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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: [email protected], [email protected]
NEELAM SAIKIA
Affiliation:
Department of Mathematics, Indian Institute of Technology Guwahati, North Guwahati, Guwahati-781039, Assam, India. e-mail: [email protected], [email protected]
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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 

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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