Taylor series associated with a differential-difference operator on the real line

Abstract

We extend the classical theory of Taylor series to a first-order differential-difference operator Λ on the real line which includes as a particular case the Dunkl operator associated with the reflection group Z2 on R. More precisely, we establish first a generalized Taylor formula with integral remainder, and then specify sufficient conditions for a function on R to be expanded as a generalized Taylor series. Moreover, we provide a criterion of analyticity for functions on R involving the differential-difference operator Λ.