Faulhaber polynomials and reciprocal Bernoulli polynomials

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.