Rogers 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 Rogers (1892, 1893, 1894) 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 (Macdonald 2003, p.156).

Not to be confused with Rogers–Szegő polynomials.

Askey & Ismail (1983) and Gasper & Rahman (2004, 7.4) 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

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

where x = cos(θ).

References

• Askey, Richard; Ismail, Mourad E. H. (1983), "A generalization of ultraspherical polynomials", in Erdős, Paul (ed.), Studies in pure mathematics. To the memory of Paul Turán., Basel, Boston, Berlin: Birkhäuser, pp. 55–78, ISBN 978-3-7643-1288-6, MR 0820210
• Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, ISBN 978-0-521-83357-8, MR 2128719
• Macdonald, I. G. (2003), Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, 157, Cambridge University Press, doi:10.1017/CBO9780511542824, ISBN 978-0-521-82472-9, MR 1976581
• Rogers, L. J. (1892), "On the expansion of some infinite products", Proc. London Math. Soc., 24 (1): 337–352, doi:10.1112/plms/s1-24.1.337, JFM 25.0432.01
• Rogers, L. J. (1893), "Second Memoir on the Expansion of certain Infinite Products", Proc. London Math. Soc., 25 (1): 318–343, doi:10.1112/plms/s1-25.1.318
• Rogers, L. J. (1894), "Third Memoir on the Expansion of certain Infinite Products" (PDF), Proc. London Math. Soc., 26 (1): 15–32, doi:10.1112/plms/s1-26.1.15