Advertisement

q-Analogues of the (E.2) and (F.2) supercongruences of Van Hamme

  • Victor J. W. Guo
Article
First Online:
  • 20 Downloads

Abstract

Motivated by Zudilin’s work, we give q-analogues of the Ramanujan-type supercongruences (E.2) and (F.2) of Van Hamme. Our proof utilizes the q-WZ method and properties of cyclotomic polynomials. Using the same q-WZ pair, we also give q-analogues of some similar supercongruences due to He and Swisher. Additionally, we propose several related conjectures on supercongruences or q-supercongruences.

Keywords

Ramanujan Supercongruences q-Analogues Cyclotomic polynomials q-WZ pair 

Mathematics Subject Classification

Primary 11B65 Secondary 05A10 05A30 
This is a preview of subscription content, log in to check access.

References

  1. 1.
    Andrews, G.E.: \(q\)-Analogs of the binomial coefficient congruences of Babbage, Wolstenholme and Glaisher. Discret. Math. 204, 15–25 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Gasper, G., Rahman, M.: Basic hypergeometric series. Encyclopedia of Mathematics and Its Applications, vol. 96, 2nd edn. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  3. 3.
    Guillera, J.: Generators of some Ramanujan formulas. Ramanujan J. 11, 41–48 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Guo, V.J.W.: A \(q\)-analogue of a Ramanujan-type supercongruence involving central binomial coefficients. J. Math. Anal. Appl. 458, 590–600 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Guo, V.J.W., Pan, H., Zhang, Y.: The Rodriguez–Villegas type congruences for truncated \(q\)-hypergeometric functions. J. Number Theory 174, 358–368 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Guo, V.J.W., Zeng, J.: Some congruences involving central \(q\)-binomial coefficients. Adv. Appl. Math. 45, 303–316 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Guo, V.J.W., Zeng, J.: Some \(q\)-analogues of supercongruences of Rodriguez–Villegas. J. Number Theory 145, 301–316 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Guo, V.J.W., Zeng, J.: Some \(q\)-supercongruences for truncated basic hypergeometric series. Acta Arith. 171, 309–326 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    He, B.: Some congruences on truncated hypergeometric series. Proc. Am. Math. Soc. 143, 5173–5180 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Liu, J., Pan, H., Zhang, Y.: A generalization of Morley’s congruence. Adv. Differ. Equ. 2015, 254 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Long, L.: Hypergeometric evaluation identities and supercongruences. Pac. J. Math. 249, 405–418 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Mortenson, E.: A \(p\)-adic supercongruence conjecture of van Hamme. Proc. Am. Math. Soc. 136, 4321–4328 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Osburn, R., Zudilin, W.: On the (K.2) supercongruence of Van Hamme. J. Math. Anal. Appl. 433, 706–711 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Pan, H.: A \(q\)-analogue of Lehmer’s congruence. Acta Arith. 128, 303–318 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Sun, Z.-W.: Supercongruences involving Lucas sequences. preprint. arXiv:1610.03384v7 (2016)
  16. 16.
    Swisher, H.: On the supercongruence conjectures of van Hamme. Res. Math. Sci. 2, 18 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Tauraso, R.: \(q\)-Analogs of some congruences involving Catalan numbers. Adv. Appl. Math. 48, 603–614 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Tauraso, R.: Some \(q\)-analogs of congruences for central binomial sums. Colloq. Math. 133, 133–143 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Van Hamme, L.: Some conjectures concerning partial sums of generalized hypergeometric series. In: \(p\)-Adic Functional Analysis (Nijmegen, 1996). Lecture Notes in Pure and Applied Mathematics, vol. 192, pp. 223–236, Dekker, New York (1997)Google Scholar
  20. 20.
    Zudilin, W.: Ramanujan-type supercongruences. J. Number Theory 129, 1848–1857 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesHuaiyin Normal UniversityHuai’anPeople’s Republic of China

Personalised recommendations

Cite article

Buy options