On the Almkvist--Meurman Theorem for Bernoulli Polynomials

Almkvist and Meurman showed that if $h$ and $k$ are integers, then so is \[k^n\bigl(B_n(h/k) - B_n\bigr),\] 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.