On reciprocity formula of character Dedekind sums and the integral of products of Bernoulli polynomials

TL;DR

This paper provides a simple proof of reciprocity formulas for character Dedekind sums using the Euler-MacLaurin summation and extends results on integrals of Bernoulli polynomial products, linking these to Dedekind sums.

Contribution

It introduces a new proof method for reciprocity formulas and generalizes the integral of Bernoulli polynomial products, connecting these concepts.

Findings

Simplified proof of reciprocity formulas for character Dedekind sums.

Extended integral formulas for products of Bernoulli polynomials.

Established a link between Bernoulli polynomial sums and Dedekind sums.

Abstract

We give a simple proof for the reciprocity formulas of character Dedekind sums associated with two primitive characters, whose modulus need not to be same, by utilizing the character analogue of the Euler-MacLaurin summation formula. Moreover, we extend known results on the integral of products of Bernoulli polynomials by considering the integral \[ \int\limits_{0}^{x}B_{n_{1}}(b_{1}z+y_{1})... B_{n_{r}}(b_{r}z+y_{r}) dz, \] where $b_{l}$ $(b_{l}\neq 0)$ and $y_{l}$ $(1\leq l\leq r)$ are real numbers. As a consequence of this integral we establish a connection between the reciprocity relations of sums of products of Bernoulli polynomials and of the Dedekind sums.