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On Some Conjectures of Swisher

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Abstract

Some congruences on conjectures of van Hamme are established. These results extend some of Swisher’s conjectures.

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References

  1. Ahlgren S., Ono K.: A Gaussian hypergeometric series evaluation and Apéry number congruences. J. Reine Angew. Math. 518, 187–212 (2000)

    MathSciNet  MATH  Google Scholar 

  2. Andrews G., Askey R., Roy R.: Special Functions. Cambridge University Press, Cambridge (1999)

    Book  MATH  Google Scholar 

  3. Cohen H.: Number Theory. Vol. II. Analytic and Modern Tools. Graduate Texts in Mathematics, 240. Springer, New York (2007)

    Google Scholar 

  4. Gessel I.M.: Finding identities with the WZ method. J. Symb. Comput. 20, 537–566 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. He B.: Some congruences on truncated hypergeometric series. Proc. Am. Math. Soc. 143, 5173–5180 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. He, B.: Supercongruences on truncated hypergeometric series, submitted.

  7. Kilbourn T.: An extension of the Apéry number supercongruence. Acta Arith. 123(4), 335–348 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Long L.: Hypergeometric evaluation identities and supercongruences. Pacific J. Math. 249, 405–418 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Long, L., Ramakrishna, R.: Some supercongruences occurring in truncated hypergeometric series. arXiv:1403.5232

  10. McCarthy D., Osburn R.: A p-adic analogue of a formula of Ramanujan. Arch. Math. (Basel) 91(6), 492–504 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Osburn R., Zudilin W.: On the (K.2) supercongruence of Van Hamme. J. Math. Anal. Appl. 433, 706–711 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ramanujan S.: Modular equations and approximations to \({\pi}\). Quart. J. Math. (Oxford) 45, 350–372 (1914)

    MATH  Google Scholar 

  14. Swisher, H.: On the supercongruence conjectures of van Hamme, arXiv:1504.01028

  15. van Hamme, L.: Some conjectures concerning partial sums of generalized hypergeometric series, p-adic functional analysis (Nijmegen, 1996). In: Lecture Notes in Pure and Appl. Math., vol. 192, pp. 223–236. Dekker, New York (1997)

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

    Article  MathSciNet  MATH  Google Scholar 

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He, B. On Some Conjectures of Swisher. Results Math 71, 1223–1234 (2017). https://doi.org/10.1007/s00025-016-0584-1

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  • DOI: https://doi.org/10.1007/s00025-016-0584-1

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