Abstract
Some congruences on conjectures of van Hamme are established. These results extend some of Swisher’s conjectures.
Similar content being viewed by others
References
Ahlgren S., Ono K.: A Gaussian hypergeometric series evaluation and Apéry number congruences. J. Reine Angew. Math. 518, 187–212 (2000)
Andrews G., Askey R., Roy R.: Special Functions. Cambridge University Press, Cambridge (1999)
Cohen H.: Number Theory. Vol. II. Analytic and Modern Tools. Graduate Texts in Mathematics, 240. Springer, New York (2007)
Gessel I.M.: Finding identities with the WZ method. J. Symb. Comput. 20, 537–566 (1995)
He B.: Some congruences on truncated hypergeometric series. Proc. Am. Math. Soc. 143, 5173–5180 (2015)
He, B.: Supercongruences on truncated hypergeometric series, submitted.
Kilbourn T.: An extension of the Apéry number supercongruence. Acta Arith. 123(4), 335–348 (2006)
Long L.: Hypergeometric evaluation identities and supercongruences. Pacific J. Math. 249, 405–418 (2011)
Long, L., Ramakrishna, R.: Some supercongruences occurring in truncated hypergeometric series. arXiv:1403.5232
McCarthy D., Osburn R.: A p-adic analogue of a formula of Ramanujan. Arch. Math. (Basel) 91(6), 492–504 (2008)
Mortenson E.: A p-adic supercongruence conjecture of van Hamme. Proc. Am. Math. Soc. 136, 4321–4328 (2008)
Osburn R., Zudilin W.: On the (K.2) supercongruence of Van Hamme. J. Math. Anal. Appl. 433, 706–711 (2016)
Ramanujan S.: Modular equations and approximations to \({\pi}\). Quart. J. Math. (Oxford) 45, 350–372 (1914)
Swisher, H.: On the supercongruence conjectures of van Hamme, arXiv:1504.01028
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)
Zudilin W.: Ramanujan-type supercongruences. J. Number Theory 129, 1848–1857 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
He, B. On Some Conjectures of Swisher. Results Math 71, 1223–1234 (2017). https://doi.org/10.1007/s00025-016-0584-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-016-0584-1