Continuous q-Laguerre polynomials

In mathematics, the continuous q-Laguerre 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}^{(\alpha )}(x|q)={\frac {(q^{\alpha +1};q)_{n}}{(q;q)_{n}}}}$${\displaystyle _{3}\phi _{2}(q^{-n},q^{\alpha /2+1/4}e^{i\theta },q^{\alpha /2+1/4}e^{-i\theta };q^{\alpha +1},0|q,q)}$