$q$-Hermite polynomials, biorthogonal rational functions, and $q$-beta integrals.

*(English)*Zbl 0812.33012The 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.

Many other results are obtained while solving this problem. A generalization of the very well poised ${}_{6}{\psi}_{6}$ 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.

Reviewer: R.Askey (Madison)