$q$-Analogue of the Dunkl transform on the real line

TL;DR

This paper introduces a $q$-analogue of the Dunkl transform on the real line, establishing its fundamental properties, inversion, and Plancherel theorem, and explores related intertwining operators and their connections to existing $q$-Fourier transforms.

Contribution

It develops a new $q$-Dunkl transform framework, including its properties, inversion, Plancherel theorem, and links to other $q$-Fourier transforms, advancing the theory of $q$-special functions.

Findings

Established the inversion formula for the $q$-Dunkl transform.

Proved the Plancherel theorem for the $q$-Dunkl transform.

Connected the $q$-Dunkl transform to the $q^2$-analogue Fourier transform.

Abstract

In this paper, we consider a $q$-analogue of the Dunkl operator on $\mathbb{R}$, we define and study its associated Fourier transform which is a $q$-analogue of the Dunkl transform. In addition to several properties, we establish an inversion formula and prove a Plancherel theorem for this $q$-Dunkl transform. Next, we study the $q$-Dunkl intertwining operator and its dual via the $q$-analogues of the Riemann-Liouville and Weyl transforms. Using this dual intertwining operator, we provide a relation between the $q$-Dunkl transform and the $q^2$-analogue Fourier transform introduced and studied by R. Rubin.