Another congruence for the Apéry numbers
type: Item
instance of: scholarly article
title: Another congruence for the Apéry numbers
zbMATH Open document ID: 0614.10011
DOI: 10.1016/0022-314X(87)90025-4
published in: Journal of Number Theory
publication date: 1987-01-01T00:00:00Z
review text: In 1979 Apéry introduced the numbers \[ a_ n=\sum^{n}_{k=0}\left( \begin{matrix} n\\ k\end{matrix} \right)^ 2 \left( \begin{matrix} n+k\\ k\end{matrix} \right)^ 2 \] in his irrationality proofs for \(\zeta\) (3). Subsequently various writers discovered interesting congruence properties of these numbers (there are papers in J. Number Theory 12, 14, 16 and 21; see the citations of the present paper or Zbl 0425.10033; Zbl 0428.10008; Zbl 0482.10003; Zbl 0504.10007; Zbl 0571.10008). The author notes that the generating function for the \(a_ n\) satisfies a certain differential equation which is the symmetric square of a second order linear differential equation and that the function can be interpreted as a period of a family of K3 surfaces [see the author and \textit{C. A. M. Peters}, J. Reine Angew. Math. 351, 42-54 (1984; Zbl 0541.14007)]. In the present note the author uses an ad hoc method to give a proof for a new congruence for the \(a_ n\) by relating the generating function for the \(a_ n\) to a certain modular form; however he remarks that the congruence must arise from the interplay between the numbers \(a_ n\) and the \(\zeta\)-function of a certain algebraic threefold and it is likely that a stronger congruence holds.
Mathematics Subject Classification ID: 11B37
Mathematics Subject Classification ID: 11F33
Mathematics Subject Classification ID: 11A07
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 11F11
zbMATH DE Number: 3993674
zbMATH Keywords: Apéry numbers
zbMATH Keywords: Picard-Fuchs equation
zbMATH Keywords: congruence properties
zbMATH Keywords: modular form
author: Frits Beukers
MaRDI profile type: Publication
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cites work: Q5625265
full work available at URL: 0022-314x(87)90025-4
OpenAlex ID: W2048654814
description: scientific article; zbMATH DE number 3993674