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

${\displaystyle C_{n}(x;\beta |q)={\frac {(\beta ;q)_{n)){(q;q)_{n))}e^{in\theta }{}_{2}\phi _{1}(q^{-n},\beta ;\beta ^{-1}q^{1-n};q,q\beta ^{-1}e^{-2i\theta })}$

where x = cos(θ).