Rogers 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 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.

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

where x = cos(θ).