Formula:KLS:09.07:01

<< Dual Hahn

formula in Meixner-Pollaczek

Formula:KLS:09.07:02 >>

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

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle P^{(\alpha)}_{n}}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i

Bibliography

Equation in Section 9.7 of KLS.

URL links

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<< Dual Hahn

formula in Meixner-Pollaczek

Formula:KLS:09.07:02 >>