Formula:KLS:09.06:03

<< Formula:KLS:09.06:01

formula in Dual Hahn

Formula:KLS:09.06:04 >>

${\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{N}\frac{(2x+\gamma+\delta+1){\left(\gamma+1\right)_{x}}{\left(-N\right)_{x}}N!}{(-1)^{x}{\left(x+\gamma+\delta+1\right)_{N+1}}{\left(\delta+1\right)_{x}}x!}R_{m}\!\left(\lambda(x);\gamma,\delta,N\right)R_{n}\!\left(\lambda(x);\gamma,\delta,N\right){}=\frac{\,\delta_{m,n}}{\dbinom{\gamma+n}{n}\dbinom{\delta+N-n}{N-n}}}}}$

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
${\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}$ : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
${\displaystyle{\displaystyle{\displaystyle R_{n}}}}$ : dual Hahn polynomial : http://dlmf.nist.gov/18.25#T1.t1.r5
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 9.6 of KLS.

URL links

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formula in Dual Hahn

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