The most concise recurrence formula for the sums of integer powers

TL;DR

The paper studies sums of integer powers $S_k(n)$, showing that $S_k(n)$ can be represented as a polynomial of degree $k+1$ with zero constant term and that several classical methods—Abramovich's minimal recurrence, the integration-based approach, the Bloom-Owens coefficient transfer, and Budin-Cantor's relation—are equivalent for deriving the coefficients from $S_{k-1}(n)$. It further derives a determinantal formula for the Bernoulli numbers and connects the coefficients to Bernoulli numbers via explicit expressions, enriching the relationships between power sums and Bernoulli polynomials. These results unify disparate procedures into a single coherent framework and provide compact, computable recurrences for power sums and Bernoulli-number representations.

Abstract

For integers $n,k \geq 1$, let $S_k(n)$ denote the power sum $1^k +2^k + \cdots + n^k$. In this note, we first recall the minimal recurrence relation connecting $S_k(n)$ and $S_{k-1}(n)$ established by Abramovich (1973). We then discuss an odd algorithm to determine the coefficients of the power sum polynomial $S_k(n)$ in terms of the coefficients of $S_{k-1}(n)$ (see, e.g., Bloom (1993) and Owens (1992)). Moreover, we bring to light an explicit relationship between $S_k(n)$ and $S_{k+1}(n)$ put forward by Budin and Cantor (1972). We conclude that these procedures (including the integration formula expressing $S_k(n)$ in terms of $S_{k-1}(n)$) all constitute equivalent methods to determine $S_k(n)$ starting from $S_{k-1}(n)$. In addition, as a by-product, we provide a determinantal formula for the Bernoulli numbers involving the binomial coefficients.