Abstract
Using an identity due to Gessel and Stanton and some properties of the p-adic Gamma function, we establish a p-adic supercongruence for truncated hypergeometric series \({}_7F_6\). From it we deduce some related supercongruences, which extend certain recent results and confirm a supercongruence conjecture.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ahlgren, S., Ono, K.: A Gaussian hypergeometric series evaluation and Apéry number congruences. J. Reine Angew. Math. 518, 187–212 (2000)
Barman, R., Saikia, N.: Supercongruences for truncated hypergeometric series and \(p\)-adic gamma function, preprint, (2015), arXiv:1507.07391
Cohen, H.: Number theory. vol. II. Analytic and modern tools, Grad. Texts in Math., vol. 240, Springer, New York, (2007)
Gessel, I., Stanton, D.: Strange evaluations of hypergeometric series. SIAM J. Math. Anal. 13, 295–308 (1982)
He, B.: Supercongruences and truncated hypergeometric series. Proc. Amer. Math. Soc. 145, 501–508 (2017)
Kilbourn, T.: An extension of the Apéry number supercongruence. Acta Arith. 123, 335–348 (2006)
Long, L.: Hypergeometric evaluation identities and supercongruences. Pac. J. Math. 249, 405–418 (2011)
Long, L., Ramakrishna, R.: Some supercongruences occurring in truncated hypergeometric series. Adv. Math. 290, 773–808 (2016)
McCarthy, D., Osburn, R.: A \(p\)-adic analogue of a formula of Ramanujan. Arch. Math. (Basel) 91, 492–504 (2008)
Morita, Y.: A \(p\)-adic analogue of the \(\Gamma \)-function. J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math 22, 255–266 (1975)
Mortenson, E.: Supercongruences for truncated \({}_{n+1}F_n\) hypergeometric series with applications to certain weight three newforms. Proc. Amer. Math. Soc. 133, 321–330 (2005)
Mortenson, E.: A \(p\)-adic supercongruence conjecture of van Hamme. Proc. Amer. Math. Soc. 136, 4321–4328 (2008)
Rodriguez-Villegas, F.: Hypergeometric families of Calabi-Yau manifolds, Calabi-Yau varieties and mirror symmetry (Toronto, ON, 2001). Fields Inst. Commun. Amer. Math. Soc., Providence, RI, vol. 38, 223-231 (2003)
van Hamme, L.: Some conjectures concerning partial sums of generalized hypergeometric series, \(p\)-adic functional analysis (Nijmegen, : Lecture Notes in Pure and Appl. Math., Dekker, New York 1997 vol. 192, 223–236 (1996)
Zudilin, W.: Ramanujan-type supercongruences. J. Num. Theory 129, 1848–1857 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liu, JC. A p-adic Supercongruence for Truncated Hypergeometric Series \({}_7F_6\) . Results Math 72, 2057–2066 (2017). https://doi.org/10.1007/s00025-017-0744-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-017-0744-y