Functions with integer-valued divided differences

TL;DR

This paper characterizes when sequences with integer-valued divided differences are polynomial functions, based on growth constraints related to exponential bounds.

Contribution

It proves that such sequences are polynomial functions if they grow slower than a specific exponential threshold, extending understanding of divided differences.

Findings

Sequences with integer-valued divided differences are polynomial if they grow slower than a certain exponential bound.

The growth condition involves a specific exponential threshold related to the order of divided differences.

Provides a criterion linking growth rate and polynomiality of sequences with integer divided differences.

Abstract

Let $s_0,s_1,s_2,\ldots$ be a sequence of rational numbers whose $m$th divided difference is integer-valued. We prove that $s_n$ is a polynomial function in $n$ if $s_n \ll \theta^n$ for some positive number $\theta$ satisfying $\theta < e^{1 + \tfrac{1}{2} + \cdots+ \tfrac{1}{m}} -1$.