An algorithm for the Faulhaber polynomials

TL;DR

This paper derives explicit formulas and recursive relations for Faulhaber polynomials representing sums of powers, providing new methods to convert between polynomial forms and sums of powers.

Contribution

It introduces explicit determinant formulas and conversion techniques for Faulhaber coefficients and polynomials, advancing the understanding of power sum representations.

Findings

Derived recursive formulas for Faulhaber coefficients

Obtained explicit determinant formulas for coefficients

Developed methods to convert between polynomial forms

Abstract

Let $S_p(n)$ denote the sum of $p$th powers of the first $n$ positive integers $1^p + 2^p + \cdots + n^p$. 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 $S_1(n)$ for any arbitrary $p$, and vice versa.