On the Almkvist-Meurman theorem for Bernoulli polynomials

TL;DR

This paper provides a new, simpler proof of the Almkvist-Meurman theorem for Bernoulli polynomials using generating functions, explores related properties, and extends the theorem to Euler polynomials.

Contribution

It introduces a novel proof technique for the Almkvist-Meurman theorem and generalizes it to Euler polynomials, connecting Bernoulli and Bernoulli-Stirling numbers.

Findings

Simplified proof of the Almkvist-Meurman theorem

Generalization to Bernoulli-Stirling numbers

Extension to Euler polynomials

Abstract

Almkvist and Meurman showed that if h and k are integers, then so is $k^n(B_n(h/k) - B_n)$ where $B_n(u)$ is the Bernoulli polynomial. We give here a new and simpler proof of the Almkvist-Meurman theorem using generating functions. We describe some properties of these numbers and prove a common generalization of the Almkvist-Meurman theorem and a result of Gy on Bernoulli-Stirling numbers. We then give a simple generating function proof of an analogue of the Almkvist-Meurman theorem for Euler polynomials, due to Fox.