Rogers polynomials

Short description: Family of orthogonal 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^[1]^[2]^[3] 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.^[4]

The Rogers polynomials can be defined in terms of the q-Pochhammer symbol by

C_{n} (x; β | q) = \frac{(β; q)_{n}}{(q; q)_{n}} e^{i n θ}_{2} ϕ_{1} (q^{- n}, β; β^{- 1} q^{1 - n}; q, q β^{- 1} e^{- 2 i θ})

where $_{2} ϕ_{1}$ is the basic hypergeometric series and $x = \cos θ$ .

(Askey Ismail) and (Gasper Rahman) discuss the properties of Rogers polynomials in detail.

Notes

Askey, R.; Ismail, Mourad E. H. (1983). "A generalization of ultraspherical polynomials". in Erdős, Paul. Studies in Pure Mathematics: To the Memory of Paul Turán. Basel: Birkhäuser. pp. 55–78. ISBN 978-3-7643-1288-6.

Gasper, George; Rahman, Mizan (2004). Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications. 96 (2nd ed.). Cambridge University Press. ISBN 978-0-521-83357-8.

Macdonald, I. G. (2003). Affine Hecke Algebras and Orthogonal Polynomials. Cambridge Tracts in Mathematics. 157. Cambridge University Press. ISBN 978-0-521-82472-9.

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