Formula:KLS:09.02:26

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formula in Racah

Formula:KLS:09.02:27 >>

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

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle\textrm{if}\quad\beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N}}}$

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
${\displaystyle{\displaystyle{\displaystyle R_{n}}}}$ : Racah polynomial : http://dlmf.nist.gov/18.25#T1.t1.r4

Bibliography

Equation in Section 9.2 of KLS.

URL links

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<< Formula:KLS:09.02:25

formula in Racah

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