Formula:KLS:09.07:15

<< Formula:KLS:09.07:14

formula in Meixner-Pollaczek

Formula:KLS:09.07:16 >>

${\displaystyle{\displaystyle{\displaystyle(1-t)^{-\gamma}\,\HyperpFq{2}{1}@@{\gamma,\lambda+\mathrm{i}x}{2\lambda}{\frac{(1-{\mathrm{e}^{-2\mathrm{i}\phi}})t}{t-1}}{}=\sum_{n=0}^{\infty}\frac{{\left(\gamma\right)_{n}}}{{\left(2\lambda\right)_{n}}}\frac{P^{(\lambda)}_{n}\!\left(x;\phi\right)}{{\mathrm{e}^{\mathrm{i}n\phi}}}t^{n}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle\gamma}}}$ arbitrary

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
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}$ : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
${\displaystyle{\displaystyle{\displaystyle P^{(\alpha)}_{n}}}}$ : Meixner-Pollaczek polynomial : http://dlmf.nist.gov/18.19#P3.p1

Bibliography

Equation in Section 9.7 of KLS.

URL links

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<< Formula:KLS:09.07:14

formula in Meixner-Pollaczek

Formula:KLS:09.07:16 >>