Locally analytic functions in a local field of positive characteristic

In a recent paper (Buium et al., 2011 [3]), Buium et al. proved that f is a locally analytic function from the p-adic integers, Zp to itself if and only if it is written as a restricted power series over Zp with finitely many iterates of the Fermat quotient operator. In this paper we establish the function field analogs of their result. In addition, we deduce Wagnerʼs result, which is the function field analog of Mahlerʼs result for continuous functions on Zp.