Formula:KLS:09.08:33

<< Formula:KLS:09.08:32

formula in Jacobi: Special cases

Formula:KLS:09.08:34 >>

${\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(\cos\theta\right)={\mathrm{e}^{\mathrm{i}n\theta}}\frac{{\left(\lambda\right)_{n}}}{n!}\HyperpFq{2}{1}@@{-n,\lambda}{1-\lambda-n}{{\mathrm{e}^{-2\mathrm{i}\theta}}}}}}$

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
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}$ : 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:09.08:32

formula in Jacobi: Special cases

Formula:KLS:09.08:34 >>