Arithmetic differential operators on Z_p

TL;DR

This paper characterizes p-adic analytic functions as those representable by restricted power series involving Fermat quotient operators, providing a new perspective on p-adic analysis.

Contribution

It establishes a precise equivalence between p-adic analyticity and representation via restricted power series with Fermat quotient operators.

Findings

Characterization of analytic functions on Z_p using Fermat quotient operators

Representation of functions as restricted power series with coefficients in Z_p

Provides a new framework for understanding p-adic analyticity

Abstract

We prove that a function f from Z_p to itself is analytic if and only if it can be represented as f(x)=F(x, dx, ..., d^r x) where dx=(x-x^p)/p is the Fermat quotient operator and F is a restricted power series with coefficients in Z_p.