Abstract
We consider a singular differential-difference operator Λ on the real line which generalizes the Dunkl operator associated with the reflection group Z
2
on R. We construct transmutation operators between Λ and the first derivative operator d/dx. We exploit these transmutation operators, firstly to determine the elementary solution of certain classes of singular differential-difference operators on a product of Euclidean spaces, and secondly to introduce a generalized translation on the real line corresponding to the operator Λ