Continuous q-Jacobi polynomials
# Continuous q-Jacobi polynomials

In mathematics, the continuous q-Jacobi polynomials $P^{(α,β)}$ $_n(x|q),$ introduced by Askey & Wilson (1985), 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

    $P_{n}^{(\alpha ,\beta )}(x;q)={\frac {(q^{n+1};q)_{n}}{(q;q)_{n}}}_{4}\phi _{3}\left[{\begin{matrix}q^{-n},q^{n+\alpha +\beta +1},q^{{\frac {1}{2}}\alpha +{\frac {1}{4}}e^{i\theta }},q^{{\frac {1}{2}}\alpha +{\frac {1}{4}}e^{-i\theta }}\\q^{n+1},-q^{{\frac {1}{2}}(\alpha +\beta +1)},-q^{{\frac {1}{2}}(\alpha +\beta +2)}\end{matrix}};q,q\right]\qquad x=\cos \,\theta .$
