An algorithm for the Faulhaber polynomials

2021/03/15 by José L. Cereceda, Cereceda, José L.
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2103.08553

openalex publication_date 2021/03/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let S_p(n) denote the sum of pth powers of the first n positive integers 1^p + 2^p + ⋯ + n^p. In this paper, first we express S_p(n) in the so-called Faulhaber form, namely, as an even or odd polynomial in (n + 1/2), according as p is odd or even. Then, using the relation S_p(n) - S_p(n-1) = n^p, we derive a recursive formula for the associated Faulhaber coefficients. Applying Cramer's rule to the corresponding system of equations, we obtain an explicit determinant formula for the said coefficients. Furthermore, we show how to convert the (even or odd) Faulhaber polynomials in (n+ 1/2) into polynomials in S₁(n) for any arbitrary p, and vice versa.

Related

• Sums of powers via integration
• On the sum of k-th powers in terms of earlier sums
• Faulhaber's Theorem on Power Sums
• A simple recursive representation of the Faulhaber series
• On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat