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 "A"1 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
where "x" = cos("θ").