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 J(n)([k])(D) 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 = F-q[T], we give a more explicit formula for the J(n)([k])(F-q[T])'s and describe a new basis for Im([k])(F-q[T]) that comes from a regular basis of Int(F-q[T]) introduced by M. Car. (c) 2005 Published by Elsevier Inc.