We consider a singular differential-difference operator L on the real line which generalizes the one-dimensional Dunkl operator and construct a pair of integral transforms which turn out to be intertwining operators of L and its dual  into the first derivative operator  Further, we exploit these intertwining operators, firstly to establish a Paley-Wiener theorem for the Fourier transform associated to L, and secondly to introduce a generalized convolution on  tied to L.