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 descending Pochhammer symbol and the basic hypergeometric series by

$C_n(x;\beta|q) = \frac{(\beta;q)_n}{(q;q)_n}e^{in\theta} {}_2\phi_1(q^{-n},\beta;\beta^{-1}q^{1-n};q,q\beta^{-1}e^{-2i\theta})$

where x = cos(θ).