-n-\tfrac{1}{2}}{x}		 

(2)^((1)/(2)*(n - 1))* x*KummerU(-(1)/(2)*n +(1)/(2), (3)/(2), (1)/(2)*(x)^(2)) = exp((1)/(4)*(x)^(2))*CylinderU(- n -(1)/(2), x)

(2)^(Divide[1,2]*(n - 1))* x*HypergeometricU[-Divide[1,2]*n +Divide[1,2], Divide[3,2], Divide[1,2]*(x)^(2)] == Exp[Divide[1,4]*(x)^(2)]*ParabolicCylinderD[- 1/2 -(- n -Divide[1,2]), x]
Failure Failure Successful [Tested: 9] Successful [Tested: 9]
18.11.E5 lim n β†’ ∞ ⁑ 1 n Ξ± ⁒ P n ( Ξ± , Ξ² ) ⁑ ( 1 - z 2 2 ⁒ n 2 ) = lim n β†’ ∞ ⁑ 1 n Ξ± ⁒ P n ( Ξ± , Ξ² ) ⁑ ( cos ⁑ z n ) subscript β†’ 𝑛 1 superscript 𝑛 𝛼 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 superscript 𝑧 2 2 superscript 𝑛 2 subscript β†’ 𝑛 1 superscript 𝑛 𝛼 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑧 𝑛 {\displaystyle{\displaystyle\lim_{n\to\infty}\frac{1}{n^{\alpha}}P^{(\alpha,% \beta)}_{n}\left(1-\frac{z^{2}}{2n^{2}}\right)=\lim_{n\to\infty}\frac{1}{n^{% \alpha}}P^{(\alpha,\beta)}_{n}\left(\cos\frac{z}{n}\right)}}
\lim_{n\to\infty}\frac{1}{n^{\alpha}}\JacobipolyP{\alpha}{\beta}{n}@{1-\frac{z^{2}}{2n^{2}}} = \lim_{n\to\infty}\frac{1}{n^{\alpha}}\JacobipolyP{\alpha}{\beta}{n}@{\cos@@{\frac{z}{n}}}		 

limit((1)/((n)^(alpha))*JacobiP(n, alpha, beta, 1 -((z)^(2))/(2*(n)^(2))), n = infinity) = limit((1)/((n)^(alpha))*JacobiP(n, alpha, beta, cos((z)/(n))), n = infinity)

Limit[Divide[1,(n)^\[Alpha]]*JacobiP[n, \[Alpha], \[Beta], 1 -Divide[(z)^(2),2*(n)^(2)]], n -> Infinity, GenerateConditions->None] == Limit[Divide[1,(n)^\[Alpha]]*JacobiP[n, \[Alpha], \[Beta], Cos[Divide[z,n]]], n -> Infinity, GenerateConditions->None]
Failure Aborted Error Skipped - Because timed out
18.11.E5 lim n β†’ ∞ ⁑ 1 n Ξ± ⁒ P n ( Ξ± , Ξ² ) ⁑ ( cos ⁑ z n ) = 2 Ξ± z Ξ± ⁒ J Ξ± ⁑ ( z ) subscript β†’ 𝑛 1 superscript 𝑛 𝛼 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑧 𝑛 superscript 2 𝛼 superscript 𝑧 𝛼 Bessel-J 𝛼 𝑧 {\displaystyle{\displaystyle\lim_{n\to\infty}\frac{1}{n^{\alpha}}P^{(\alpha,% \beta)}_{n}\left(\cos\frac{z}{n}\right)=\frac{2^{\alpha}}{z^{\alpha}}J_{\alpha% }\left(z\right)}}
\lim_{n\to\infty}\frac{1}{n^{\alpha}}\JacobipolyP{\alpha}{\beta}{n}@{\cos@@{\frac{z}{n}}} = \frac{2^{\alpha}}{z^{\alpha}}\BesselJ{\alpha}@{z}		 β„œ ⁑ ( ( Ξ± ) + k + 1 ) > 0 𝛼 π‘˜ 1 0 {\displaystyle{\displaystyle\Re((\alpha)+k+1)>0}} 
limit((1)/((n)^(alpha))*JacobiP(n, alpha, beta, cos((z)/(n))), n = infinity) = ((2)^(alpha))/((z)^(alpha))*BesselJ(alpha, z)

Limit[Divide[1,(n)^\[Alpha]]*JacobiP[n, \[Alpha], \[Beta], Cos[Divide[z,n]]], n -> Infinity, GenerateConditions->None] == Divide[(2)^\[Alpha],(z)^\[Alpha]]*BesselJ[\[Alpha], z]
Failure Aborted Error Skipped - Because timed out
18.11.E6 lim n β†’ ∞ ⁑ 1 n Ξ± ⁒ L n ( Ξ± ) ⁑ ( z n ) = 1 z 1 2 ⁒ Ξ± ⁒ J Ξ± ⁑ ( 2 ⁒ z 1 2 ) subscript β†’ 𝑛 1 superscript 𝑛 𝛼 Laguerre-polynomial-L 𝛼 𝑛 𝑧 𝑛 1 superscript 𝑧 1 2 𝛼 Bessel-J 𝛼 2 superscript 𝑧 1 2 {\displaystyle{\displaystyle\lim_{n\to\infty}\frac{1}{n^{\alpha}}L^{(\alpha)}_% {n}\left(\frac{z}{n}\right)=\frac{1}{z^{\frac{1}{2}\alpha}}J_{\alpha}\left(2z^% {\frac{1}{2}}\right)}}
\lim_{n\to\infty}\frac{1}{n^{\alpha}}\LaguerrepolyL[\alpha]{n}@{\frac{z}{n}} = \frac{1}{z^{\frac{1}{2}\alpha}}\BesselJ{\alpha}@{2z^{\frac{1}{2}}}		 β„œ ⁑ ( ( Ξ± ) + k + 1 ) > 0 𝛼 π‘˜ 1 0 {\displaystyle{\displaystyle\Re((\alpha)+k+1)>0}} 
limit((1)/((n)^(alpha))*LaguerreL(n, alpha, (z)/(n)), n = infinity) = (1)/((z)^((1)/(2)*alpha))*BesselJ(alpha, 2*(z)^((1)/(2)))

Limit[Divide[1,(n)^\[Alpha]]*LaguerreL[n, \[Alpha], Divide[z,n]], n -> Infinity, GenerateConditions->None] == Divide[1,(z)^(Divide[1,2]*\[Alpha])]*BesselJ[\[Alpha], 2*(z)^(Divide[1,2])]
Missing Macro Error Aborted -
Failed [21 / 21]
Result: Plus[Complex[-0.5130891006146308, 0.11628471920726866], Limit[Times[Power[n, -1.5], LaguerreL[n, 1.5, Times[Complex[0.8660254037844387, 0.49999999999999994], Power[n, -1]]]], Rule[n, DirectedInfinity[1]], Rule[GenerateConditions, None]]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ±, 1.5]}


Result: Plus[Complex[-0.5517607501957961, 0.2594860904083832], Limit[Times[Power[n, -0.5], LaguerreL[n, 0.5, Times[Complex[0.8660254037844387, 0.49999999999999994], Power[n, -1]]]], Rule[n, DirectedInfinity[1]], Rule[GenerateConditions, None]]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ±, 0.5]}

... skip entries to safe data
18.11.E7 lim n β†’ ∞ ⁑ ( - 1 ) n ⁒ n 1 2 2 2 ⁒ n ⁒ n ! ⁒ H 2 ⁒ n ⁑ ( z 2 ⁒ n 1 2 ) = 1 Ο€ 1 2 ⁒ cos ⁑ z subscript β†’ 𝑛 superscript 1 𝑛 superscript 𝑛 1 2 superscript 2 2 𝑛 𝑛 Hermite-polynomial-H 2 𝑛 𝑧 2 superscript 𝑛 1 2 1 superscript πœ‹ 1 2 𝑧 {\displaystyle{\displaystyle\lim_{n\to\infty}\frac{(-1)^{n}n^{\frac{1}{2}}}{2^% {2n}n!}H_{2n}\left(\frac{z}{2n^{\frac{1}{2}}}\right)=\frac{1}{\pi^{\frac{1}{2}% }}\cos z}}
\lim_{n\to\infty}\frac{(-1)^{n}n^{\frac{1}{2}}}{2^{2n}n!}\HermitepolyH{2n}@{\frac{z}{2n^{\frac{1}{2}}}} = \frac{1}{\pi^{\frac{1}{2}}}\cos@@{z}		 

limit(((- 1)^(n)* (n)^((1)/(2)))/((2)^(2*n)* factorial(n))*HermiteH(2*n, (z)/(2*(n)^((1)/(2)))), n = infinity) = (1)/((Pi)^((1)/(2)))*cos(z)

Limit[Divide[(- 1)^(n)* (n)^(Divide[1,2]),(2)^(2*n)* (n)!]*HermiteH[2*n, Divide[z,2*(n)^(Divide[1,2])]], n -> Infinity, GenerateConditions->None] == Divide[1,(Pi)^(Divide[1,2])]*Cos[z]
Failure Aborted Successful [Tested: 7] Skipped - Because timed out
18.11.E8 lim n β†’ ∞ ⁑ ( - 1 ) n 2 2 ⁒ n ⁒ n ! ⁒ H 2 ⁒ n + 1 ⁑ ( z 2 ⁒ n 1 2 ) = 2 Ο€ 1 2 ⁒ sin ⁑ z subscript β†’ 𝑛 superscript 1 𝑛 superscript 2 2 𝑛 𝑛 Hermite-polynomial-H 2 𝑛 1 𝑧 2 superscript 𝑛 1 2 2 superscript πœ‹ 1 2 𝑧 {\displaystyle{\displaystyle\lim_{n\to\infty}\frac{(-1)^{n}}{2^{2n}n!}H_{2n+1}% \left(\frac{z}{2n^{\frac{1}{2}}}\right)=\frac{2}{\pi^{\frac{1}{2}}}\sin z}}
\lim_{n\to\infty}\frac{(-1)^{n}}{2^{2n}n!}\HermitepolyH{2n+1}@{\frac{z}{2n^{\frac{1}{2}}}} = \frac{2}{\pi^{\frac{1}{2}}}\sin@@{z}		 

limit(((- 1)^(n))/((2)^(2*n)* factorial(n))*HermiteH(2*n + 1, (z)/(2*(n)^((1)/(2)))), n = infinity) = (2)/((Pi)^((1)/(2)))*sin(z)

Limit[Divide[(- 1)^(n),(2)^(2*n)* (n)!]*HermiteH[2*n + 1, Divide[z,2*(n)^(Divide[1,2])]], n -> Infinity, GenerateConditions->None] == Divide[2,(Pi)^(Divide[1,2])]*Sin[z]
Failure Aborted Error Skipped - Because timed out
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