Exact Euler–Maclaurin formulas for simple lattice polytopes

Euler–Maclaurin formulas for a polytope express the sum of the values of a function over the lattice points in the polytope in terms of integrals of the function and its derivatives over faces of the polytope or its expansions. Exact Euler–Maclaurin formulas [A.G. Khovanskii, A.V. Pukhlikov, Algebra and Analysis 4 (1992) 188–216; S.E. Cappell, J.L. Shaneson, Bull. Amer. Math. Soc. 30 (1994) 62–69; C. R. Acad. Sci. Paris Sér. I Math. 321 (1995) 885–890; V. Guillemin, J. Differential Geom. 45 (1997) 53–73; M. Brion, M. Vergne, J. Amer. Math. Soc. 10 (2) (1997) 371–392] apply to exponential or polynomial functions; Euler–Maclaurin formulas with remainder [Y. Karshon, S. Sternberg, J. Weitsman, Proc. Natl. Acad. Sci. 100 (2) (2003) 426–433; Duke Math. J. 130 (3) (2005) 401–434] apply to more general smooth functions.In this paper we review these results and present proofs of the exact formulas obtained by these authors, using elementary methods. We then use an algebraic formalism due to Cappell and Shaneson to relate the different formulas.