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On reciprocity formula of character Dedekind sums and the integral of products of Bernoulli polynomials (1412.7363v2)
Published 23 Dec 2014 in math.NT
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.