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.