Skip to main content
SpringerLink
Log in

q-Supercongruences on Triple and Quadruple Sums

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

Inspired by the recent work of El Bachraoui, we present some new q-supercongruences on triple and quadruple sums of basic hypergeometric series. In particular, we give a q-supercongruence modulo the fifth power of a cyclotomic polynomial, which is a q-analogue of the quadruple sum of Van Hamme’s supercongruence (G.2).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. El Bachraoui, M.: On supercongruences for truncated sums of squares of basic hypergeometric series. Ramanujan J. 54, 415–426 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  2. El Bachraoui, M.: \(N\)-tuple sum analogues for Ramanujan-type congruences. Proc. Am. Math. Soc. 151, 1–16 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berndt, B.C., Rankin, R.A.: Ramanujan: Letters and commentary, History of Mathematics, vol. 9, American Mathematical Society, Providence/London Mathematical Society, London (1995)

  4. Guo, V.J.W.: A new extension of the (A.2) supercongruence of Van Hamme, Results Math. 77, Art. 96 (2022)

  5. Guo, V.J.W., Li, L.: \(q\)-Supercongruences from squares of basic hypergeometric series, arXiv:2112.12076

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

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

    Article  MathSciNet  MATH  Google Scholar 

  8. He, B.: Supercongruences on truncated hypergeometric series. Res. Math. 72, 303–317 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, L.: Some \(q\)-supercongruences for truncated forms of squares of basic hypergeometric series. J. Differ. Equ. Appl. 27, 16–25 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  10. Liu, Y., Wang, X.: \(q\)-Analogues of the (G.2) supercongruence of Van Hamme, Rocky Mountain. J. Math. 51, 1329–1340 (2021)

  11. Liu, Y., Wang, X.: \(q\)-Analogues of two Ramanujan-type supercongrucences. J. Math. Anal. Appl. 502, Art. 125238 (2021)

  12. Liu, Y., Wang, X.: Further \(q\)-analogues of the (G.2) supercongruence of Van Hamme. Ramanujan J. (2022). https://doi.org/10.1007/s11139-022-00597-x

  13. Song, H., Wang, C.: Some \(q\)-supercongruences modulo the fifth power of a cyclotomic polynomial from squares of \(q\)-hypergeometric series. Res. Math. 76, Art. 222 (2021)

  14. Van Hamme, L.: Proof of a conjecture of Beukers on Apéry numbers. In: Proceedings of the conference on \(p\)-Adic Analysis (Houthalen, 1987), pp. 189–195. Vrije Universiteit Brussel, Brussels (1986)

  15. 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)

  16. Wang, C.: A new \(q\)-extension of the (H.2) congruence of Van Hamme for primes \(p\equiv 1~(mod \; 4)\). Res. Math. 76, Art. 205 (2021)

  17. Wei, C.: Some \(q\)-supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial, arXiv:2104.07025

  18. Wei, C.: \(q\)-Supercongruences from Gasper and Rahman’s summation formula. Adv. Appl. Math. 139, Art. 102376 (2022)

  19. Xu, C., Wang, X.: Proofs of Guo and Schlosser’s two conjectures. Period. Math. Hungar. (2022). https://doi.org/10.1007/s10998-022-00452-y

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

    Xiaoxia Wang & Chang Xu

Authors
  1. Xiaoxia Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chang Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chang Xu.

Ethics declarations

Conflict of Interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

This work is supported by Natural Science Foundation of Shanghai (Grant No. 22ZR1424100).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, X., Xu, C. q-Supercongruences on Triple and Quadruple Sums. Results Math 78, 27 (2023). https://doi.org/10.1007/s00025-022-01801-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-022-01801-6

Keywords

Mathematics Subject Classification

Search

Navigation