Formula:KLS:09.07:14

<< Formula:KLS:09.07:13

formula in Meixner-Pollaczek

Formula:KLS:09.07:15 >>

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

Proof

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Symbols List

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

Bibliography

Equation in Section 9.7 of KLS.

URL links

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<< Formula:KLS:09.07:13

formula in Meixner-Pollaczek

Formula:KLS:09.07:15 >>