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Supercongruences for sporadic sequences

Published online by Cambridge University Press:  17 December 2015

Robert Osburn ,
Brundaban Sahu  and
Armin Straub
Robert Osburn
Affiliation:
School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland ([email protected])
Brundaban Sahu
Affiliation:
School of Mathematical Sciences, National Institute of Science Education and Research, Bhubaneswar 751005, India ([email protected])
Armin Straub*
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA ([email protected])
*
* Present address: Max-Planck-Institut für Mathematik, 53111 Bonn, Germany
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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.

Keywords

sporadic sequencessupercongruencesÁpery-like numbersPrimary 11A0711B38
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 2 , May 2016 , pp. 503 - 518
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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References

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. 1–12 (Kluwer Academic, Dordrecht, 2001).Google Scholar
2. Ahlgren, S. and Ono, K., A Gaussian hypergeometric series evaluation and Apéry number congruences, J. Reine Angew. Math. 518 (2000), 187–212.Google Scholar
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), 273–295.CrossRefGoogle Scholar
4. Beukers, F., Some congruences for the Apéry numbers, J. Number Theory 21(2) (1985), 141–155.CrossRefGoogle Scholar
5. Beukers, F., Another congruence for the Apéry numbers, J. Number Theory 25(2) (1987), 201–210.CrossRefGoogle Scholar
6. Beukers, F., On B. Dwork’s accessory parameter problem, Math. Z. 241(2) (2002), 425– 444.CrossRefGoogle Scholar
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. 42–54 (Johan Grundt Tanums Forlag, Oslo, 1952).Google Scholar
8. Chamberland, M. and Dilcher, K., Divisibility properties of a class of binomial sums, J. Number Theory 120(2) (2006), 349–371.Google Scholar
9. Chamberland, M. and Dilcher, K., A binomial sum related to Wolstenholme’s theorem, J. Number Theory 129(11) (2009), 2659–2672.CrossRefGoogle Scholar
10. Chan, H. and Cooper, S., Rational analogues of Ramanujan’s series for 1/π, Math. Proc. Camb. Phil. Soc. 153(2) (2012), 361–383.CrossRefGoogle Scholar
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), 9–31.Google Scholar
12. Cooper, S., Sporadic sequences, modular forms and new series for 1/π, Ramanujan J. 29(1-3) (2012), 163–183.Google Scholar
13. Coster, M., Supercongruences, PhD thesis, Universiteit Leiden, 1988.Google Scholar
14. Coster, M. and Hamme, L. Van, Supercongruences of Atkin and Swinnerton-Dyer type for Legendre polynomials, J. Number Theory 38(3) (1991), 265–286.Google Scholar
15. Gessel, I., Some congruences for Apéry numbers, J. Number Theory 14(3) (1982), 362–368.Google Scholar
16. Gessel, I., Some congruences for generalized Euler numbers, Can. J. Math. 35(4) (1983), 687–709.CrossRefGoogle Scholar
17. Gessel, I., Super ballot numbers, J. Symb. Computat. 14(2-3) (1992), 179–194.Google Scholar
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).Google Scholar
19. Greene, J., Hypergeometric functions over finite fields, Trans. Am. Math. Soc. 301(1) (1987), 77–101.CrossRefGoogle Scholar
20. Guillera, J., Mosaic supercongruences of Ramanujan type, Exp. Math. 21(1) (2012), 65–68.Google Scholar
21. Guillera, J. and Zudilin, W., ‘Divergent’ Ramanujan-type supercongruences, Proc. Am. Math. Soc. 140(3) (2012), 765–777.Google Scholar
22. Kibelbek, J., Long, L., Moss, K., Sheller, B. and Yuan, H., Supercongruences and complex multiplication, Preprint (arxiv.org/abs/1210.4489; 2012).Google Scholar
23. Kilbourn, T., An extension of the Apéry number supercongruence, Acta Arith. 123 (2006), 335–348.Google Scholar
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).Google Scholar
25. Loh, P. and Rhoades, R., p-adic and combinatorial properties of modular form coefficients, Int. J. Number Theory 2(2) (2006), 305–328.Google Scholar
26. Long, L., Hypergeometric evaluation identities and supercongruences, Pac. J. Math. 249(2) (2011), 405–418.Google Scholar
27. McCarthy, D., On a supercongruence conjecture of Rodriguez-Villegas, Proc. Am. Math. Soc. 140(7) (2012), 2241–2254.CrossRefGoogle Scholar
28. McCarthy, D. and Osburn, R., A p-adic analogue of a formula of Ramanujan, Arch. Math. 91(6) (2008), 492–504.CrossRefGoogle Scholar
29. Mortenson, E., Supercongruences between truncated 2 F 1 hypergeometric functions and their Gaussian analogs, Trans. Am. Math. Soc. 355(3) (2003), 987–1007.CrossRefGoogle Scholar
30. Mortenson, E., A supercongruence conjecture of Rodriguez-Villegas for a certain truncated hypergeometric function, J. Number Theory 99(1) (2003), 139–147.CrossRefGoogle Scholar
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), 321–330.CrossRefGoogle Scholar
32. Mortenson, E., Modularity of a certain Calabi-Yau threefold and combinatorial congruences, Ramanujan J. 11(1) (2006), 5–39.Google Scholar
33. Mortenson, E., A p-adic supercongruence conjecture of van Hamme, Proc. Am. Math. Soc. 136(12) (2008), 4321–4328.Google Scholar
34. Osburn, R. and Sahu, B., Supercongruences for Apéry-like numbers, Adv. Appl. Math. 47(3) (2011), 631–638.Google Scholar
35. Osburn, R. and Sahu, B., A supercongruence for generalized Domb numbers, Funct. Approx. Comment. Math. 48(1) (2013), 29–36.CrossRefGoogle Scholar
36. Osburn, R. and Schneider, C., Gaussian hypergeometric series and supercongruences, Math. Comp. 78(265) (2009), 275–292.Google Scholar
37. Scholl, A., Modular forms and de Rham cohomology; Atkin–Swinnerton-Dyer congruences, Invent. Math. 79(1) (1985), 49–77.Google Scholar
38. Straub, A., Multivariate Apéry numbers and supercongruences of rational functions, Alg. Number Theory 8(8) (2014), 1985–2007.CrossRefGoogle Scholar
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. 223–236 (Dekker, New York, 1997).Google Scholar
40. Zagier, D., Integral solutions of Apéry-like recurrence equations, in Groups and symmetries, CRM Proceedings and Lecture Notes, Volume 47, pp. 349–366 (American Mathematical Society/Centre de Recherches Mathématiques, 2009).CrossRefGoogle Scholar
41. Zudilin, W., Ramanujan-type supercongruences, J. Number Theory 129(8) (2009), 1848–1857.CrossRefGoogle Scholar