Asymptotic Euler-Maclaurin formula over lattice polytopes

Formulas for the Riemann sums over lattice polytopes determined by the lattice points in the polytopes are often called Euler-Maclaurin formulas. An asymptotic Euler-Maclaurin formula, by which we mean an asymptotic expansion formula for Riemann sums over lattice polytopes, was first obtained by Guillemin and Sternberg (2007) [11]. Then, the problem is to find a concrete formula for each term of the expansion. In this paper, an asymptotic Euler-Maclaurin formula of the Riemann sums over general lattice polytopes is given. The formula given here is an asymptotic form of the so-called local Euler-Maclaurin formula of Berline and Vergne (2007) [3]. For Delzant polytopes, our proof given here is independent of the local Euler-Maclaurin formula. Furthermore, a concrete description of differential operators which appear in each term of the asymptotic expansion for Delzant lattice polytopes is given. By using this description, when the polytopes are Delzant lattice, a concrete formula for each term of the expansion in two dimension and a formula for the third term of the expansion in arbitrary dimension are given.