Local Euler–Maclaurin formula for polytopes

We prove a local Euler–Maclaurin formula for rational convex polytopes in a rational Euclidean space. For every affine rational polyhedral cone $\mathfrak c$ in $V$, we construct a differential operator of infinite order $D(\mathfrak c)$ on $V$ with constant rational coefficients. Then for every convex rational polytope $\mathfrak p$ in $V$ and every polynomial function $h(x)$ on $V$, the sum of the values of $h(x)$ at the integral points of $\mathfrak p$ is equal to the sum, for all faces f of $\mathfrak p$, of the integral over $\mathfrak f$ of the function $D(\mathfrak t(\mathfrak{p,f}))\cdot h$ where we denote by $\mathfrak t(\mathfrak{p,f})$ the transverse cone of $\mathfrak p$ along $\mathfrak f$, an affine cone of dimension equal to the codimension of $\mathfrak f$. Applications to numerical computations when $\mathfrak p$ is a polygon are given.