Continuous q-Legendre polynomials

In mathematics, the continuous q-Legendre polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Koekoek, Lesky & Swarttouw (2010) give a detailed list of their properties.

Definition

The polynomials are given in terms of basic hypergeometric functions by

$P_n(x|q)={}_4{\varphi }_3(q^{-n},q^{n+1},q^{\frac{1}{4}}{e}^{i\theta },q^{\frac{1}{4}}{e}^{-i\theta };q,-q^{-\frac{1}{2}},-q;q,q),x=\cos \theta .$