Formula:KLS:09.08:38

<< Formula:KLS:09.08:37

formula in Jacobi: Special cases

Formula:KLS:09.08:39 >>

${\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(\cos\theta\right)=\frac{2^{\lambda}\Gamma\left(\lambda+\frac{1}{2}\right)}{{\pi^{\frac{missing}{missing}}}12\Gamma\left(\lambda+1\right)}\frac{{\left(2\lambda\right)_{n}}}{{\left(\lambda+1\right)_{n}}}(\sin\theta)^{-\lambda}\Re{{\mathrm{e}^{-\frac{1}{2}\mathrm{i}\lambda\pi}}{\mathrm{e}^{\mathrm{i}(n+\lambda)\theta}}\HyperpFq{2}{1}@@{\lambda,1-\lambda}{1+\lambda+n}{\frac{{\mathrm{e}^{\mathrm{i}\theta}}}{2\mathrm{i}\sin\theta}}}}}}$

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
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}$ : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
${\displaystyle{\displaystyle{\displaystyle\mathrm{sin}}}}$ : sine function : http://dlmf.nist.gov/4.14#E1
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#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
${\displaystyle{\displaystyle{\displaystyle{{}_{p}F_{q}}}}}$ : 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:09.08:37

formula in Jacobi: Special cases

Formula:KLS:09.08:39 >>