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Supercongruences for sporadic sequences

  • Robert Osburn (a1), Brundaban Sahu (a2) and Armin Straub (a3)
Abstract

We prove two-term supercongruences for generalizations of recently discovered sporadic sequences of Cooper. We also discuss recent progress and future directions concerning other types of supercongruences.

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* Present address: Max-Planck-Institut für Mathematik, 53111 Bonn, Germany
References
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1. Ahlgren, S., Gaussian hypergeometric series and combinatorial congruences, in Symbolic computation, number theory, special functions, physics and combinatorics, Developments in Mathematics, Volume 4, pp. 112 (Kluwer Academic, Dordrecht, 2001).
2. Ahlgren, S. and Ono, K., A Gaussian hypergeometric series evaluation and Apéry number congruences, J. Reine Angew. Math. 518 (2000), 187212.
3. Almkvist, G., Straten, D. van and Zudilin, W., Generalizations of Clausen’s formula and algebraic transformations of Calabi–Yau differential equations, Proc. Edinb. Math. Soc. 54(2) (2011), 273295.
4. Beukers, F., Some congruences for the Apéry numbers, J. Number Theory 21(2) (1985), 141155.
5. Beukers, F., Another congruence for the Apéry numbers, J. Number Theory 25(2) (1987), 201210.
6. Beukers, F., On B. Dwork’s accessory parameter problem, Math. Z. 241(2) (2002), 425– 444.
7. Brun, V., Stubban, J., Fjeldstad, J., Tambs, L., Aubert, K., Ljunggren, W. and Jacobsthal, E., On the divisibility of the difference between two binomial coefficients, in Den 11te Skandinaviske Mathematiker Kongress, Trondheim, 1949, pp. 4254 (Johan Grundt Tanums Forlag, Oslo, 1952).
8. Chamberland, M. and Dilcher, K., Divisibility properties of a class of binomial sums, J. Number Theory 120(2) (2006), 349371.
9. Chamberland, M. and Dilcher, K., A binomial sum related to Wolstenholme’s theorem, J. Number Theory 129(11) (2009), 26592672.
10. Chan, H. and Cooper, S., Rational analogues of Ramanujan’s series for 1/π, Math. Proc. Camb. Phil. Soc. 153(2) (2012), 361383.
11. Chisholm, S., Deines, A., Long, L., Nebe, G. and Swisher, H., p-adic analogues of Ramanujan type formulas for 1/π, Math. 1 (2013), 931.
12. Cooper, S., Sporadic sequences, modular forms and new series for 1/π, Ramanujan J. 29(1-3) (2012), 163183.
13. Coster, M., Supercongruences, PhD thesis, Universiteit Leiden, 1988.
14. Coster, M. and Hamme, L. Van, Supercongruences of Atkin and Swinnerton-Dyer type for Legendre polynomials, J. Number Theory 38(3) (1991), 265286.
15. Gessel, I., Some congruences for Apéry numbers, J. Number Theory 14(3) (1982), 362368.
16. Gessel, I., Some congruences for generalized Euler numbers, Can. J. Math. 35(4) (1983), 687709.
17. Gessel, I., Super ballot numbers, J. Symb. Computat. 14(2-3) (1992), 179194.
18. Granville, A., Arithmetic properties of binomial coefficients, I, Binomial coefficients modulo prime powers, in Organic mathematics, CMS Conference Proceedings, Volume 20, pp. 253-276 (American Mathematical Society, Providence, RI, 1997).
19. Greene, J., Hypergeometric functions over finite fields, Trans. Am. Math. Soc. 301(1) (1987), 77101.
20. Guillera, J., Mosaic supercongruences of Ramanujan type, Exp. Math. 21(1) (2012), 6568.
21. Guillera, J. and Zudilin, W., ‘Divergent’ Ramanujan-type supercongruences, Proc. Am. Math. Soc. 140(3) (2012), 765777.
22. Kibelbek, J., Long, L., Moss, K., Sheller, B. and Yuan, H., Supercongruences and complex multiplication, Preprint (arxiv.org/abs/1210.4489; 2012).
23. Kilbourn, T., An extension of the Apéry number supercongruence, Acta Arith. 123 (2006), 335348.
24. Li, W.-C. W. and Long, L., Atkin and Swinnerton-Dyer congruences and noncongruence modular forms, in Algebraic number theory and related topics 2012, Volume B51, pp. 269-299 (RIMS Kôkyûroku Bessatsu, Kyoto, 2014).
25. Loh, P. and Rhoades, R., p-adic and combinatorial properties of modular form coefficients, Int. J. Number Theory 2(2) (2006), 305328.
26. Long, L., Hypergeometric evaluation identities and supercongruences, Pac. J. Math. 249(2) (2011), 405418.
27. McCarthy, D., On a supercongruence conjecture of Rodriguez-Villegas, Proc. Am. Math. Soc. 140(7) (2012), 22412254.
28. McCarthy, D. and Osburn, R., A p-adic analogue of a formula of Ramanujan, Arch. Math. 91(6) (2008), 492504.
29. Mortenson, E., Supercongruences between truncated 2 F 1 hypergeometric functions and their Gaussian analogs, Trans. Am. Math. Soc. 355(3) (2003), 9871007.
30. Mortenson, E., A supercongruence conjecture of Rodriguez-Villegas for a certain truncated hypergeometric function, J. Number Theory 99(1) (2003), 139147.
31. Mortenson, E., Supercongruences for truncated n+1 F n hypergeometric series with applications to certain weight three newforms, Proc. Am. Math. Soc. 133(2) (2005), 321330.
32. Mortenson, E., Modularity of a certain Calabi-Yau threefold and combinatorial congruences, Ramanujan J. 11(1) (2006), 539.
33. Mortenson, E., A p-adic supercongruence conjecture of van Hamme, Proc. Am. Math. Soc. 136(12) (2008), 43214328.
34. Osburn, R. and Sahu, B., Supercongruences for Apéry-like numbers, Adv. Appl. Math. 47(3) (2011), 631638.
35. Osburn, R. and Sahu, B., A supercongruence for generalized Domb numbers, Funct. Approx. Comment. Math. 48(1) (2013), 2936.
36. Osburn, R. and Schneider, C., Gaussian hypergeometric series and supercongruences, Math. Comp. 78(265) (2009), 275292.
37. Scholl, A., Modular forms and de Rham cohomology; Atkin–Swinnerton-Dyer congruences, Invent. Math. 79(1) (1985), 4977.
38. Straub, A., Multivariate Apéry numbers and supercongruences of rational functions, Alg. Number Theory 8(8) (2014), 19852007.
39. Hamme, L. Van, Some conjectures concerning partial sums of generalized hypergeometric series, in p-adic functional analysis, Lecture Notes in Pure and Applied Mathematics, Volume 192, pp. 223236 (Dekker, New York, 1997).
40. Zagier, D., Integral solutions of Apéry-like recurrence equations, in Groups and symmetries, CRM Proceedings and Lecture Notes, Volume 47, pp. 349366 (American Mathematical Society/Centre de Recherches Mathématiques, 2009).
41. Zudilin, W., Ramanujan-type supercongruences, J. Number Theory 129(8) (2009), 18481857.
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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
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