Residue formulae for vector partitions and Euler-MacLaurin sums

Let V be an n-dimensional real vector space endowed with a rank-n lattice Γ . The dual lattice Γ ∗ = Hom(Γ.Z) is naturally a subset of the dual vector space V ∗. Let Φ = [β1, β2, . . . , βN ] be a sequence of not necessarily distinct elements of Γ ∗, which span V ∗ and lie entirely in an open halfspace of V ∗. In what follows, the order of elements in the sequence will not matter. The closed cone C(Φ) generated by the elements of Φ is an acute convex cone, divided into open conic chambers by the (n− 1)-dimensional cones generated by linearly independent (n−1)-tuples of elements of Φ . Denote by ZΦ the sublattice of Γ ∗ generated by Φ . Pick a vector a ∈ V ∗ in the cone C(Φ), and denote by ΠΦ(a) ⊂ R+ the convex polytope consisting of all solutions x = (x1, x2, . . . , xN) of the equation ∑Nk=1 xkβk = a in nonnegative real numbers xk . This is a closed convex polytope called the partition polytope associated to Φ and a. Conversely, any closed convex polytope can be realized as a partition polytope. If λ ∈ Γ ∗, then the vertices of the partition polytope ΠΦ(λ) have rational coordinates. We denote by ιΦ(λ) the number of points with integral coordinates in ΠΦ(λ). Thus ιΦ(λ) is the number of solutions of the equation ∑N k=1 xkβk = λ in nonnegative integers xk . The function λ → ιΦ(λ) is called the vector partition function associated to Φ . Obviously, ιΦ(λ) vanishes if λ does not belong to C(Φ) ∩ZΦ .