Results in Mathematics

, 74:131 | Cite as

Common q-Analogues of Some Different Supercongruences

  • Victor J. W. GuoEmail author


We prove some q-congruences modulo the cube of a product of two cyclotomic polynomials by using Watson’s \(_8\phi _7\) transformation formula and the creative microscoping method, recently devised by the author in collaboration with Wadim Zudilin. When n is an odd prime power, we deduce different supercongruences from each of these q-congruences by taking the limit as \(q\rightarrow 1\) or \(q\rightarrow -1\). As a conclusion, we confirm the \(m=3\) case of Conjecture 1.1 in Guo (Integral Transform Spec Funct 28:888–899, 2017) and partially confirm the \(m=3\) case of Conjecture 4.3 in the same paper. We also raise several related conjectures on q-congruences and supercongruences.


Basic hypergeometric series Watson’s transformation q-congruences supercongruences cyclotomic polynomial 

Mathematics Subject Classification

33D15 Secondary 11A07 11F33 



The author thanks the anonymous referee for very useful comments that helped to improve the quality of the article. Thanks also to Michael J. Schlosser for a careful reading of the introduction. This work was partially supported by the National Natural Science Foundation of China (Grant 11771175).


  1. 1.
    Gasper, G., Rahman, M.: Basic Hypergeometric Series, Second Edition, Encyclopedia of Mathematics and Its Applications 96. Cambridge University Press, Cambridge (2004)Google Scholar
  2. 2.
    Gorodetsky, O.: \(q\)-Congruences, with applications to supercongruences and the cyclic sieving phenomenon. Int. J. Number Theory (2019) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Guillera, J.: WZ pairs and \(q\)-analogues of Ramanujan series for \(1/\pi \). J. Differ. Equ. Appl. 24, 1871–1879 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Guo, V.J.W.: Some generalizations of a supercongruence of van Hamme. Integral Transforms Spec. Funct. 28, 888–899 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Guo, V.J.W.: A \(q\)-analogue of a Ramanujan-type supercongruence involving central binomial coefficients. J. Math. Anal. Appl. 458, 590–600 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Guo, V.J.W.: A \(q\)-analogue of the (I.2) supercongruence of Van Hamme. Int. J. Number Theory 15, 29–36 (2019)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Guo, V.J.W.: Proof of a \(q\)-congruence conjectured by Tauraso. Int. J. Number Theory 15, 37–41 (2019)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Guo, V.J.W.: Factors of some truncated basic hypergeometric series. J. Math. Anal Appl. 476, 851–859 (2019)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Guo, V.J.W.: \(q\)-Analogues of the (E.2) and (F.2) supercongruences of Van Hamme. Ramanujan J. (2018). MathSciNetCrossRefGoogle Scholar
  10. 10.
    Guo, V.J.W.: \(q\)-Analogues of two “divergent” Ramanujan-type supercongruences, Ramanujan J. (2019).
  11. 11.
    Guo, V.J.W.: A \(q\)-analogue of a curious supercongruence of Guillera and Zudilin. J. Differ. Equ. Appl. 25, 342–350 (2019)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Guo, V.J.W.: Some \(q\)-congruences with parameters, Acta Arith., to appearGoogle Scholar
  13. 13.
    Guo, V.J.W., Liu, J.-C.: Some congruences related to a congruence of Van Hamme, preprint (2019). arXiv:1903.03766
  14. 14.
    Guo, V.J.W., Schlosser, M.J.: Some new \(q\)-congruences for truncated basic hypergeometric series. Symmetry 11(2), 268 (2019)CrossRefGoogle Scholar
  15. 15.
    Guo, V.J.W., Schlosser, M.J.: Proof of a basic hypergeometric supercongruence modulo the fifth power of a cyclotomic polynomial. J. Differ. Equ. Appl. (2019). MathSciNetCrossRefGoogle Scholar
  16. 16.
    Guo, V.J.W., Schlosser, M.J.: Some \(q\)-supercongruences from transformation formulas for basic hypergeometric series. Preprint arXiv:1812.06324 (2018)
  17. 17.
    Guo, V.J.W., Wang, S.-D.: Factors of sums and alternating sums of products of \(q\)-binomial coefficients and powers of \(q\)-integers. Taiwan. J. Math. 23, 11–27 (2019)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Guo, V.J.W., Wang, S.-D.: Some congruences involving fourth powers of central \(q\)-binomial coefficients. Proc. R. Soc. Edinburgh Sect. A. (2019).
  19. 19.
    Guo, V.J.W., Zudilin, W.: A \(q\)-microscope for supercongruences. Adv. Math. 346, 329–358 (2019)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Guo, V.J.W., Zudilin, W.: On a \(q\)-deformation of modular forms. J. Math. Anal. Appl. 475, 1636–646 (2019)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Jana, A., Kalita, G., Supercongruences for sums involving rising factorial \((\frac{1}{\ell })_k^3\). Integral Transforms Spec. Funct. in press. MathSciNetCrossRefGoogle Scholar
  22. 22.
    Liu, J.-C.: Semi-automated proof of supercongruences on partial sums of hypergeometric series. J. Symb. Comput. 93, 221–229 (2019)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Long, L.: Hypergeometric evaluation identities and supercongruences. Pacific J. Math. 249, 405–418 (2011)MathSciNetCrossRefGoogle Scholar
  24. 24.
    McCarthy, D., Osburn, R.: A \(p\)-adic analogue of a formula of Ramanujan. Arch. Math. 91, 492–504 (2008)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Mortenson, E.: A \(p\)-adic supercongruence conjecture of van Hamme. Proc. Am. Math. Soc. 136, 4321–4328 (2008)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Ni, H.-X., Pan, H.: On a conjectured \(q\)-congruence of Guo and Zeng. Int. J. Number Theory 14, 1699–1707 (2018)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Straub, A.: Supercongruences for polynomial analogs of the Apéry numbers. Proc. Am. Math. Soc. 147, 1023–1036 (2019)CrossRefGoogle Scholar
  28. 28.
    Sun, Z.-W.: A refinement of a congruence result by van Hamme and Mortenson. Illinois J. Math. 56, 967–979 (2012)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Swisher, H.: On the supercongruence conjectures of van Hamme. Res. Math. Sci. 2, 8 (2015)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Tauraso, R.: Some \(q\)-analogs of congruences for central binomial sums. Colloq. Math. 133, 133–143 (2013)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Van Hamme, L.: Some conjectures concerning partial sums of generalized hypergeometric series. In: \(p\)-Adic functional analysis (Nijmegen, 1996), Lecture Notes in Pure and Appl. Math. vol. 192, pp. 223–236. Dekker, New York (1997)Google Scholar
  32. 32.
    Wang, S.-D.: Some supercongruences involving \({2k\atopwithdelims ()k}^4\). J. Differ. Equ. Appl. 24, 1375–1383 (2018)CrossRefGoogle Scholar
  33. 33.
    Wilf, H.S., Zeilberger, D.: An algorithmic proof theory for hypergeometric (ordinary and “\(q\)”) multisum/integral identities. Invent. Math. 108, 575–633 (1992)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Zudilin, W.: Ramanujan-type supercongruences. J. Number Theory 129, 1848–1857 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

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

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