We continue our study of intermediate sums over polyhedra, interpolating between integrals and discrete sums, which were introduced by Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex. Math. Comp. 75 (2006), 1449–1466]. By well-known decompositions, it is sufficient to consider the case of affine cones
is an arbitrary real vertex and
is a rational polyhedral cone. For a given rational subspace
, we define the intermediate generating functions
by integrating an exponential function over all lattice slices of the affine cone
parallel to the subspace
and summing up the integrals. We expose the bidegree structure in parameters
, which was implicitly used in the algorithms in our papers [Computation of the highest coefficients of weighted Ehrhart quasi-polynomials of rational polyhedra. Found. Comput. Math.
12 (2012), 435–469] and [Intermediate sums on polyhedra: computation and real Ehrhart theory. Mathematika
59 (2013), 1–22]. The bidegree structure is key to a new proof for the Baldoni–Berline–Vergne approximation theorem for discrete generating functions [Local Euler–Maclaurin expansion of Barvinok valuations and Ehrhart coefficients of rational polytopes. Contemp. Math.
452 (2008), 15–33], using the Fourier analysis with respect to the parameter
and a continuity argument. Our study also enables a forthcoming paper, in which we study intermediate sums over multi-parameter families of polytopes.