Newton representation of functions over natural integers having integral difference ratios

Abstract

Different questions lead to the same class of functions from natural integers to integers: those which have integral difference ratios, i.e. verifying $f(a)-f(b)\equiv 0(\mathrm{mod}(a-b))$ for all $a>b$. We characterize this class of functions via their representations as Newton series. This class, which obviously contains all polynomials with integral coefficients, also contains unexpected functions, for instance all functions $x\mapsto \lfloor e^{1/a}{a}^{x}x!\rfloor$, with $a\in \mathbb{Z}\setminus \{0,1\}$, and a function equal to $\lfloor ex!\rfloor$ except on 0. Finally, to study the complement class, we look at functions $\N\to\RR$ which are not uniformly close to any function having integral difference ratios.