e-mail: Khlifa.Trimeche@fst.rnu.tn

**Harmonic Analysis associated with a singular differential-difference
operator generalizing the Dunkl operator on the real line**

**Abstract**

We consider a singular differential-difference operator *A* on
the real line which includes as particular case the Dunkl operator assiciated
with the reflection group **Z**_2 on **R**. We give a Laplace integral
representation for the eigenfunctions of the operator *A*. From this
representation we constract a pair of integral transforms which turn out
to be transmutation operators of *A* into the first derivative operator
*d/dx*.

We exploit these transmutation operators to develop a new commutative
harmonic analysid on the real line corresponding to the operator *A*
(convolution product, Fourier transform, inversion formula, Paley-Wiener
theorem,...).