On the integral of the product of four and more Bernoulli polynomials

TL;DR

This paper derives explicit formulas for the integrals of products of any number of Bernoulli polynomials, extending previous results and addressing a long-standing challenge in the field.

Contribution

It provides a general explicit formula for the integral of the product of multiple Bernoulli polynomials, unifying and extending prior special cases.

Findings

Explicit formulas for integrals of multiple Bernoulli polynomials

Unification of previous special case results

Addresses longstanding open problem in Bernoulli polynomial integrals

Abstract

In 1958, L.J. Mordell provided the formula for the integral of the product of two Bernoulli polynomials, he also remarked: "The integrals containing the product of more than two Bernoulli polynomials do not appear to lead to simple results." In this paper, we provide explicit formulas for the integral of the product of $r$ Bernoulli polynomials, where $r$ is any positive integer. Many authors' results in this direction, including N\"orlund, Mordell, Carlitz, Agoh and Dilcher are special cases of the formulas given in this paper.