Formula:KLS:09.08:39

<< Formula:KLS:09.08:38

formula in Jacobi: Special cases

Formula:KLS:09.08:40 >>

${\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(\cos\theta\right)=\frac{2^{2\lambda}\Gamma\left(\lambda+\frac{1}{2}\right)}{{\pi^{\frac{missing}{missing}}}12\Gamma\left(\lambda+1\right)}\frac{{\left(2\lambda\right)_{n}}}{{\left(\lambda+1\right)_{n}}}\sum_{k=0}^{\infty}\frac{{\left(\lambda\right)_{k}}{\left(1-\lambda\right)_{k}}}{{\left(1+\lambda+n\right)_{k}}k!}\frac{\cos\left((n+k+\lambda)\theta-\frac{1}{2}(k+\lambda)\pi\right)}{(2\sin\theta)^{k+\lambda}}}}}$

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
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
${\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}$ : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\mathrm{sin}}}}$ : sine function : http://dlmf.nist.gov/4.14#E1

Bibliography

Equation in Section 9.8 of KLS.

URL links

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<< Formula:KLS:09.08:38

formula in Jacobi: Special cases

Formula:KLS:09.08:40 >>