Rogers polynomials

Not to be confused with Rogers–Szegő 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 ${}_{2}\phi _{1}$ is the basic hypergeometric series and $x=\cos \theta$.

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

References

• Askey, R.; 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: Birkhäuser. pp. 55–78. ISBN 978-3-7643-1288-6. MR 0820210.

• Gasper, George; Rahman, Mizan (2004) [1990]. Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications. Vol. 96 (2nd ed.). Cambridge University Press. ISBN 978-0-521-83357-8. MR 2128719.

• Macdonald, I. G. (2003). Affine Hecke Algebras and Orthogonal Polynomials. Cambridge Tracts in Mathematics. Vol. 157. Cambridge University Press. ISBN 978-0-521-82472-9. MR 1976581.

• Rogers, L. J. (1893). "On the expansion of some infinite products". Proceedings of the London Mathematical Society. 24: 337–352. doi:10.1112/plms/s1-24.1.337. JFM 25.0432.01.

• Rogers, L. J. (1894). "Second memoir on the expansion of certain infinite products". Proceedings of the London Mathematical Society. 25: 318–343. doi:10.1112/plms/s1-25.1.318.

• Rogers, L. J. (1894b). "Third memoir on the expansion of certain infinite products". Proceedings of the London Mathematical Society. 26: 15–32. doi:10.1112/plms/s1-26.1.15.