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We provide a multidimensional weighted Euler--MacLaurin summation formula onpolytopes and a multidimensional generalization of a result due to L. J.Mordell on the series expansion in Bernoulli polynomials. These results areconsequences of a more general series expansion; namely, if $\chi_{\tau\mathcal{P}}$ denotes the characteristic function of a dilated integerconvex polytope $\mathcal{P}$ and $q$ is a function with suitable regularity,we prove that the periodization of $q\chi_{\tau\mathcal{P}}$ admits anexpansion in terms of multivariate Bernoulli polynomials. These multivariatepolynomials are related to the Lerch Zeta function. In order to prove ourresults we need to carefully study the asymptotic expansion of$\widehat{q\chi_{\tau\mathcal{P}}}$, the Fourier transform of $q\chi_{\tau\mathcal{P}}$.

Euler-MacLaurin summation formula on polytopes and expansions in multivariate Bernoulli polynomials