Formula:KLS:09.08:31

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∑ n = 0 ∞ λ + n λ n ! ( 2 ⁢ λ ) n r n C n λ ⁡ ( x ) C n λ ⁡ ( y ) = 1 - r 2 ( 1 - 2 ⁢ r ⁢ x ⁢ y + r 2 ) λ + 1 \HyperpFq 21 @ @ 1 2 ( λ + 1 ) , 1 2 ( λ + 2 ) λ + 1 2 4 ⁢ r 2 ⁢ ( 1 - x 2 ) ⁢ ( 1 - y 2 ) ( 1 - 2 ⁢ r ⁢ x ⁢ y + r 2 ) 2 ( r ∈ ( - 1 , 1 ) , x , y ∈ [ - 1 , 1 ] ) fragments superscript subscript 𝑛 0 𝜆 𝑛 𝜆 𝑛 Pochhammer-symbol 2 𝜆 𝑛 superscript 𝑟 𝑛 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑦 1 superscript 𝑟 2 superscript 1 2 𝑟 𝑥 𝑦 superscript 𝑟 2 𝜆 1 \HyperpFq 21 @ @ 1 2 fragments ( λ 1 ) , 1 2 fragments ( λ 2 ) λ 1 2 4 superscript 𝑟 2 1 superscript 𝑥 2 1 superscript 𝑦 2 superscript 1 2 𝑟 𝑥 𝑦 superscript 𝑟 2 2 fragments ( r fragments ( 1 , 1 ) , x , y fragments [ 1 , 1 ] ) {\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

Σ Σ {\displaystyle{\displaystyle{\displaystyle\Sigma}}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
( a ) n subscript 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
C n μ subscript superscript 𝐶 𝜇 𝑛 {\displaystyle{\displaystyle{\displaystyle C^{\mu}_{n}}}}  : ultraspherical/Gegenbauer polynomial : http://dlmf.nist.gov/18.3#T1.t1.r5
F q p subscript subscript 𝐹 𝑞 𝑝 {\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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