Irrational proofs for three theorems of Stanley

We give new proofs of three theorems of Stanley on generating functions for the integer points in rational cones. The first, Stanley’s Reciprocity Theorem, relates the rational generating function σv+K(x) := ∑ m∈v+K∩Z x m, where K is a rational cone and v ∈ R, with σ−v+K◦(1/x). The second, Stanley’s Positivity Theorem, asserts that the generating function of the Ehrhart quasipolynomial LP(n) := # (