Integral and Series Representations of $q$-Polynomials and Functions: Part I

Abstract

By applying an integral representation for $q^{k^2}$ we systematically derive a large number of new Fourier and Mellin transform pairs and establish new integral representations for a variety of $q$-functions and polynomials that naturally arise from combinatorics, analysis, and orthogonal polynomials corresponding to indeterminate moment problems. These functions include $q$-Bessel functions, the Ramanujan function, Stieltjes--Wigert polynomials, $q$-Hermite and $q^{-1}$-Hermite polynomials, and the $q$-exponential functions $e_q$, $E_q$ and ${\mathcal{E}}_q$. Their representations are in turn used to derive many new identities involving $q$-functions and polynomials. In this work we also present contour integral representations for the above mentioned functions and polynomials.

61 pages

References in corpus (1)

• Random matrix central limit theorems for nonintersecting random walks

Cited by in corpus (1)

• Integrals and Series Representations of $q$-Polynomials and Functions: Part II Schur Polynomials and the Rogers-Ramanujan Identities