The most concise recurrence formula for the sums of integer powers

2026/01/26 by José L. Cereceda · 1 voice
Mathematics · #math.HO #math.NT

paper · pdf · doi:10.1017/mag.2025.10073

arxiv published 2026/01/26 · arxiv updated 2026/01/29

Abstract

For integers n,k ≥ 1, let S_k(n) denote the power sum 1^k +2^k + ⋯ + 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.

Citations

• A Short Proof of a Known Relation for Consecutive Power Sums

Discussions

• Jos\'e L. Cereceda: The most concise recurrence formula for the sums of integer powers https://arxiv.org/abs/2601.18855 https://arxiv.org/pdf/2601.18855 https://arxiv.org/html/2601.18855 [bsky, 0 points, 0 comments]

Related

• Sums of powers of integers via differentiation
• On the sum of k-th powers in terms of earlier sums
• On an exponential power sum
• A remark on an explicit formula for the sums of powers of integers
• On sums of powers of natural numbers