An Euler operator approach to Ehrhart series

Abstract

We use the ordinary Euler operator to compute the Ehrhart series for an arbitrary lattice polytope. The resulting formula involves the coefficients of the Ehrhart polynomial, combined via Eulerian numbers. We use this to compute $h_{d-1}^{\ast }$ in terms of the coefficients of the Ehrhart polynomial, resulting in a new linear inequality satisfied by the coefficents of the Ehrhart polynomial.