On the integral of products of higher-order Bernoulli and Euler polynomials

TL;DR

This paper derives formulas for integrals involving products of higher-order Bernoulli and Euler polynomials, explores their connections to generalized sums, and provides the Laplace transform of Euler polynomials.

Contribution

It introduces new integral formulas for higher-order Bernoulli and Euler polynomials and links these results to generalized Dedekind and Hardy--Berndt sums.

Findings

Formulas for integrals of products of higher-order Euler polynomials

Relations for higher-order Bernoulli and Euler polynomials

Connection to generalized Dedekind and Hardy--Berndt sums

Abstract

In this paper, we derive a formula on the integral of products of the higher-order Euler polynomials. By the same way, similar relations are obtained for $l$ higher-order Bernoulli polynomials and $r$ higher-order Euler polynomials. Moreover, we establish the connection between the results and the generalized Dedekind sums and Hardy--Berndt sums. Finally, the Laplace transform of Euler polynomials is given.