Finite differences in finite characteristic

Let D be a Dedekind domain with characteristic p>0. In this paper, we are interested in the D-algebras Int[k](D) of integer-valued polynomials with all their k-first finite differences. Mainly, we describe the characteristic ideals of Int[k](D). When D=V is the ring of a discrete valuation domain, we correct an old formula given by Barsky and we construct bases of the V-module Int[k](V). When D=Fq[T], we give a more explicit formula for the 's and describe a new basis for Int[k](Fq[T]) that comes from a regular basis of Int(Fq[T]) introduced by M. Car.