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

$P_n^{(\alpha ,\beta )}(x;q)=\frac{(q^{n+1};q{)}_n}{(q;q{)}_n}{}_4{\varphi }_3[\begin{array}{c}{q}^{-n},q^{n+\alpha +\beta +1},q^{\frac{1}{2}\alpha +\frac{1}{4}{e}^{i\theta }},q^{\frac{1}{2}\alpha +\frac{1}{4}{e}^{-i\theta }}\\ q^{n+1},-q^{\frac{1}{2}(\alpha +\beta +1)},-q^{\frac{1}{2}(\alpha +\beta +2)}\end{array};q,q]x=\cos \theta .$

References

• .