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).

(Askey Ismail) and (Gasper Rahman) 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

[math]\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}) }[/math]

where x = cos(θ).

References

• Askey, Richard; Ismail, Mourad E. H. (1983), "A generalization of ultraspherical polynomials", in Erdős, Paul, Studies in pure mathematics. To the memory of Paul Turán., Basel, Boston, Berlin: Birkhäuser, pp. 55–78, ISBN 978-3-7643-1288-6, https://books.google.com/books?id=WePuAAAAMAAJ 
• 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, doi:10.1017/CBO9780511542824, ISBN 978-0-521-82472-9, http://www.numdam.org/item/SB_1994-1995__37__189_0/ 
• 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, https://zenodo.org/record/1433380 
• 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, https://zenodo.org/record/1447732 
• Rogers, L. J. (1894), "Third Memoir on the Expansion of certain Infinite Products", Proc. London Math. Soc. 26 (1): 15–32, doi:10.1112/plms/s1-26.1.15, https://zenodo.org/record/1447734