RESIDUE FORMULAS, VECTOR PARTITION-FUNCTIONS AND LATTICE POINTS IN RATIONAL POLYTOPES

We obtain residue formulae for certain functions of several variables. As an application, we obtain closed formulae for vector partition fun ctions and for their continuous analogs. They imply an Euler-MacLaurin summation formula for vector partition functions, and for rational co nvex polytopes as well: we express the sum of values of a polynomial f unction at all lattice points of a rational convex polytope in terms o f the variation of the integral of the function over the deformed poly tope.