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

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 $b_l$ $(b_l\ne 0)$ and $y_l$ $(1\le l\le r)$ 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.

References in corpus (1)

• Character analogues of certain Hardy-Berndt sums

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