Abstract
This paper deals with a new singular differential-difference operator Y
t, A
on the real line
including, as a particular case, the Dunkl, Dunkl-Heckman and Dunkl-Cherednik operators. We establish some results of Harmonic analysis related to this operator, such that a product formula for the related eigenfunction G
λ
is given as integral with an explicit kernel. This product formula is an important tool to define the generalized translation operator which is used to set up a convolution structure. Next, we establish an inversion formula and prove a generalized Plancherel theorem for this operator. As a direct application, we give a maximum principle of the operators
and we solve the heat equation associated with the generalized Dunkl operator on the real line.