On the coefficients of power sums of arithmetic progressions

TL;DR

This paper explores the coefficients of power sums of arithmetic progressions, expressing them through Stirling and Whitney numbers, and establishes conditions for their integrality.

Contribution

It provides explicit formulas for these coefficients using special number sequences and characterizes when these coefficients are integers.

Findings

Coefficients are expressed via Stirling and Whitney numbers.

A necessary and sufficient condition for integrality is established.

The results connect power sums with combinatorial number theory.

Abstract

We investigate the coefficients of the polynomial \[ S_{m,r}^n(\ell)=r^n+(m+r)^n+(2m+r)^n+\cdots+((\ell-1)m+r)^n. \] We prove that these can be given in terms of Stirling numbers of the first kind and $r$-Whitney numbers of the second kind. Moreover, we prove a necessary and sufficient condition for the integrity of these coefficients.