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

TL;DR

This paper develops new integral and series representations for various $q$-functions and polynomials, enhancing understanding of their properties and relationships through Fourier, Mellin, and contour integral methods.

Contribution

It introduces systematic integral representations for $q$-functions and polynomials, leading to new identities and contour integral formulas, expanding analytical tools in $q$-series.

Findings

Derived new Fourier and Mellin transform pairs for $q$-functions.

Established novel integral representations for $q$-Bessel, Ramanujan, and orthogonal polynomials.

Presented contour integral formulas for key $q$-functions.

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.