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