Formula:KLS:09.06:31

<< Formula:KLS:09.06:30

formula in Dual Hahn

Formula:KLS:09.07:01 >>

${\displaystyle{\displaystyle{\displaystyle\sum_{n=0}^{N}\binomial{\gamma+n}{n}\binomial{\delta+N-n}{N-n}\dualHahn{n}@{\lambda(x)}{\gamma}{\delta}{N}\dualHahn{n}@{\lambda(y)}{\gamma}{\delta}{N}{}=\frac{(-1)^{x}\pochhammer{x+\gamma+\delta+1}{N+1}\pochhammer{\delta+1}{x}x!}{(2x+\gamma+\delta+1)\pochhammer{\gamma+1}{x}\pochhammer{-N}{x}N!}\Kronecker{x}{y}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle x,y\in\{0,1,2,\ldots,N\}}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}$

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
${\displaystyle{\displaystyle{\displaystyle R_{n}}}}$ : dual Hahn polynomial : http://dlmf.nist.gov/18.25#T1.t1.r5
${\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle\in}}}$ : element of : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r9

Bibliography

Equation in Section 9.6 of KLS.

URL links

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<< Formula:KLS:09.06:30

formula in Dual Hahn

Formula:KLS:09.07:01 >>