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SUPERCONGRUENCES INVOLVING $p$-ADIC GAMMA FUNCTIONS

Part of: Elementary number theory Algebraic number theory: local and $p$-adic fields Elementary classical functions

Published online by Cambridge University Press:  30 May 2018

JI-CAI LIU Open the ORCID record for JI-CAI LIU [Opens in a new window]
JI-CAI LIU*
Affiliation:
Department of Mathematics, Wenzhou University, Wenzhou 325035, PR China email [email protected]
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Abstract

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We establish some supercongruences for the truncated $_{2}F_{1}$ and $_{3}F_{2}$ hypergeometric series involving the $p$-adic gamma functions. Some of these results extend the four Rodriguez-Villegas supercongruences on the truncated $_{3}F_{2}$ hypergeometric series. Related supercongruences modulo $p^{3}$ are proposed as conjectures.

Keywords

supercongruencestruncated hypergeometric seriesp-adic gamma functions

MSC classification

Primary: 11A07: Congruences; primitive roots; residue systems
Secondary: 11S80: Other analytic theory (analogues of beta and gamma functions, $p$-adic integration, etc.) 33C20: Generalized hypergeometric series, $_pF_q$ 33B15: Gamma, beta and polygamma functions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 98 , Issue 1 , August 2018 , pp. 27 - 37
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

References

Ahlgren, S., ‘Gaussian hypergeometric series and combinatorial congruences’, in: Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics (Gainesville, FL, 1999), Developments in Mathematics, 4 (Kluwer, Dordrecht, 2001), 1–12.Google Scholar
Bailey, W. N., Generalized Hypergeometric Series (Stechert-Hafner, New York, 1964).Google Scholar
Cohen, H., Number Theory. Vol. II. Analytic and Modern Tools, Graduate Texts in Mathematics, 240 (Springer, New York, 2007).Google Scholar
Guo, V. J. W. and Zeng, J., ‘Some q-supercongruences for truncated basic hypergeometric series’, Acta Arith. 171 (2015), 309–326.CrossRefGoogle Scholar
Ishikawa, T., ‘Super congruence for the Apéry numbers’, Nagoya Math. J. 118 (1990), 195–202.Google Scholar
Kilbourn, T., ‘An extension of the Apéry number supercongruence’, Acta Arith. 123 (2006), 335–348.Google Scholar
Long, L. and Ramakrishna, R., ‘Some supercongruences occurring in truncated hypergeometric series’, Adv. Math. 290 (2016), 773–808.Google Scholar
Mortenson, E., ‘Supercongruences for truncated n+1 F n hypergeometric series with applications to certain weight three newforms’, Proc. Amer. Math. Soc. 133 (2005), 321–330.Google Scholar
Rodriguez-Villegas, F., ‘Hypergeometric families of Calabi-Yau manifolds’, in: Calabi-Yau Varieties and Mirror Symmetry (Toronto, ON, 2001), Fields Institute Communications, 38 (American Mathematical Society, Providence, RI, 2003), 223–231.Google Scholar
Schneider, C., ‘Symbolic summation assists combinatorics’, Sém. Lothar. Combin. 56 (2007), B56b.Google Scholar
Sun, Z.-H., ‘Congruences concerning Legendre polynomials’, Proc. Amer. Math. Soc. 139 (2011), 1915–1929.Google Scholar
Sun, Z.-H., ‘Generalized Legendre polynomials and related supercongruences’, J. Number Theory 143 (2014), 293–319.CrossRefGoogle Scholar
Sun, Z.-H., ‘Note on super congruences modulo $p^{2}$ ’, Preprint, 2015, arXiv:1503.03418.Google Scholar
Sun, Z.-W., ‘Super congruences and Euler numbers’, Sci. China Math. 54 (2011), 2509–2535.Google Scholar
Sun, Z.-W., ‘On sums involving products of three binomial coefficients’, Acta Arith. 156 (2012), 123–141.Google Scholar
Sun, Z.-W., ‘Supercongruences involving products of two binomial coefficients’, Finite Fields Appl. 22 (2013), 24–44.Google Scholar
Tauraso, R., ‘An elementary proof of a Rodriguez-Villegas supercongruence’, Preprint, 2009, arXiv:0911.4261.Google Scholar
van Hamme, L., ‘Proof of a conjecture of Beukers on Apéry numbers’, in: Proceedings of the Conference on p-Adic Analysis (Houthalen, 1987) (Vrije Universiteit, Brussels, 1987), 189–195.Google Scholar