Continuous q-Hermite Polynomials When q > 1

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The continuous q-Hermite polynomials have a reasonably well known orthogonality relation when — 1 < q < 1. When q > 1 the change of variables xix lends to a set of orthogonal polynomials. The same happens for the continuous q-ultraspherical polynomials, but now only finitely many polynomials are orthogonal with respect to a positive measure. A positive measure is found in each of these cases.

Written while a visiting professor at the University of Minnesota at the Institute for Mathematics and its Applications. Also partially supported by sabbatical leave from the University of Wisconsin-Madison.