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 $\phi \text{\hspace{0.17em}⁣}:V\to \text{CC}$ be a homogeneous polynomial function of degree $d$ (i.e., an element of ${\text{Sym}}^d(V^{\ast })$). For $q\in {\text{ZZ}}_{>0}$ and any face $F$ of $P$, let $D_{\phi ,F}(q)$ be the sum of $\phi$ over the lattice points in the dilate $qF$. We define a generating function $G_{\phi }(q,y)\in \text{QQ}[q][y]$ 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 $G_{\phi }$ can be computed using an analogue of Brion–Vergne’s Euler–Maclaurin summation formula.