The most concise recurrence formula for the sums of integer powers

TL;DR

This paper reviews and connects various methods for deriving the minimal recurrence relation and explicit formulas for sums of integer powers, highlighting their equivalence and providing a new determinantal formula for Bernoulli numbers.

Contribution

It clarifies the relationships among different algorithms for computing power sum polynomials and introduces a new determinantal formula for Bernoulli numbers.

Findings

Recurrence relation connecting $S_k(n)$ and $S_{k-1}(n)$ is minimal and well-established.

Various methods for deriving $S_k(n)$ are equivalent, including recurrence, coefficient algorithms, and integral formulas.

A new determinantal formula for Bernoulli numbers involving binomial coefficients is provided.

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 old 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.