Meixner–Pollaczek polynomials

In mathematics, the Meixner–Pollaczek polynomials are a family of orthogonal polynomials P^(λ)
_n(x,φ) introduced by Meixner (1934), which up to elementary changes of variables are the same as the Pollaczek polynomials P^λ
_n(x,a,b) rediscovered by Pollaczek (1949) in the case λ=1/2, and later generalized by him.

They are defined by

${\displaystyle P_{n}^{(\lambda )}(x;\phi )={\frac {(2\lambda )_{n}}{n!}}e^{in\phi }{}_{2}F_{1}\left({\begin{array}{c}-n,~\lambda +ix\\2\lambda \end{array}};1-e^{-2i\phi }\right)}$

${\displaystyle P_{n}^{\lambda }(\cos \phi ;a,b)={\frac {(2\lambda )_{n}}{n!}}e^{in\phi }{}_{2}F_{1}\left({\begin{array}{c}-n,~\lambda +i(a\cos \phi +b)/\sin \phi \\2\lambda \end{array}};1-e^{-2i\phi }\right)}$