Euler Maclaurin with Remainder for a Simple Integral Polytope

Because ∫ b+h2 a−h1 f(x)dx is a polynomial in h1 and h2 if f is a polynomial in x, applying the infinite order differential operator L( ∂ ∂hi ) then yields a finite sum, so the right hand side of (2) is well defined when f is a polynomial. A polytope in Rn is called integral, or a lattice polytope, if its vertices are in the lattice Zn; it is called simple if exactly n edges emanate from each vertex; it is called regular if, additionally, the edges emanating from each vertex lie along lines which are generated by a Z-basis of the lattice Zn. Khovanskii and Pukhlikov [KP1, KP2], following Khovanskii [Kh1, Kh2], generalized the classical Euler Maclaurin formula to give a formula for the sums of the values of polynomial or exponential functions on the lattice points in higher dimensional convex polytopes ∆ which are integral and regular. This formula was generalized to simple integral polytopes by Cappell and Shaneson [CS1, CS2, CS3, S], and subsequently by Guillemin [Gu2] and by Brion-Vergne [BV]. All of these generalizations involve “corrections” to the Khovanskii-Pukhlikov formula when the simple polytope is not regular. When applied to the constant function f ≡ 1, these Euler Maclaurin formulas compute the number of lattice points in ∆ in terms of the volumes of “dilations” of ∆. A small