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7 - Some q-Orthogonal Polynomials

Published online by Cambridge University Press:  14 September 2020

Edited by
Mourad E. H. Ismail
Assisted by
Walter Van Assche
Mourad E. H. Ismail
Affiliation:
University of Central Florida
Walter Van Assche
Affiliation:
Katholieke Universiteit Leuven, Belgium
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Summary

The continuous q-ultraspherical and continuous q-Hermite polynomials first appeared in Rogers’ work on the Rogers–Ramanujan identities in 1893–95 (Askey and Ismail, 1983). They belong to the Fejér class of polynomials having a generating function of the form

∑n=0∞ϕn(cosθ)tn=|F(reiθ)|2, (7.0.1)

where F(z) is analytic in a neighborhood of z=0. Feldheim (1941) and Lanzewizky (1941) independently proved that the only orthogonal generalized polynomials in the Fejér class are either the ultraspherical polynomials or the q-ultraspherical polynomials or special cases of them. They proved that F has to be F1 or F2, or some limiting cases of them, where

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Encyclopedia of Special Functions: The Askey-Bateman Project , pp. 157 - 177
Publisher: Cambridge University Press
Print publication year: 2020

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