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

M. Cihat Dağlı, Mümün Can

Published 2014-12-23Version 1

In this paper, we study on two subjects. We first give a new proof for the reciprocity formula of character Dedekind sums with the help of the character analogue of the Euler-MacLaurin summation formula. Secondly, we extend known results on the integral of products of Bernoulli polynomials by considering the integral \[\int_{0}^{x} B_{n_{1}}\left( b_{1}z+y_{1}\right) \cdots B_{n_{r}}\left( b_{r}z+y_{r}\right) dz.\] 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. As applications we present some integrals involving periodic Bernoulli polynomials.