A new extension of the (H.2) supercongruence of Van Hamme for primes \(p\equiv 3\pmod {4}\)

Abstract

Using Andrews’ multiseries generalization of Watson’s \(_8\phi _7\) transformation, we give a new extension of the (H.2) supercongruence of Van Hamme for primes \(p\equiv 3\pmod {4}\), as well as its q-analogue. Meanwhile, applying the method of ‘creative microscoping’, recently introduced by the author and Zudilin, we establish some further q-supercongruences modulo \(\Phi _n(q)^3\), where \(\Phi _n(q)\) denotes the nth cyclotomic polynomial in q.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Andrews, G.E.: Problems and prospects for basic hypergeometric functions. In: Askey, R.A. (ed.) Theory and Application for Basic Hypergeometric Functions, pp. 191–224. Math. Res. Center, Univ. Wisconsin, Publ. No. 35, Academic Press, New York (1975)

    Google Scholar 

  2. 2.

    Gasper, G., Rahman, M.: Basic Hpergeometric Series, 2nd Edition, Encyclopedia of Mathematics and Its Applications 96. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  3. 3.

    Gorodetsky, O.: \(q\)-Congruences, with applications to supercongruences and the cyclic sieving phenomenon. Int. J. Number Theory 15, 1919–1968 (2019)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Guillera, J.: WZ pairs and \(q\)-analogues of Ramanujan series for \(1/\pi \). J. Difference Equ. Appl. 24, 1871–1879 (2018)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Guo, V.J.W.: Common \(q\)-analogues of some different supercongruences. Results Math. 74, Art. 131 (2019)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Guo, V.J.W.: A family of \(q\)-congruences modulo the square of a cyclotomic polynomial. Electron. Res. Arch. 28, 1031–1036 (2020)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Guo, V.J.W.: Proof of a generalization of the (B.2) supercongruence of Van Hamme through a \(q\)-microscope. Adv. Appl. Math. 116, Art. 102016 (2020)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Guo, V.J.W.: \(q\)-Supercongruences modulo the fourth power of a cyclotomic polynomial via creative microscoping. Adv. Appl. Math. 120, Art. 102078 (2020)

    Article  Google Scholar 

  10. 10.

    Guo, V.J.W.: \(q\)-Analogues of Dwork-type supercongruences. J. Math. Anal. Appl. 487, Art. 124022 (2020)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Guo, V.J.W. A further \(q\)-analogue of Van Hamme’s (H.2) supercongruence for primes \(p\equiv 3\) (mod 4). Int. J. Number Theory, in press. https://doi.org/10.1142/S1793042121500329

  12. 12.

    Guo, V.J.W., Schlosser, M.J.: A new family of \(q\)-supercongruences modulo the fourth power of a cyclotomic polynomial. Results Math. 75, Art. 155 (2020)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Guo, V.J.W. and Schlosser, M.J. A family of \(q\)-hypergeometric congruences modulo the fourth power of a cyclotomic polynomial, Israel J. Math., in press. https://doi.org/10.1007/s11856-020-2081-1

  14. 14.

    Guo, V.J.W., Schlosser, M.J.: Some q-supercongruences from transformation formulas for basic hypergeometric series. Constr. Approx. (2020). https://doi.org/10.1007/s00365-020-09524-z

    Article  Google Scholar 

  15. 15.

    Guo, V.J.W., Zeng, J.: Some \(q\)-supercongruences for truncated basic hypergeometric series. Acta Arith. 171, 309–326 (2015)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Guo, V.J.W., Zudilin, W.: A \(q\)-microscope for supercongruences. Adv. Math. 346, 329–358 (2019)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Guo, V.J.W., Zudilin, W.: On a \(q\)-deformation of modular forms. J. Math. Anal. Appl. 475, 1636–646 (2019)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Guo, V.J.W., Zudilin, W.: A common \(q\)-analogue of two supercongruences. Results Math. 75, Art. 46 (2020)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Guo, V.J.W., Zudilin, W.: Dwork-type supercongruences through a creative \(q\)-microscope. J. Combin. Theory, Ser. A 178, Art. 105362 (2021)

  20. 20.

    Li, L., Wang, S.-D.: Proof of a \(q\)-supercongruence conjectured by Guo and Schlosser. Rev. R. Acad. Cienc. Exactas Fs. Nat., Ser. A Mat. RACSAM 114, Art. 190 (2020)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Liu, J.-C.: Some supercongruences on truncated \(_3F_2\) hypergeometric series. J. Difference Equ. Appl. 24, 438–451 (2018)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Liu, J.-C.: On Van Hamme’s (A.2) and (H.2) supercongruences. J. Math. Anal. Appl. 471, 613–622 (2019)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Liu, J.-C.: On a congruence involving \(q\)-Catalan numbers. C. R. Math. Acad. Sci. Paris 358, 211–215 (2020)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Liu, J.-C., Petrov, F.: Congruences on sums of \(q\)-binomial coefficients. Adv. Appl. Math. 116, Art. 102003 (2020)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Long, L., Ramakrishna, R.: Some supercongruences occurring in truncated hypergeometric series. Adv. Math. 290, 773–808 (2016)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Mao, G.-S., Pan, H.: On the divisibility of some truncated hypergeometric series. Acta Arith. 195, 199–206 (2020)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Mortenson, E.: A \(p\)-adic supercongruence conjecture of van Hamme. Proc. Amer. Math. Soc. 136, 4321–4328 (2008)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Ni, H.-X., Pan, H.: Divisibility of some binomial sums. Acta Arith. 194, 367–381 (2020)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Ni, H.-X., Pan, H.: Some symmetric \(q\)-congruences modulo the square of a cyclotomic polynomial. J. Math. Anal. Appl. 481, Art. 123372 (2020)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Sun, Z.-H.: Generalized Legendre polynomials and related supercongruences. J. Number Theory 143, 293–319 (2014)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Sun, Z.-W.: On sums of Apéry polynomials and related congruences. J. Number Theory 132, 2673–2690 (2012)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Swisher, H.: On the supercongruence conjectures of van Hamme. Res. Math. Sci. 2, Art. 18 (2015)

    MathSciNet  Article  Google Scholar 

  33. 33.

    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. 192, Dekker, New York, pp. 223–236 (1997)

  34. 34.

    Wang, X., Yue, M.: A \(q\)-analogue of the (A.2) supercongruence of Van Hamme for any prime \(p\equiv 3\) (mod 4). Int. J. Number Theory 16, 1325–1335 (2020)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Wang, X., Yue, M.: Some \(q\)-supercongruences from Watson’s \(_8\phi _7\) transformation formula. Results Math. 75, Art. 71 (2020)

    Article  Google Scholar 

  36. 36.

    Zudilin, W.: Ramanujan-type supercongruences. J. Number Theory 129, 1848–1857 (2009)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Zudilin, W.: Congruences for \(q\)-binomial coefficients. Ann. Combin. 23, 1123–1135 (2019)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

We thank the anonymous referee for valuable comments.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Victor J. W. Guo.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The author was partially supported by the National Natural Science Foundation of China (Grant 11771175)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Guo, V.J.W. A new extension of the (H.2) supercongruence of Van Hamme for primes \(p\equiv 3\pmod {4}\). Ramanujan J (2021). https://doi.org/10.1007/s11139-020-00369-5

Download citation

Keywords

  • Cyclotomic polynomial
  • q-Congruence
  • Supercongruence
  • Andrews’ transformation
  • Watson’s transformation

Mathematics Subject Classification

  • 33D15
  • 11A07
  • 11B65