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Latest revision as of 00:34, 6 March 2017


∑ n = 0 ∞ 2 ⁢ n + α + β + 1 n + α + β + 1 ⁢ ( α + β + 2 ) n ⁢ n ! ( α + 1 ) n ⁢ ( β + 1 ) n ⁢ r n ⁢ P n ( α , β ) ⁡ ( x ) ⁢ P n ( α , β ) ⁡ ( y ) = 1 - r ( 1 + r ) α + β + 2 ⁢ \AppellFiv ⁢ @ ⁢ 1 2 ⁢ ( α + β + 2 ) ⁢ 1 2 ⁢ ( α + β + 3 ) ⁢ α + 1 ⁢ β + 1 ⁢ r ⁢ ( 1 - x ) ⁢ ( 1 - y ) ( 1 + r ) 2 ⁢ r ⁢ ( 1 + x ) ⁢ ( 1 + y ) ( 1 + r ) 2 superscript subscript 𝑛 0 2 𝑛 𝛼 𝛽 1 𝑛 𝛼 𝛽 1 Pochhammer-symbol 𝛼 𝛽 2 𝑛 𝑛 Pochhammer-symbol 𝛼 1 𝑛 Pochhammer-symbol 𝛽 1 𝑛 superscript 𝑟 𝑛 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑦 1 𝑟 superscript 1 𝑟 𝛼 𝛽 2 \AppellFiv @ 1 2 𝛼 𝛽 2 1 2 𝛼 𝛽 3 𝛼 1 𝛽 1 𝑟 1 𝑥 1 𝑦 superscript 1 𝑟 2 𝑟 1 𝑥 1 𝑦 superscript 1 𝑟 2 {\displaystyle{\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{2n+\alpha+% \beta+1}{n+\alpha+\beta+1}\frac{{\left(\alpha+\beta+2\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-r}{(1+r)^{\alpha+% \beta+2}}\AppellFiv@{\frac{1}{2}(\alpha+\beta+2)}{\frac{1}{2}(\alpha+\beta+3)}% {\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

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

Bibliography

Equation in Section of KLS.

URL links

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