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On the supercongruence conjectures of van Hamme

  • Holly SwisherEmail author
Open Access
Research

Abstract

In 1997, van Hamme developed \(p\)–adic analogs, for primes p, of several series which relate hypergeometric series to values of the gamma function, originally studied by Ramanujan. These analogs relate truncated sums of hypergeometric series to values of the \(p\)–adic gamma function, and are called Ramanujan-type supercongruences. In all, van Hamme conjectured 13 such formulas, three of which were proved by van Hamme himself, and five others have been proved recently using a wide range of methods. Here, we explore four of the remaining five van Hamme supercongruences, revisit some of the proved ones, and provide some extensions.

Keywords

Ramanujan-type supercongruences Hypergeometric series 

Mathematics subject classification

Primary 33C20 Secondary 44A20 

Notes

Acknowledgements

The author would like to thank Ling Long for numerous helpful conversations which provided the ideas behind this work, and Robert Osburn for many inspiring conversations and for introducing her to the van Hamme conjectures. The author also thanks Tulane University for hosting her while working on this project, and Sage Days 56, where she ran computations related to this project with Ling Long and Hao Chen.

References

  1. 1.
    Ahlgren, S., Ono, K.: Modularity of a certain calabi-yau threefold. Monatshefte für. Mathematik. 129(3), 177–190 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Andrews, G.E., Askey, R., Roy, R.: Special functions. Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press, Cambridge (1999)Google Scholar
  3. 3.
    Borwein, J.M., Borwein, P.B.: Pi and the AGM. Canadian Mathematical Society Series of Monographs and Advanced Texts. A study in analytic number theory and computational complexity. A Wiley-Interscience Publication, New York (1987)Google Scholar
  4. 4.
    Chisholm, S., Deines, A., Long, L., Nebe, G., Swisher, H.: \(p\)-adic analogues of Ramanujan type formulas for 1/\(\pi \). Mathematics 1(1), 9–30 (2013)CrossRefzbMATHGoogle Scholar
  5. 5.
    Chudnovsky, D.V., Chudnovsky, G.V.: Approximations and complex multiplication according to Ramanujan. In Ramanujan revisited (Urbana-Champaign, Ill., 1987), pp. 375–472. Academic Press, Boston (1988)Google Scholar
  6. 6.
    Dwork, B.: A note on the \(p\)-adic gamma function. In Study group on ultrametric analysis, 9th year: 1981/82, No. 3 (Marseille, 1982), pages Exp. No. J5, 10. Inst. Henri Poincaré, Paris (1983)Google Scholar
  7. 7.
    He, B.: On a \(p\)-adic supercongruence conjecture of l. van hamme. Proc. Amer. Math. Soc. (2015, to appear)Google Scholar
  8. 8.
    He B.: Some congruences on truncated hypergeometric series (2015, preprint)Google Scholar
  9. 9.
    Karlsson, PW.: Clausen’s hypergeometric function with variable \(-1/8\) or \(-8\). Math. Sci. Res. Hot-Line 4(7), 25–33 (2000)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Kilbourn, T: An extension of the Apéry number supercongruence. Acta. Arith. 123(4), 335–348 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Long, L: Hypergeometric evaluation identities and supercongruences. Pacific J. Math. 249(2), 405–418 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Long, L., Ramakrishna, R.: Some supercongruences occurring in truncated hypergeometric series. preprint. http://arxiv.org/abs/1403.5232
  13. 13.
    McCarthy, D., Osburn, R.: A \(p\)-adic analogue of a formula of Ramanujan. Arch. Math. (Basel) 91(6), 492–504 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Morita, Y.: A p-adic analogue of the-function. J. Fac. Sci. Univ. Tokyo 22, 255–266 (1975)zbMATHGoogle Scholar
  15. 15.
    Mortenson, E.: A \(p\)-adic supercongruence conjecture of van Hamme. Proc. Amer. Math. Soc. 136(12), 4321–4328 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Osburn, R., Zudilin, W.: On the (k.2) supercongruence of van hamme (2015, preprint). http://arxiv.org/abs/1504.01976
  17. 17.
    Rogers, M., Wan, J.G., Zucker, I.J.: Moments of elliptic integrals and critical \(l\)-values. Ramanujan J. 37(1), 113–130 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    van Hamme, L.: Some conjectures concerning partial sums of generalized hypergeometric series. Lect. Notes Pure Appl. Math. 192, 223–236 (1997)MathSciNetGoogle Scholar
  19. 19.
    Vangeemen, B., Nygaard, N.O.: On the geometry and arithmetic of some siegel modular threefolds. J. Number Theor. 53(1), 45–87 (1995)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Verrill, H.A.: Arithmetic of a certain Calabi-Yau threefold. Number theory (Ottawa, ON, 1996), volume 19 of CRM Proc. Lecture Notes, pp. 333–340. Amer. Math. Soc, Providence, RI (1999)Google Scholar
  21. 21.
    Whipple, F.J.W.: On well-poised series, generalized hypergeometric series having parameters in pairs, each pair with the same sum. Proc. London Math. Soc. 24(1), 247–263 (1926)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Zudilin, W.: Ramanujan-type supercongruences. J. Number Theor. 129(8), 1848–1857 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Swisher. 2015

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Mathematics Oregon State UniversityCorvallisUSA

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