Integral and series representations of q-polynomials and functions: Part I

By applying an integral representation for [Formula: see text], we systematically derive a large number of new Fourier and Mellin transform pairs and establish new integral representations for a variety of [Formula: see text]-functions and polynomials that naturally arise from combinatorics, analysis, and orthogonal polynomials corresponding to indeterminate moment problems. These functions include [Formula: see text]-Bessel functions, the Ramanujan function, Stieltjes–Wigert polynomials, [Formula: see text]-Hermite and [Formula: see text]-Hermite polynomials, and the [Formula: see text]-exponential functions [Formula: see text], [Formula: see text] and [Formula: see text]. Their representations are in turn used to derive many new identities involving [Formula: see text]-functions and polynomials. In this paper, we also present contour integral representations for the above mentioned functions and polynomials.

A Family of Heat Functions as Solutions of Indeterminate Moment Problems

We construct a family of functions satisfying the heat equation and show how they can be used to generate solutions to indeterminate moment problems. The following cases are considered: log-normal, generalized Stieltjes-Wigert, andq-Laguerre.