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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.

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