A short proof of the Almkvist-Meurman theorem

Ira M. Gessel

paper

A short proof of the Almkvist-Meurman theorem

arXiv:2310.15312

Abstract

We give a short generating function proof of the Almkvist-Meurman theorem: For integers $h$ and $k \neq = 0$ , define the numbers $M_{n} (h, k)$ by $k x (e^{h x} - 1) / (e^{k x} - 1) = \sum_{n = 0}^{\infty} M_{n} (h, k) x^{n} / n!$ . Equivalently, $M_{n} (h, k) = k^{n} (B_{n} (h / k) - B_{n})$ , where $B_{n} (u)$ is the Bernoulli polynomial. Then $M_{n} (h, k)$ is an integer. The proof is related to Postnikov's functional equation for the generating function for intransitive trees.