Abstract: In this paper, we present some results for vector-valued fractional difference equations. We are successful to completely characterize the maximal regularity of solutions for the problem in Lebesgue vector-valued spaces defined on the set Z+. Our approach use as main ingredients Blunck’s operator valued multiplier theorem, and the introduction of a special sequence of bounded operators, that we called α-resolvent families, which will play a central role in the representation of the solution of the problem by means of a kind of discrete variation of parameters formula.