Yes, this is a big area of research. I'll add some references to the ones Dmitri provides.
Here are references (including some that Dmitri has already put, for completeness), see also my from a question about Moment map for toric actions:
- Riemann-Roch for toric orbifolds by Victor Guillemin
- Residues formulae for volumes and Ehrhart polynomials of convex polytopes, arXiv:math/0103097
- Local Euler-Maclaurin formula for polytopes
- to learn about toric geometry, arXiv:math/0507256
- Paradan's wall crossing formula for partition functions and Khovanski-Pukhlikov differential operator
- other papers draft of a book Toric Varieties by Michèle VergneCox et al
More on the topic itself:
- Riemann sums over polytopes by Victor Guillemin and Shlomo Sternberg
- Exact Euler Maclaurin formulas for simple lattice polytopes by Shlomo Sternberg et al.
- the original paper by Pukhlikov and Khovanskii (abstract page in English, full text in Russian)
And to learn about toric geometry
A series of papers on arXiv by Michèle Vergne, especially:
- draft
- Residues formulae for volumes and Ehrhart polynomials of a book Toric Varietiesconvex polytopes, arXiv:math/0103097
- Local Euler-Maclaurin formula for polytopes, arXiv:math/0507256
- Paradan's wall crossing formula for partition functions and Khovanski-Pukhlikov differential operator
Also papers by Cox et alBrion and Vergne, which seem to be missing from arXiv (Google Scholar, thanks to Steve).
