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

2022/07/28 by José L. Cereceda, Cereceda, José L.
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Mathematical Theories and Applications #FOS: Mathematics #Graph theory and applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2207.14188

openalex publication_date 2022/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 mth 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≥ 0 and m ≥ 1, S_m^(r)(n) is proportional to S₁^(r)(n) times the determinant of a lower Hessenberg matrix of order m-1 involving the Bernoulli numbers and the variable N_r = n + (r)/(2). Furthermore, whenever r≥ 1, evaluating this determinant gives us S_m^(r)(n) as S₁^(r)(n) times an even or odd polynomial in N_r of degree m-1, depending on whether m is odd or even.

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