Formula:KLS:09.06:22

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formula in Dual Hahn

Formula:KLS:09.06:23 >>

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

Substitution(s)

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

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle{{}_{p}F_{q}}}}}$ : generalized hypergeometric function : http://dlmf.nist.gov/16.2#E1
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : 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

Bibliography

Equation in Section 9.6 of KLS.

URL links

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formula in Dual Hahn

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