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Rogers polynomials
Family of orthogonal polynomials
Family of orthogonal 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).
References
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