Weighted Lattice Point Sums in Lattice Polytopes, Unifying Dehn–Sommerville and Ehrhart–Macdonald

Let \(V\) be a real vector space of dimension \(n\) and let \(M\subset V\) be a lattice. Let \(P\subset V\) be an \(n\)-dimensional polytope with vertices in \(M\), and let \(\varphi\colon V\rightarrow \mathbb{C}\) be a homogeneous polynomial function of degree \(d\). For \(q\in \mathbb{Z}_{>0}\) and any face \(F\) of \(P\), let \(D_{\varphi, F}(q)\) be the sum of \(\varphi\) over the lattice points in the dilate \(qF\). We define a generating function \(G_{\varphi}(q,y) \in \mathbb{Q}[q][y]\) packaging together the various \(D_{\varphi ,F}(q)\), and show that it satisfies a functional equation that simultaneously generalizes Ehrhart–Macdonald reciprocity and the Dehn–Sommerville relations. When \(P\) is a simple lattice polytope (i.e., each vertex meets \(n\) edges), we show how \(G_{\varphi}\) can be computed using an analogue of Brion–Vergne’s Euler–Maclaurin summation formula.