Rogers polynomials

Not to be confused with Rogers–Szegő polynomials.

In mathematics, the Rogers polynomials, also called Rogers–Askey–Ismail polynomials and continuous q-ultraspherical polynomials, are a family of orthogonal polynomials introduced by Rogers (1892, 1893, 1894) in the course of his work on the Rogers–Ramanujan identities. They are q-analogs of ultraspherical polynomials, and are the Macdonald polynomials for the special case of the A₁ affine root system (Macdonald 2003, p.156).

Askey & Ismail (1983) and Gasper & Rahman (2004, 7.4) discuss the properties of Rogers polynomials in detail.

Definition

The Rogers polynomials can be defined in terms of the q-Pochhammer symbol and the basic hypergeometric series by

$C_n(x;\beta |q)=\frac{(\beta ;q{)}_n}{(q;q{)}_n}{e}^{in\theta }{}_2{\varphi }_1(q^{-n},\beta ;{\beta }^{-1}{q}^{1-n};q,q{\beta }^{-1}{e}^{-2i\theta })$

where x = cos(θ).