From Eulerian to Bernoulli Numbers via Ehrhart Polynomials

2026/07/24 by Luis Ferroni
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebra over a field #Bernoulli number #Bernoulli polynomials #Bernoulli's principle #Eulerian path #History and Theory of Mathematics

paper · doi:10.1080/00029890.2026.2697721

openalex publication_date 2026/07/24 · openalex created_date 2026/07/25 · openalex updated_date 2026/07/26

Abstract

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.

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