Another congruence for the Apéry numbers
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.
scientific article; zbMATH DE number 3993674