Formula:KLS:09.09:13

<< Formula:KLS:09.09:12

formula in Pseudo Jacobi

Formula:KLS:09.09:14 >>

${\displaystyle{\displaystyle{\displaystyle\left[\HyperpFq{0}{1}@@{-}{-N+\mathrm{i}\nu}{(x+\mathrm{i})t}\,\HyperpFq{0}{1}@@{-}{-N-\mathrm{i}\nu}{(x-\mathrm{i})t}\right]_{N}{}=\sum_{n=0}^{N}\frac{{\left(n-2N-1\right)_{n}}}{{\left(-N+\mathrm{i}\nu\right)_{n}}{\left(-N-\mathrm{i}\nu\right)_{n}}n!}P_{n}\!\left(x;\nu,N\right)t^{n}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle{{}_{p}F_{q}}}}}$ : generalized hypergeometric function : http://dlmf.nist.gov/16.2#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}$ : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
${\displaystyle{\displaystyle{\displaystyle P_{n}}}}$ : pseudo Jacobi polynomomal : http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi

Bibliography

Equation in Section 9.9 of KLS.

URL links

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<< Formula:KLS:09.09:12

formula in Pseudo Jacobi

Formula:KLS:09.09:14 >>