Asymptotic Euler–Maclaurin formula over lattice polytopes

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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.