From Eulerian to Bernoulli Numbers via Ehrhart Polynomials

The Bernoulli numbers are pervasive in analysis and number theory, and appear naturally when studying the Riemann zeta function. The Eulerian numbers are equally relevant in enumerative combinatorics and discrete geometry. We provide a simple and surprising proof of an identity relating Bernoulli and Eulerian numbers by means of the Ehrhart theory of polytopes. The ideas in this note can be used to generalize this identity in many ways.