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formula in Jacobi: Special cases

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${\displaystyle{\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{{\left(\alpha+\beta+1\right)_{n}}n!}{{\left(\alpha+1\right)_{n}}{\left(\beta+1\right)_{n}}}r^{n}P^{(\alpha,\beta)}_{n}\left(x\right)P^{(\alpha,\beta)}_{n}\left(y\right)=\frac{1}{(1+r)^{\alpha+\beta+1}}\AppellFiv@{\frac{1}{2}(\alpha+\beta+1)}{\frac{1}{2}(\alpha+\beta+2)}{\alpha+1}{\beta+1}{\frac{r(1-x)(1-y)}{(1+r)^{2}}}{\frac{r(1+x)(1+y)}{(1+r)^{2}}}}}}$

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
${\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}}}}$ : Jacobi polynomial : http://dlmf.nist.gov/18.3#T1.t1.r3
${\displaystyle{\displaystyle{\displaystyle{F_{4}}}}}$ : Appell function ${\displaystyle{\displaystyle{\displaystyle F_{4}}}}$ : http://dlmf.nist.gov/16.13#E4

Bibliography

Equation in Section of KLS.

URL links

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formula in Jacobi: Special cases

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