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 where x = cos(θ). References Orthogonal polynomials Q-analogs