## Another congruence for the Apéry numbers.(English)Zbl 0614.10011

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.

### 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
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### 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 P2 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., () [11] {\scC. A. M. Peters}, Monodromy and Picard-Fuchs equations for families of K3-surfaces and elliptic curves, preprint. · 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., () [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 Princeton
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