Rogers polynomials

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

${\displaystyle C_{n}(x;\beta |q)={\frac {(\beta ;q)_{n}}{(q;q)_{n}}}e^{in\theta }{}_{2}\phi _{1}(q^{-n},\beta$ ;\beta ^{-1}q^{1-n};q,q\beta ^{-1}e^{-2i\theta })}

where ${\displaystyle {}_{2}\phi _{1}}$ is the basic hypergeometric series and ${\displaystyle x=\cos \theta }$.

Askey & Ismail (1983) and Gasper & Rahman (2004, section 7.4) discuss the properties of Rogers polynomials in detail.

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References

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