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Generalizations of the (F.2) supercongruence of Van Hamme

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In 2015, Swisher (Res Math Sci 2:18, 2015) verified Van Hamme’s (F.2) supercongruence conjecture, where \(p\equiv 1 \pmod {4}\) is a prime. She also gave a similar formula, where \(p\equiv 3 \pmod {4}\) is any prime, in the same paper. With the help of the creative microscoping method introduced by Guo and Zudilin, we obtain one-parameter generalizations of Van Hamme’s (F.2) supercongruence and Swisher’s result.

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References

  1. Gasper, G., Rahman, M.: Basic Hypergeometric Series, 2nd edn. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Guo, V.J.W.: \(q\)-Analogues of the (E.2) and (F.2) supercongruences of Van Hamme. Ramanujan J. 49, 531–544 (2019)

    Article  MathSciNet  Google Scholar 

  4. Guo, V.J.W., Li, L.: \(q\)-Supercongruences from squares of basic hypergeometric series. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 117, 26 (2023)

  5. Guo, V.J.W., Ni, H.-X.: Further generalizations of four supercongruences of Rodriguez–Villegas. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 117, 49 (2023)

  6. Guo, V.J.W., Ni, H.-X.: Proof of some supercongruences through a \(q\)-microscope, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 117, 147 (2023)

  7. Guo, V.J.W., Schlosser, M.J.: A family of \(q\)-hypergeometric congruences modulo the fourth power of a cyclotomic polynomial. Israel J. Math. 240, 821–835 (2020)

    Article  MathSciNet  Google Scholar 

  8. Guo, V.J.W., Schlosser, M.J.: Three families of \(q\)-supercongruences modulo the square and cube of a cyclotomic polynomial. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 117, 9 (2023)

  9. Guo, V.J.W., Wang, S.-D.: Factors of certain sums involving central \(q\)-binomial coefficients. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 116, 46 (2022)

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

  13. Liu, J.-C.: A variation of the \(q\)-Wolstenholme theorem. Ann. Mat. Pura Appl. 201, 1993–2000 (2022)

    Article  MathSciNet  Google Scholar 

  14. Liu, J.-C.: On the divisibility of \(q\)-trinomial coefficients. Ramanujan J. 60, 455–462 (2023)

    Article  MathSciNet  Google Scholar 

  15. Ni, H.-X., Wang, L.-Y.: Two \(q\)-supercongruences from Watson’s transformation. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 116, 30 (2022)

  16. Sun, Z.-W.: Super congruences and Euler numbers. Sci. China Math. 54, 2509–2535 (2011)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. Tang, N.: A new \(q\)-supercongruence modulo the fourth power of a cyclotomic polynomial. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 117, 101 (2023)

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

  20. Wei, C.: Some \(q\)-supercongruences modulo the fourth power of a cyclotomic polynomial. J. Combin. Theory Ser. A 182, 105469 (2021)

    Article  MathSciNet  Google Scholar 

  21. Wei, C.: A \(q\)-supercongruence from a \(q\)-analogue of Whipple’s \(_3F_2\) summation formula. J. Combin. Theory Ser. A 194, 105705 (2023)

    Article  Google Scholar 

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Acknowledgements

The work is supported by the Hainan Natural Science Foundation of China (No. 621QN246).

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  1. School of Biomedical Information and Engineering, Hainan Medical University, Haikou, 571199, China

    Chuanan Wei

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  1. Chuanan Wei
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Correspondence to Chuanan Wei.

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Wei, C. Generalizations of the (F.2) supercongruence of Van Hamme. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 33 (2024). https://doi.org/10.1007/s13398-023-01532-5

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