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 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 $$\underset{0}{\overset{x}{\int }}{B}_{n_1}(b_{1}z+y_1)\mathrm{.}\mathrm{.}\mathrm{.}{B}_{n_r}(b_{r}z+y_r)dz,$$ where $b_l$ $(b_l\mathrm{\ne }0)$ 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.