Q-Meixner–Pollaczek polynomials

Not to be confused with q-Meixner polynomials.

In mathematics, the q-Meixner–Pollaczek polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.

Definition

The polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by ：^[1]

${\displaystyle P_{n}(x;a\mid q)=a^{-n}e^{in\phi }{\frac {(a^{2};q)_{n}}{(q;q)_{n}}}{}_{3}\phi _{2}(q^{-n},ae^{i(\theta +2\phi )},ae^{-i\theta };a^{2},0\mid q;q),\quad x=\cos(\theta +\phi ).}$