Orthogonality and Completeness of $q$-Fourier Type Systems

Abstract

We establish orthogonality and completeness of the system of $q$-exponential functions {${\mathcal{E}}_q(\cdot ;i{\omega }_n)$} using orthogonality and dual orthogonality of a $q$-analogue of Lommel polynomials. We also set up a very general procedure by which one can produce similar orthogonal systems using bilinear generating functions formed by products of two complete orthogonal function systems.