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Abstract

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 \CC $ be a homogeneous polynomial function
of degree $d$ (i.e., an element of $\Sym^{d} (V^{*})$). For $q\in \ZZ_{>0}$ 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
$G_{\varphi}(q,y) \in \QQ [q] [y]$ 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 $G_{\varphi}$ can be computed using an analogue of Brion--Vergne's
Euler--Maclaurin summation formula.