Several properties of summatory Ehrhart polynomials and series of convex lattice polytopes

Abstract

In this article, for a convex lattice polytope, we further investigate the summatory function of its Ehrhart polynomial, which is called the summatory Ehrhart polynomial, and introduce its summatory Ehrhart series. We prove several fundamental properties of these invariants. In particular, we derive a summatory analogue of the classical Ehrhart--Macdonald reciprocity law, which establishes a signed functional equation between the polytope and its relative interior via the substitution $t \mapsto 1-t.$