A note on another approach on power sums

TL;DR

This paper reviews a recent novel approach to power sums, compares it with the original method, and provides additional insights into the coefficients involved, including a new way to compute them and a conjecture on their form.

Contribution

It introduces an alternative matrix inversion method for calculating coefficients and offers supplementary facts and a conjecture related to the power sums approach.

Findings

Coefficients can be obtained by inverting a binomial coefficient matrix.

Compared to the original, the new method involves a different matrix structure.

A conjecture is proposed regarding the functional form of certain coefficients.

Abstract

In this note, we first review the novel approach to power sums put forward recently by Muschielok in arXiv:2207.01935v1, which can be summarized by the formula $S_m^{(a)}(n) = \sum_{k} c_{mk} \psi_k^{(a)}(n)$, where the $c_{mk}$'s are the expansion coefficients and where the basis functions $\psi_m^{(a)}(n)$ fulfil the recursive property $\psi_m^{(a+1)}(n)= \sum_{i=1}^n \psi_m^{(a)}(i)$. Then, we point out a number of supplementary facts concerning the said approach not contemplated explicitly in Muschielok's paper. In particular, we show that, for any given $m$, the values of the $c_{mk}$'s can be obtained by inverting a matrix involving only binomial coefficients. This may be compared with the original approach of Muschielok, where the values of the $c_{mk}$'s can be obtained by inverting a lower triangular matrix involving the Stirling numbers of the first kind. Also, we make a conjecture about the functional form of the coefficients $c_{m\, m-k}$.