Arithmetic differential operators on Zp

Given a prime p, we let δx=(x−xp)/p be the Fermat quotient operator over Zp. We prove that a function f:Zp→Zp is analytic if, and only if, there exists m such that f can be represented as f(x)=F(x,δx,…,δmx), where F is a restricted power series with Zp-coefficients in m+1 variables.