Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-14T07:17:13.889Z Has data issue: false hasContentIssue false
  • English
  • Français

Supercongruences for truncated hypergeometric series and p-adic gamma function

Published online by Cambridge University Press:  10 August 2018

RUPAM BARMAN  and
NEELAM SAIKIA
RUPAM BARMAN
Affiliation:
Department of Mathematics, Indian Institute of Technology Guwahati, North Guwahati, Guwahati-781039, Assam, India. e-mail: rupam@iitg.ernet.in, nlmsaikia1@gmail.com
NEELAM SAIKIA
Affiliation:
Department of Mathematics, Indian Institute of Technology Guwahati, North Guwahati, Guwahati-781039, Assam, India. e-mail: rupam@iitg.ernet.in, nlmsaikia1@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove three more general supercongruences between truncated hypergeometric series and p-adic gamma function from which some known supercongruences follow. A supercongruence conjectured by Rodriguez--Villegas and proved by E. Mortenson using the theory of finite field hypergeometric series follows from one of our more general supercongruences. We also prove a supercongruence for 7F6 truncated hypergeometric series which is similar to a supercongruence proved by L. Long and R. Ramakrishna.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 1 , January 2020 , pp. 171 - 195
Copyright
Copyright © Cambridge Philosophical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Ahlgren, S.. Gaussian hypergeometric series and combinatorial congruences. Symbolic computation, number theory, special functions, physics and combinatorics. Dev. Math. 4 (Kluwer, Dodrecht 2001).Google Scholar
[2] Ahlgren, S. and Ono, K. A Gaussian hypergeometric series evaluation and Apéry number congruences. J. Reine Angew. Math. 518 (2000), 187–212.Google Scholar
[3] Andrews, G., Askey, R. and Roy, R. Special Functions. Encyc. Math. Appl. 71 (Cambridge University Press, Cambridge, 1999).Google Scholar
[4] Bailey, W.. Generalised Hypergeometric Series (Cambridge University Press, Cambridge, 1935).Google Scholar
[5] Barman, R. and Kalita, G. Hypergeometric functions and a family of algebraic curves. Ramanujan J. 28 (2012), 175–185.Google Scholar
[6] Beukers, F. Some congruence for the Apéry numbers. J. Number Theory 21 (1985), 241–155.Google Scholar
[7] Beukers, F. Another congruence for the Apéry numbers. J. Number Theory 25 (1987), 201–210.Google Scholar
[8] Chan, H.-H., Cooper, S. and Sica, F.. Congruences satisfied by Apéry-like numbers. Int. J. Number Theory 6 (2010), 89–97.Google Scholar
[9] Deines, A., Fuselier, J., Long, L., Swisher, H. and Tu, F. Hypergeometric series, truncated hypergeometric series, and Gaussian hypergeometric functions. Directions in Number Theory: Proceedings for Women in Number Theory 3 workshop. Association for Women in Mathematics series (Springer, 2016).Google Scholar
[10] Fuselier, J. and McCarthy, D.. Hypergeometric type identities in the p-adic setting and modular forms. Proc. Amer. Math. Soc. 144 (2016), 1493–1508.Google Scholar
[11] Gasper, G. and Rahman, M. Basic hypergeometric series. Encycl. Math. Appl. 96 (Cambridge University Press, Cambridge, second edition, 2004), with a foreword by Askey, Richard.Google Scholar
[12] Greene, J. Hypergeometric functions over finite fields. Trans. Amer. Math. Soc. 301 (1987), 77–101.Google Scholar
[13] van Hamme, L.. Some conjectures concerning partial sums of generalized hypergeometric series. p-adic functional analysis (Nijmegen, 1996), 223–236. Lecture Notes in Pure and Appl. Math., 192 (Dekker, New York, 1997).Google Scholar
[14] Kibelbek, J., Long, L., Moss, K., Sheller, B. and Yuan, H.. Supercongruences and complex multiplication. J. Number Theory 164 (2016), 166–178.Google Scholar
[15] Kilbourn, T. An extension of the Apéry number supercongruence. Acta Arith. 123 (2006), 335–348.Google Scholar
[16] Koblitz, N. p-adic analysis: a short course on recent work, London Math. Soc. Lecture Note Series 46 (Cambridge University Press, Cambridge-New York, 1980).Google Scholar
[17] Long, L. Hypergeometric evaluation identities and supercongruences. Pacific J. Math. 249 (2011), 405–418.Google Scholar
[18] Long, L. and Ramakrishna, R. Some supercongruences occurring in truncated hypergeometric series. Adv. Math. 290 (2016), 773–808.Google Scholar
[19] McCarthy, D. 3F 2 Hypergeometric series and periods of elliptic curves. Int. J. Number Theory 6 (2010), 461–470.Google Scholar
[20] McCarthy, D.. Extending Gaussian hypergeometric series to the p-adic setting. Int. J. Number Theory 8 (2012), 1581–1612.Google Scholar
[21] McCarthy, D.. The trace of Frobenius of elliptic curves and the p-adic gamma function. Pacific J. Math. 261 (2013), 219–236.Google Scholar
[22] McCarthy, D. and Osburn, R.. A p-adic analogue of a formula of Ramanujan. Arch. Math. (Basel) 91 (2008), 492–504.Google Scholar
[23] Mortenson, E. A supercongruence conjecture of Rodriguez–Villegas for a certain truncated hypergeometric function. J. Number Theory 99 (2003), 139–147.Google Scholar
[24] Mortenson, E. Supercongruences for truncated n+1F n hypergeometric series with applications to certain weight three newforms. Proc. Amer. Math. Soc. 133 (2005), 321–330.Google Scholar
[25] Osburn, R., Sahu, B. and Straub, A. Supercongruences for sporadic sequences. Pro. Edinburgh Math. Soc. 59 (2016), 503–518.Google Scholar
[26] Selberg, A. and Chowla, S. On Epstein's zeta-function. J. Reine Angew. Math. 227 (1967), 86–110.Google Scholar
[27] Slater, L. J. Generalised Hypergeometric Functions (Cambridge University Press, Cambridge, 1966).Google Scholar
[28] Sun, Z.–W. On congruences related to central binomial coefficients. J. Number Theory 131 (2011), 2219–2238.Google Scholar
[29] Sun, Z.–W. Super congruences and Euler numbers. Sci. China Math. 54 (2011), 2509–2535.Google Scholar
[30] Swisher, H. On the supercongruence conjectures of van Hamme. Research in the Mathematical Sciences 2 (2015), 18.Google Scholar
[31] Rodriguez–Villegas, F.. Hypergeometric families of Calabi–Yau manifolds. Calabi–Yau varieties and mirror symmetry. (Toronto, ON, 2001), 223–243, Fields Inst. Commun. 38 (Amer. Math. Soc., Providence, RI, 2003).Google Scholar
[32] Zudilin, W. Ramanujan-type supercongruences. J. Number Theory 129 (2009), 1848–1857.Google Scholar