Asymptotic Euler-Maclaurin formula over lattice polytopes

An asymptotic expansion formula of Riemann sums over lattice polytopes is given. The formula is an asymptotic form of the local Euler-Maclaurin formula due to Berline-Vergne. The proof given here for Delzant lattice polytopes is independent of the local Euler-Maclaurin formula. But we use it for general lattice polytopes. As corollaries, an explicit formula for each term in the expansion over Delzant polytopes in two dimension and an explicit formula for the third term of the expansion for Delzant polytopes in arbitrary dimension are given. Moreover, some uniqueness results are given.Comment: 35 pages. Results in the previous version are generalized to lattice polytopes. Some further results are added. The title is changed. The organization is changed to clarify the discussion

Riemann sums over polytopes

We show that the Euler-MacLaurin formula for Riemann sums has an n-dimensional analogue in which intervals on the line get replaced by convex polytopes.Comment: 13 page

Resurgence of the Euler-MacLaurin summation formula

The Euler-MacLaurin summation formula relates a sum of a function to a corresponding integral, with a remainder term. The remainder term has an asymptotic expansion, and for a typical analytic function, it is a divergent (Gevrey-1) series. Under some decay assumptions of the function in a half-plane (resp. in the vertical strip containing the summation interval), Hardy (resp. Abel-Plana) prove that the asymptotic expansion is a Borel summable series, and give an exact Euler-MacLaurin summation formula. Using a mild resurgence hypothesis for the function to be summed, we give a Borel summable transseries expression for the remainder term, as well as a Laplace integral formula, with an explicit integrand which is a resurgent function itself. In particular, our summation formula allows for resurgent functions with singularities in the vertical strip containing the summation interval. Finally, we give two applications of our results. One concerns the construction of solutions of linear difference equations with a small parameter. And another concerns the problem of proving resurgence of formal power series associated to knotted objects.Comment: AMS-LaTeX, 15 pages with 2 figure