Another Approach on Power Sums

TL;DR

This paper introduces explicit polynomial forms to efficiently compute power sums of any order, providing a new basis for expressing monomials and sums with nested summations.

Contribution

It develops explicit forms for polynomials that serve as a basis to express monomials and power sums, simplifying calculations of nested power sums.

Findings

Explicit polynomial forms for power sums are derived.

Power sums can be expressed as linear combinations of these polynomials.

The method simplifies the computation of nested power sums.

Abstract

We show that explicit forms for certain polynomials~$\psi^{(a)}_m(n)$ with the property \[ \psi^{(a+1)}_m(n) = \sum_{\nu=1}^n \psi_m^{(a)}(\nu) \] can be found (here, $a,m,n\in\mathbb{N}_0$). We use these polynomials as a basis to express the monomials~$n^m$. Once the expansion coefficients are determined, we can express the $m$-th power sums~$S^{(a)}_m(n)$ of any order $a$, \[ S^{(a)}_m(n) = \sum_{\nu_a = 1}^n \cdots \sum_{\nu_2 = 1}^{\nu_3} \sum_{\nu_1=1}^{\nu_2} \nu_1^m, \] in a very convenient way by exploiting the summation property of the $\psi_m^{(a)}$, \[ S^{(a)}_m(n) = \sum_k c_{mk} \psi_k^{(a)}(n). \]