Formula:KLS:09.08:11

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( 1 - x 2 ) λ - 1 2 ⁢ C n λ ⁡ ( x ) = ( 2 ⁢ λ ) n ⁢ ( - 1 ) n ( λ + 1 2 ) n ⁢ 2 n ⁢ n ! ⁢ ( d d ⁢ x ) n ⁢ [ ( 1 - x 2 ) λ + n - 1 2 ] superscript 1 superscript 𝑥 2 𝜆 1 2 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 Pochhammer-symbol 2 𝜆 𝑛 superscript 1 𝑛 Pochhammer-symbol 𝜆 1 2 𝑛 superscript 2 𝑛 𝑛 superscript 𝑑 𝑑 𝑥 𝑛 delimited-[] superscript 1 superscript 𝑥 2 𝜆 𝑛 1 2 {\displaystyle{\displaystyle{\displaystyle(1-x^{2})^{\lambda-\frac{1}{2}}C^{% \lambda}_{n}\left(x\right)=\frac{{\left(2\lambda\right)_{n}}(-1)^{n}}{{\left(% \lambda+\frac{1}{2}\right)_{n}}2^{n}n!}\left(\frac{d}{dx}\right)^{n}\left[(1-x% ^{2})^{\lambda+n-\frac{1}{2}}\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

Bibliography

Equation in Section 9.8 of KLS.

URL links

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