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

We give a simple proof for the reciprocity formulas of character Dedekind sums associated with two primitive characters, whose modulus need not to be the 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∫0xBn1(b1z+y1)⋯Bnr(brz+yr)dz, where blbl(bl≠0)(bl≠0) and ylyl(1≤l≤r)(1≤l≤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.