Factorization of and Determinant Expressions for the Hypersums of Powers of Integers

TL;DR

This paper presents a compact determinant formula for hypersum polynomials of powers of integers, enabling efficient calculation and factorization of these sums in terms of N and L.

Contribution

It introduces a novel determinant expression for hypersum polynomials, simplifying their computation and revealing their factorization properties.

Findings

Derived a determinant formula for hypersum polynomials

Provided explicit factorization of hypersum polynomials

Enhanced computational efficiency for sums of powers

Abstract

We derive a compact determinant formula for calculating and factorizing the hypersum polynomials S^{(L)}_k(N) \equiv \sum_{n_1=1}^N ...\sum_{n_{L+1}=1}^{n_{L}}(n_{L+1})^k expressed in the variable N(N+L+1)