Formula:KLS:09.05:20

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\HyperpFq 20 @ @ - x , - x + β + N + 1 - - t \HyperpFq 20 @ @ x - N , x + α + 1 - t = ∑ n = 0 N ( - N ) n ⁢ ( α + 1 ) n n ! Q n ⁡ ( x ; α , β , N ) t n fragments \HyperpFq 20 @ @ x , x β N 1 t \HyperpFq 20 @ @ x N , x α 1 t superscript subscript 𝑛 0 𝑁 Pochhammer-symbol 𝑁 𝑛 Pochhammer-symbol 𝛼 1 𝑛 𝑛 Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\HyperpFq{2}{0}@@{-x,-x+\beta+N+1}{-% }{-t}\,\HyperpFq{2}{0}@@{x-N,x+\alpha+1}{-}{t}{}=\sum_{n=0}^{N}\frac{{\left(-N% \right)_{n}}{\left(\alpha+1\right)_{n}}}{n!}Q_{n}\!\left(x;\alpha,\beta,N% \right)t^{n}}}}

Proof

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Symbols List

F q p subscript subscript 𝐹 𝑞 𝑝 {\displaystyle{\displaystyle{\displaystyle{{}_{p}F_{q}}}}}  : generalized hypergeometric function : http://dlmf.nist.gov/16.2#E1
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
Q n subscript 𝑄 𝑛 {\displaystyle{\displaystyle{\displaystyle Q_{n}}}}  : Hahn polynomial : http://dlmf.nist.gov/18.19#T1.t1.r3

Bibliography

Equation in Section 9.5 of KLS.

URL links

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