Formula:KLS:09.08:24

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formula in Jacobi: Special cases

Formula:KLS:09.08:25 >>

${\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(x\right)=\sum_{\ell=0}^{\lfloor n/2\rfloor}\frac{(-1)^{\ell}{\left(\lambda\right)_{n-\ell}}}{\ell!\;(n-2\ell)!}(2x)^{n-2\ell}}}}$

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
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}$ : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii

Bibliography

Equation in Section 9.8 of KLS.

URL links

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formula in Jacobi: Special cases

Formula:KLS:09.08:25 >>