The most concise recurrence formula for the sums of integer powers

Abstract

For integers $n,k\ge 1$, 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.

6 pages; corrects some minor misprints in the published version