Formula:KLS:09.08:28

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C n λ ⁡ ( x ) C n λ ⁡ ( 1 ) = ( 2 ⁢ λ ) n ( λ + 1 2 ) n ⁢ P n ( λ - 1 2 , λ - 1 2 ) ⁡ ( x ) ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 1 Pochhammer-symbol 2 𝜆 𝑛 Pochhammer-symbol 𝜆 1 2 𝑛 Jacobi-polynomial-P 𝜆 1 2 𝜆 1 2 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle\frac{C^{\lambda}_{n}\left(x\right)}% {C^{\lambda}_{n}\left(1\right)}=\frac{{\left(2\lambda\right)_{n}}}{{\left(% \lambda+\frac{1}{2}\right)_{n}}}P^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}_% {n}\left(x\right)}}}

Proof

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Symbols List

C n μ subscript superscript 𝐶 𝜇 𝑛 {\displaystyle{\displaystyle{\displaystyle C^{\mu}_{n}}}}  : ultraspherical/Gegenbauer polynomial : http://dlmf.nist.gov/18.3#T1.t1.r5
( a ) n subscript 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
P n ( α , β ) subscript superscript 𝑃 𝛼 𝛽 𝑛 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}}}}  : Jacobi polynomial : http://dlmf.nist.gov/18.3#T1.t1.r3

Bibliography

Equation in Section 9.8 of KLS.

URL links

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