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Zbl 0812.33012
Ismail, M.E.H.; Masson, D.R.
$q$-Hermite polynomials, biorthogonal rational functions, and $q$-beta integrals. (English)
[J] Trans. Am. Math. Soc. 346, No.1, 63-116 (1994). ISSN 0002-9947; ISSN 1088-6850/e

The continuous $q$-Hermite polynomials are orthogonal when $-1<q<1$. When $q>1$, they are orthogonal with respect to a positive measure on the imaginary axis. In this case the moment problem is indeterminate. The class of positive measures for the orthogonality of these polynomials is convex. The authors determine all of the extreme points of this convex set. This is the first case where all extreme measures have been found.\par Many other results are obtained while solving this problem. A generalization of the very well poised ${}\sb 6\psi\sb 6$ is obtained. Some new biorthogonal rational functions are found. Finally a new proof of the triple product for the theta function function is obtained from these orthogonal polynomials.
[R.Askey (Madison)]

MSC 2000:: *33D45 Basic hypergeometric functions and integrals in several variables
33D05 q-gamma functions, q-beta functions and integrals
42A65 Completeness of sets of functions
30E05 Moment problems, etc.

Keywords: indeterminate moment problem; $q$-Hermite polynomials; biorthogonal rational functions

Cited in: Zbl 1089.33016 Zbl 0905.33010

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