Differential operators on monomial curves

Let k be an algebraically closed field of characteristic 0, let be a numerical semigroup, and let A=k[Γ] be the corresponding semigroup algebra. We give an explicit description of the ring D=D(A) of k-linear differential operators on A, the associated graded ring GrD(A) and the module of derivations Derk(A). We also classify all graded D-modules which are finitely generated torsion-free A-modules of rank 1, considered as modules over the sub-ring A D(A).