A note on the coefficients of power sums of arithmetic progressions

TL;DR

This paper presents a simple, explicit formula for the coefficients of power sums of arithmetic progressions, utilizing binomial coefficients and Stirling numbers, simplifying previous complex expressions.

Contribution

It introduces a straightforward double-sum formula for these coefficients and provides an explicit Bernoulli polynomial formula involving Stirling numbers.

Findings

Derived a simple double-sum formula for power sum coefficients

Provided an explicit Bernoulli polynomial formula with Stirling numbers

Simplified the computation of power sums for arithmetic progressions

Abstract

In this note we show a simple formula for the coefficients of the polynomial associated with the sums of powers of the terms of an arbitrary arithmetic progression. This formula consists of a double sum involving only ordinary binomial coefficients and binomial powers. Arguably, this is the simplest formula that can probably be found for the said coefficients. Furthermore, we give an explicit formula for the Bernoulli polynomials involving the Stirling numbers of the first and second kind.