A short proof of the Almkvist-Meurman theorem

2023/10/23 by Ira M. Gessel, Gessel, Ira M.
Mathematics · #11B68 (Primary) 05A15(Secondary) #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Number Theory (math.NT) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2310.15312

openalex publication_date 2023/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a short generating function proof of the Almkvist-Meurman theorem: For integers h and k≠0, define the numbers M_n(h,k) by kx(e^hx-1)/(e^kx-1)=∑_n=0^∞ M_n(h,k) xⁿ/n!. Equivalently, M_n(h,k) = kⁿ(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.

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