In mathematics, the Rogers polynomials, also called RogersâÂÂAskeyâÂÂIsmail polynomials and continuous q-ultraspherical polynomials, are a family of orthogonal polynomials introduced by in the course of his work on the RogersâÂÂRamanujan identities. They are q-analogs of ultraspherical polynomials, and are the Macdonald polynomials for the special case of the A<sub>1</sub> affine root system .
and discuss the properties of Rogers polynomials in detail.
The Rogers polynomials can be defined in terms of the q-Pochhammer symbol and the basic hypergeometric series by
where x = cos(ø).