Beukers, F. Another congruence for the Apéry numbers. (English) Zbl 0614.10011 J. Number Theory 25, 201-210 (1987). 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 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. Reviewer: A.J.van der Poorten Cited in 6 ReviewsCited in 62 Documents MSC: 11B37 Recurrences 11F33 Congruences for modular and \(p\)-adic modular forms 11A07 Congruences; primitive roots; residue systems 14J30 \(3\)-folds 11F11 Holomorphic modular forms of integral weight Keywords:Apéry numbers; Picard-Fuchs equation; congruence properties; modular form Citations:Zbl 0425.10033; Zbl 0428.10008; Zbl 0482.10003; Zbl 0504.10007; Zbl 0571.10008; Zbl 0541.14007 PDFBibTeX XMLCite \textit{F. Beukers}, J. Number Theory 25, 201--210 (1987; Zbl 0614.10011) Full Text: DOI Online Encyclopedia of Integer Sequences: Apéry numbers: a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(n+k,k). Apery (Apéry) numbers: Sum_{k=0..n} (binomial(n,k)*binomial(n+k,k))^2. a(n) = Sum_{k = 0..n} binomial(n,k)^4. Coefficients in expansion of Eisenstein series E_2 (also called E_1 or G_2). Expansion of (phi(-q^3) * psi(q))^3 / (phi(-q) * psi(q^3)) in powers of q where phi(), psi() are Ramanujan theta functions. Expansion of a cusp form of weight 8 for Gamma_1(6). a(n) = A005259(n) mod 2*n+1. References: [1] Beukers, F., Some congruences for the Apéry numbers, J. Number Theory, 21, 141-155 (1985) · Zbl 0571.10008 [2] Beukers, F., Irrationality proofs using modular forms, Journées arithmétiques Besançon (1985), Astérisque, to appear · Zbl 0613.10031 [3] Beauville, A., Les familles stables de courbes elliptiques sur \(P^2\) admettant quatre fibres singulières, C. R. Acad. Sci. Paris, 294, 657 (1982) · Zbl 0504.14016 [4] Beukers, F.; Peters, C. A.M., A family of \(K3\) surfaces and ζ(3), J. Reine Angew. Math., 351, 42-54 (1984) · Zbl 0541.14007 [5] Chowla, S.; Cowles, J.; Cowles, M., Congruence properties of Apéry numbers, J. Number Theory, 12, 188-190 (1980) · Zbl 0428.10008 [6] Dwork, B., On Apéry’s differential operator, “Groupe d”étude d’analyse ultramétrique”, Paris (1980/1981) [7] Fano, G., Uber lineare homogene Differentialgleichungen, Math. Anal., 53, 493-590 (1900) · JFM 31.0342.01 [8] Gessel, I., Some congruences for the Apéry numbers, J. Number Theory, 14, 362-368 (1982) · Zbl 0482.10003 [9] Koike, M., On McKay’s conjecture, Nagoya Math. J., 95, 85-89 (1984) · Zbl 0548.10018 [10] Ogg, A., (Modular Forms and Dirichlet Series (1969), Benjamin: Benjamin New York) · Zbl 0191.38101 [11] C. A. M. Peters\(K\); C. A. M. Peters\(K\) · Zbl 0612.14006 [12] van der Poorten, A. J., A proof that Euler missed… Apéry’s proof of the irrationality of ζ(3), Math. Intelligencer, 1, 195-203 (1979) · Zbl 0409.10028 [13] Rademacher, H., (Topics in Analytic Number Theory (1973), Springer: Springer New York/Berlin) · Zbl 0253.10002 [14] Stienstra, J.; Beukers, F., On the Picard-Fuchs equation and the formal Brauer group of certain elliptic \(K3\) surfaces, Math. Ann., 271, 269-304 (1985) · Zbl 0539.14006 [15] Shimura, G., Introduction to the arithmetic theory of automorphic forms (1971), Iwanami Shoten: Iwanami Shoten Princeton This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.