Faulhaber polynomials and reciprocal Bernoulli polynomials

TL;DR

This paper explores Faulhaber polynomials related to power sums, revealing new recurrence relations and expressing coefficients via derivatives of reciprocal Bernoulli polynomials, simplifying their computation and uncovering Bernoulli number symmetries.

Contribution

It introduces a novel method to compute Faulhaber polynomial coefficients using derivatives of reciprocal Bernoulli polynomials, simplifying previous formulas.

Findings

Recurrences between Faulhaber polynomials are described by a differential operator.

Coefficients of Faulhaber polynomials can be expressed through derivatives of reciprocal Bernoulli polynomials.

New recurrences of Bernoulli numbers are derived from symmetry properties.

Abstract

About four centuries ago, Johann Faulhaber developed formulas for the power sum $1^n + 2^n + \cdots + m^n$ in terms of $m(m+1)/2$. The resulting polynomials are called the Faulhaber polynomials. We first give a short survey of Faulhaber's work and discuss the results of Jacobi (1834) and the less known ones of Schr\"oder (1867), which already imply some results published afterwards. We then show, for suitable odd integers $n$, the following properties of the Faulhaber polynomials $F_n$. The recurrences between $F_n$, $F_{n-1}$, and $F_{n-2}$ can be described by a certain differential operator. Furthermore, we derive a recurrence formula for the coefficients of $F_n$ that is the complement of a formula of Gessel and Viennot (1989). As a main result, we show that these coefficients can be expressed and computed in different ways by derivatives of generalized reciprocal Bernoulli polynomials, whose values can also be interpreted as central coefficients. This new approach finally leads to a simplified representation of the Faulhaber polynomials. As an application, we obtain some recurrences of the Bernoulli numbers, which are induced by symmetry properties.