Formula:KLS:09.05:02

<< Formula:KLS:09.05:01

formula in Hahn

Formula:KLS:09.05:03 >>

${\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{N}\binomial{\alpha+x}{x}\binomial{\beta+N-x}{N-x}\Hahn{m}@{x}{\alpha}{\beta}{N}\Hahn{n}@{x}{\alpha}{\beta}{N}{}=\frac{(-1)^{n}\pochhammer{n+\alpha+\beta+1}{N+1}\pochhammer{\beta+1}{n}n!}{(2n+\alpha+\beta+1)\pochhammer{\alpha+1}{n}\pochhammer{-N}{n}N!}\,\Kronecker{m}{n}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle Q_{n}}}}$ : Hahn polynomial : http://dlmf.nist.gov/18.19#T1.t1.r3
${\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}$ : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 9.5 of KLS.

URL links

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formula in Hahn

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