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

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

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

The first few Meixner–Pollaczek polynomials are

${\displaystyle P_{0}^{(\lambda )}(x;\phi )=1}$

${\displaystyle P_{1}^{(\lambda )}(x;\phi )=2(\lambda \cos \phi +x\sin \phi )}$

${\displaystyle P_{2}^{(\lambda )}(x;\phi )=x^{2}+\lambda ^{2}+(\lambda ^{2}+\lambda -x^{2})\cos(2\phi )+(1+2\lambda )x\sin(2\phi ).}$

The Meixner–Pollaczek polynomials P_m^(λ)(x;φ) 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 relation is given by¹

${\displaystyle \int _{-\infty }^{\infty }P_{n}^{(\lambda )}(x;\phi )P_{m}^{(\lambda )}(x;\phi )w(x;\lambda ,\phi )dx={\frac {2\pi \Gamma (n+2\lambda )}{(2\sin \phi )^{2\lambda }n!}}\delta _{mn},\quad \lambda >0,\quad 0<\phi <\pi .}$

The sequence of Meixner–Pollaczek polynomials satisfies the recurrence relation²

${\displaystyle (n+1)P_{n+1}^{(\lambda )}(x;\phi )=2{\bigl (}x\sin \phi +(n+\lambda )\cos \phi {\bigr )}P_{n}^{(\lambda )}(x;\phi )-(n+2\lambda -1)P_{n-1}(x;\phi ).}$

The Meixner–Pollaczek polynomials are given by the Rodrigues-like formula³

${\displaystyle P_{n}^{(\lambda )}(x;\phi )={\frac {(-1)^{n}}{n!\,w(x;\lambda ,\phi )}}{\frac {d^{n}}{dx^{n}}}w\left(x;\lambda +{\tfrac {1}{2}}n,\phi \right),}$

where w(x;λ,φ) is the weight function given above.

The Meixner–Pollaczek polynomials have the generating function⁴

${\displaystyle \sum _{n=0}^{\infty }t^{n}P_{n}^{(\lambda )}(x;\phi )=(1-e^{i\phi }t)^{-\lambda +ix}(1-e^{-i\phi }t)^{-\lambda -ix}.}$