  We study a natural set of refinements of the Ehrhart series of a closed
polytope, first considered by Chapoton. We compute the refined series in full
generality for a simplex of dimension d, a cross-polytope of dimension d,
respectively a hypercube of dimension d<4, using commutative algebra. We deduce
summation formulae for products of q-integers with different arguments,
generalizing a classical identity due to MacMahon and Carlitz. We also present
a characterisation of a certain refined Eulerian polynomial in algebraic terms.
