Rogers polynomials

Template:Short description Template:Distinguish In mathematics, the Rogers polynomials, also called Rogers–Askey–Ismail polynomials and continuous q-ultraspherical polynomials, are a family of orthogonal polynomials introduced by Template:Harvs 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 Template:Harv.

Template:Harvtxt and Template:Harvtxt 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

$C_n(x;\beta |q)=\frac{(\beta ;q{)}_n}{(q;q{)}_n}{e}^{in\theta }{}_2{\varphi }_1(q^{-n},\beta ;{\beta }^{-1}{q}^{1-n};q,q{\beta }^{-1}{e}^{-2i\theta })$

where x = cos(θ).

References

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