# Meixner–Pollaczek polynomials

Not to be confused with Meixner 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 Pλ
n
(x,a,b) rediscovered by Pollaczek (1949) in the case λ=1/2, and later generalized by him.

They are defined by

${\displaystyle P_{n}^{(\lambda )}(x;\phi )={\frac {(2\lambda )_{n}}{n!}}e^{in\phi }{}_{2}F_{1}(-n,\lambda +ix;2\lambda ;1-e^{-2i\phi })}$
${\displaystyle P_{n}^{\lambda }(\cos \phi ;a,b)={\frac {(2\lambda )_{n}}{n!}}e^{in\phi }{}_{2}F_{1}(-n,\lambda +i(a\cos \phi +b)/\sin \phi ;2\lambda ;1-e^{-2i\phi })}$

They are orthogonal on the real line with respect to the weight function

${\displaystyle w(x;\lambda ,\phi )=|\Gamma (\lambda +ix)|^{2}e^{(2\phi -\pi )x}}$

and the orthogonality is given by

${\displaystyle \int _{-\infty }^{\infty }P_{n}^{(\lambda )}(x;\phi )P_{m}^{(\lambda )}(x;\phi )w(x;\lambda ,\phi )dx={\frac {2\pi \Gamma (n+2\lambda )}{(2\sin \phi )^{2\lambda }n!}}\delta _{mn}}$