THE BRENKE TYPE GENERATING FUNCTIONS AND EXPLICIT FORMS OF MRM-TRIPLES BY MEANS OF Q-HYPERGEOMETRIC SERIES

Abstract

An MRM-triple (h(x), ρ(t), B(t)) gives a generating function B(t)h(ρ(t)x) of some orthogonal polynomials on ℝ. In particular, B(t)h(tx) is called the Brenke type.19 In this paper, we shall determine all MRM-triples and associated Jacobi-Szegö parameters of this type with showing very careful computations in detail. (h(x), t, B(t)) is classified into four categories. In any case, h(x) and B(t) can be expressed in terms of two kinds of q-hypergeometric series, old basic and basic hypergeometric series, rΦs and rϕs, respectively. As examples, our results contain generating functions of the Al-Salam-Carlitz (I and II), little q-Laguerre, q-Laguerre, and discrete q-Hermite (I and II) polynomials. Our results are more complete and general than those of Refs. 20 and 21 by Chihara. The following are special cases of our results in each class. Here {αn, ωn} are the Jacobi-Szegö parameters. [Formula: see text]