(q\)-Analogue of the Dunkl transform on the real line

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.