Note on the coefficients of rational Ehrhart quasi-polynomials of Minkowski-sums

By extending former results of Ehrhart, it was shown by Peter McMullen that the number of lattice points in the Minkowski-sum of dilated rational polytopes is a quasipolynomial function in the dilation factors. Here we take a closer look at the coefficients of these quasi-polynomials and show that they are piecewise polynomials themselves and that they are related to each other by a simple differential equation. As a corollary, we obtain a refinement of former results on lattice points in vector dilated polytopes