Formula:KLS:09.08:31

<< Formula:KLS:09.08:30

formula in Jacobi: Special cases

Formula:KLS:09.08:32 >>

${\displaystyle{\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{\lambda+n}{\lambda}\frac{n!}{{\left(2\lambda\right)_{n}}}r^{n}C^{\lambda}_{n}\left(x\right)C^{\lambda}_{n}\left(y\right)=\frac{1-r^{2}}{(1-2rxy+r^{2})^{\lambda+1}}\HyperpFq{2}{1}@@{\frac{1}{2}(\lambda+1),\frac{1}{2}(\lambda+2)}{\lambda+\frac{1}{2}}{\frac{4r^{2}(1-x^{2})(1-y^{2})}{(1-2rxy+r^{2})^{2}}}(r\in(-1,1),\;x,y\in[-1,1])}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}$ : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
${\displaystyle{\displaystyle{\displaystyle C^{\mu}_{n}}}}$ : ultraspherical/Gegenbauer polynomial : http://dlmf.nist.gov/18.3#T1.t1.r5
${\displaystyle{\displaystyle{\displaystyle{{}_{p}F_{q}}}}}$ : generalized hypergeometric function : http://dlmf.nist.gov/16.2#E1
${\displaystyle{\displaystyle{\displaystyle\in}}}$ : element of : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r9

Bibliography

Equation in Section 9.8 of KLS.

URL links

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<< Formula:KLS:09.08:30

formula in Jacobi: Special cases

Formula:KLS:09.08:32 >>