A determinantal formula for the hyper-sums of powers of integers

TL;DR

This paper presents a new determinantal formula for hyper-sums of powers of integers, expressing these sums in terms of determinants involving Bernoulli numbers and providing polynomial representations depending on parity.

Contribution

The paper introduces a novel determinantal expression for hyper-sums of powers, linking them to lower Hessenberg matrices and Bernoulli numbers, expanding the analytical tools for these sums.

Findings

Derived a determinantal formula for hyper-sums of powers.

Expressed hyper-sums as polynomials in shifted variable $N_r$.

Connected hyper-sums to Bernoulli numbers through matrix determinants.

Abstract

For non-negative integers $r$ and $m$, let $S_m^{(r)}(n)$ denote the $r$-fold summation (or hyper-sum) over the first $n$ positive integers to the $m$th powers, with the initial condition $S_m^{(0)}(n) =n^m$. In this paper, we derive a new determinantal formula for $S_m^{(r)}(n)$. Specifically, we show that, for all integers $r\geq 0$ and $m \geq 1$, $S_m^{(r)}(n)$ is proportional to $S_1^{(r)}(n)$ times the determinant of a lower Hessenberg matrix of order $m-1$ involving the Bernoulli numbers and the variable $N_r = n + \frac{r}{2}$. Furthermore, whenever $r\geq 1$, evaluating this determinant gives us $S_m^{(r)}(n)$ as $S_1^{(r)}(n)$ times an even or odd polynomial in $N_r$ of degree $m-1$, depending on whether $m$ is odd or even.