In mathematics, the Rogers polynomials, also called the Rogers–Askey–Ismail polynomials or continuous q-ultraspherical polynomials, are a family of orthogonal polynomials introduced by Leonard James Rogers 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.
The Rogers polynomials can be defined in terms of the q-Pochhammer symbol by
where is the basic hypergeometric series and .
and discuss the properties of Rogers polynomials in detail.
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