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Supercongruences for the Catalan–Larcombe–French numbers

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In this short note, we develop the Stienstra–Beukers theory of supercongruences in the setting of the Catalan–Larcombe–French sequence. We also give some applications to other sequences.

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  1. Apostol, T.M.: Modular Functions and Dirichlet Series in Number Theory. Graduate Texts in Mathematics, vol. 41. Springer, New York (1976)

    MATH  Google Scholar 

  2. Beukers, F.: Another congruence for the Apéry numbers. J. Number Theory 25, 201–210 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  3. Borwein, J.M., Borwein, P.B.: Pi and the AGM. Wiley–Interscience, New York (1987)

    MATH  Google Scholar 

  4. Catalan, E.: Sur les nombres de Segner. Rend. Circ. Mat. Palermo 1, 190–201 (1887)

    Article  Google Scholar 

  5. Conway, J.H., Norton, S.P.: Monstrous moonshine. Bull. Lond. Math. Soc. 11, 308–339 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cusick, T.W.: Recurrences for sums of powers of binomial coefficients. J. Comb. Theory, Ser. A 52, 77–83 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fang, L., Hoffman, J.W., Linowitz, B., Rupinski, A., Verrill, H.A.: Modular forms on noncongruence subgroups and Atkin–Swinnerton–Dyer relations. arXiv:0805.2144 [math.NT]

  8. Fine, N.J.: Basic Hypergeometric Series and Applications. With a foreword by George E. Andrews. Mathematical Surveys and Monographs, vol. 27. Am. Math. Soc., Providence (1988). xvi+124 pp. ISBN:0-8218-1524-5

    MATH  Google Scholar 

  9. Granville, A.: Arithmetic properties of binomial coefficients I. Binomial coefficients modulo prime powers. In: Organic Mathematics, Burnaby, BC, 1995. CMS Conf. Proc., vol. 20, pp. 253–276. Am. Math. Soc., Providence (1997)

    Google Scholar 

  10. Jarvis, A.F., Larcombe, P., French, D.: Applications of the A.G.M. of Gauss: some new properties of the Catalan–Larcombe–French sequence. Congr. Numer. 161, 151–162 (2003). Proceedings of the Thirty-Fourth Southeastern International Conference on Combinatorics, Graph Theory and Computing

    MathSciNet  MATH  Google Scholar 

  11. Jarvis, A.F., Larcombe, P., French, D.: On small prime divisibility of the Catalan–Larcombe–French numbers. Indian J. Math. 47, 159–181 (2005)

    MathSciNet  MATH  Google Scholar 

  12. Larcombe, P., French, D.: On the ‘other’ Catalan numbers: a historical formulation re-examined. Congr. Numer. 143, 33–64 (2000)

    MathSciNet  MATH  Google Scholar 

  13. Martin, Y.: Multiplicative η-quotients. Trans. Am. Math. Soc. 348, 4825–4856 (1996)

    Article  MATH  Google Scholar 

  14. Shimura, G.: Introduction to the Arithmetic Theory of Automorphic Functions. Publications of the Mathematical Society of Japan, vol. 11. Princeton University Press, Princeton (1994), xiv+271 pp. Reprint of the 1971 original. ISBN:0-691-08092-5.

    MATH  Google Scholar 

  15. Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences, published electronically at

  16. Stienstra, J., Beukers, F.: On the Picard–Fuchs equation and the formal Brauer group of certain elliptic K3-surfaces. Math. Ann. 271, 271–304 (1985)

    Article  MathSciNet  Google Scholar 

  17. Verrill, H.A.: Picard–Fuchs Equations of Some Families of Elliptic Curves, Proceedings on Moonshine and Related Topics, Montréal, Québec, 1999. CRM Proc. Lecture Notes, vol. 30, pp. 253–268. Am. Math. Soc., Providence (2001)

    Google Scholar 

  18. Verrill, H.A.: Some Congruences Related to modular forms. Max Planck Institut für Mathematik preprint 26 (1999)

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Correspondence to Frazer Jarvis.

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Jarvis, F., Verrill, H.A. Supercongruences for the Catalan–Larcombe–French numbers. Ramanujan J 22, 171–186 (2010).

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