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

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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.

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References

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

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

  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. 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  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We thank the anonymous referee for valuable comments.

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  1. School of Mathematics and Statistics, Huaiyin Normal University, Huai’an, 223300, Jiangsu, People’s Republic of China

    Victor J. W. Guo

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  1. Victor J. W. Guo
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Correspondence to Victor J. W. Guo.

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

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Guo, V.J.W. A new extension of the (H.2) supercongruence of Van Hamme for primes \(p\equiv 3\pmod {4}\). Ramanujan J 57, 1387–1398 (2022). https://doi.org/10.1007/s11139-020-00369-5

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