Formula:KLS:09.08:02

<< Formula:KLS:09.08:01

formula in Jacobi: Special cases

Formula:KLS:09.08:03 >>

${\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(x\right)=\frac{{\left(2\lambda\right)_{n}}}{n!}\,\HyperpFq{2}{1}@@{-n,n+2\lambda}{\lambda+\frac{1}{2}}{\frac{1-x}{2}}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle C^{\mu}_{n}}}}$ : ultraspherical/Gegenbauer polynomial : http://dlmf.nist.gov/18.3#T1.t1.r5
${\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}$ : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
${\displaystyle{\displaystyle{\displaystyle{{}_{p}F_{q}}}}}$ : generalized hypergeometric function : http://dlmf.nist.gov/16.2#E1

Bibliography

Equation in Section 9.8 of KLS.

URL links

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<< Formula:KLS:09.08:01

formula in Jacobi: Special cases

Formula:KLS:09.08:03 >>