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Further q-analogues of the (G.2) supercongruence of Van Hamme

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Abstract

In 2015, Swisher generalized the (G.2) supercongruence of Van Hamme to the modulus \(p^4\). In this paper, we first propose two q-analogues of Swisher’s supercongruence, and then a parameter extension of them is presented. Furthermore, we prove a q-congruence modulo the fourth power of a cyclotomic polynomial, which was conjectured by the authors earlier.

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References

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

    Book  Google Scholar 

  2. Guo, V.J.W.: Proof of some \(q\)-supercongruences modulo the fourth power of a cyclotomic polynomial. Results Math. 75, 77 (2020)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Guo, V.J.W.: Proof of a generalization of the (C.2) supercongruence of Van Hamme. Rev. R. Acad. Cienc. Exactas Fís. Nat. A 115, 45 (2021)

    MathSciNet  MATH  Google Scholar 

  5. Guo, V.J.W.: Some variations of a ‘divergent’ Ramanujan-type \(q\)-supercongruence. J. Differ. Equ. Appl. 27, 376–388 (2021)

    Article  MathSciNet  Google Scholar 

  6. Guo, V.J.W., Schlosser, M.J.: Some new \(q\)-congruences for truncated basic hypergeometric series: even power. Results Math. 75, 1 (2020)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Guo, V.J.W., Schlosser, M.J.: A family of \(q\)-supercongruences modulo the cube of a cyclotomic polynomial. Bull. Aust. Math. Soc. 105, 296–302 (2022)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Liu, Y., Wang, X.: \(q\)-Analogues of two Ramanujan-type supercongruences. J. Math. Anal. Appl. 502(1), 125238 (2021)

    Article  MathSciNet  Google Scholar 

  14. Liu, Y., Wang, X.: \(q\)-Analogues of the (G.2) supercongruence of Van Hamme. Rocky Mt. J. Math. 51(4), 1329–1340 (2021)

    Article  MathSciNet  Google Scholar 

  15. Liu, Y., Wang, X.: Some \(q\)-supercongruences from Rahman’s summation formula. Results Math. 77(1), 44 (2022)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Morita, Y.: A \(p\)-adic supercongruence of the \(\Gamma \) function. J. Fac. Sci. Univ. Tokyo 22, 255–266 (1975)

    MATH  Google Scholar 

  18. Ni, H.-X., Pan, H.: On a conjectured \(q\)-congruence of Guo and Zeng. Int. J. Number Theory 14(6), 1699–1707 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Rahman, M.: Some quadratic and cubic summation formulas for basic hypergeometric series. Can. J. Math. 45(2), 394–411 (1993)

    Article  MathSciNet  Google Scholar 

  21. Sun, Z.-W.: A new series for \(\pi ^3\) and related congruences. Int. J. Math. 26(8), 1550055 (2019)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. 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)

  24. Wang, C., Pan, H.: Supercongruences concerning truncated hypergeometric series. Math. Z. 300, 161–177 (2022)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  26. Wang, X., Yue, M.: A \(q\)-analogue of a Dwork-type supercongruence. Bull. Aust. Math. Soc. 103(2), 303–310 (2021)

    Article  MathSciNet  Google Scholar 

  27. Wang, X., Yu, M.: Some new \(q\)-congruences on double sums. Rev. R. Acad. Cienc. Exactas Fís. Nat. A 115, 9 (2021)

    MathSciNet  MATH  Google Scholar 

  28. Yu, M., Wang, X.: Proof of two conjectures of Guo and Schlosser. Ramanujan J. 58(1), 239–252 (2022)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors thank the anonymous referee for helpful comments that helped improve the exposition of this article.

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Authors and Affiliations

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

    Yudong Liu & Xiaoxia Wang

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  1. Yudong Liu
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  2. Xiaoxia Wang
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Correspondence to Xiaoxia Wang.

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This work is supported by Natural Science Foundation of Shanghai (22ZR1424100)

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Liu, Y., Wang, X. Further q-analogues of the (G.2) supercongruence of Van Hamme. Ramanujan J 59, 791–802 (2022). https://doi.org/10.1007/s11139-022-00597-x

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  • DOI: https://doi.org/10.1007/s11139-022-00597-x

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