Intertwining Operators Associated to a Family of Differential-Reflection Operators

We introduce a family of differential-reflection operators Λ _{A , ε} acting on smooth functions defined on R. Here, A is a Sturm–Liouville function with additional hypotheses and - 1 ≤ ε≤ 1. For special pairs (A, ε) , we recover Dunkl’s, Heckman’s and Cherednik’s operators (in one dimension). The spectral problem for the operators Λ _{A , ε} is studied. In particular, we obtain suitable growth estimates for the eigenfunctions of Λ _{A , ε}. As the operators Λ _{A , ε}, are a mixture of d / d x and reflection operators, we prove the existence of an intertwining operator V_{A , ε} between Λ _{A , ε} and the usual derivative. The positivity of V_{A , ε} is also established.