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CONGRUENCES FOR TRUNCATED HYPERGEOMETRIC SERIES $_{2}F_{1}$

Part of: Elementary number theory

Published online by Cambridge University Press:  06 March 2017

JI-CAI LIU
JI-CAI LIU*
Affiliation:
College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, PR China email [email protected]
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Abstract

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Rodriguez-Villegas conjectured four supercongruences associated to certain elliptic curves, which were first confirmed by Mortenson by using the Gross–Koblitz formula. In this paper we prove four supercongruences between two truncated hypergeometric series $_{2}F_{1}$. The results generalise the four Rodriguez-Villegas supercongruences.

Keywords

supercongruencehypergeometric seriesFermat quotient

MSC classification

Primary: 11A07: Congruences; primitive roots; residue systems
Secondary: 33C05: Classical hypergeometric functions, $_2F_1$
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 96 , Issue 1 , August 2017 , pp. 14 - 23
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

References

Apéry, R., ‘Irrationalité de 𝜁(2) et 𝜁(3)’, Astérisque 61 (1979), 11–13.Google Scholar
Babbage, C., ‘Demonstration of a theorem relating to prime numbers’, Edinburgh Philos. J. 1 (1819), 46–49.Google Scholar
Chan, H. H., Long, L. and Zudilin, W., ‘A supercongruence motivated by the Legendre family of elliptic curves’, Math. Notes 88 (2010), 599–602.CrossRefGoogle Scholar
Chowla, S. J., Cowles, J. and Cowles, M., ‘Congruence properties of Apéry numbers’, J. Number Theory 12 (1980), 188–190.CrossRefGoogle Scholar
Gessel, I., ‘Some congruences for Apéry numbers’, J. Number Theory 14 (1982), 362–368.CrossRefGoogle Scholar
Guo, V. J. W., Pan, H. and Zhang, Y., ‘The Rodriguez-Villegas type congruences for truncated q-hypergeometric functions’, J. Number Theory 174 (2017), 358–368.CrossRefGoogle Scholar
Guo, V. J. W. and Zeng, J., ‘Some q-analogues of supercongruences of Rodriguez-Villegas’, J. Number Theory 145 (2014), 301–316.CrossRefGoogle Scholar
Lehmer, E., ‘On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson’, Ann. of Math. (2) 39 (1938), 350–360.CrossRefGoogle Scholar
Mortenson, E., ‘A supercongruence conjecture of Rodriguez-Villegas for a certain truncated hypergeometric function’, J. Number Theory 99 (2003), 139–147.CrossRefGoogle Scholar
Mortenson, E., ‘Supercongruences between truncated 2 F 1 hypergeometric functions and their Gaussian analogs’, Trans. Amer. Math. Soc. 355 (2003), 987–1007.CrossRefGoogle Scholar
Osburn, R. and Schneider, C., ‘Gaussian hypergeometric series and supercongruences’, Math. Comp. 78 (2009), 275–292.CrossRefGoogle Scholar
Prodinger, H., ‘Human proofs of identities by Osburn and Schneider’, Integers 8 (2008), A10, 8 pages.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
Sun, Z.-H., ‘Generalized Legendre polynomials and related supercongruences’, J. Number Theory 143 (2014), 293–319.Google Scholar
Sun, Z.-H., ‘Supercongruences involving Euler polynomials’, Proc. Amer. Math. Soc. 144 (2016), 3295–3308.CrossRefGoogle Scholar
Sun, Z.-W., ‘Super congruences and Euler numbers’, Sci. China Math. 54 (2011), 2509–2535.CrossRefGoogle Scholar
Sun, Z.-W., ‘Supercongruences involving Lucas sequences’, Preprint, 2016, arXiv:1610.03384.Google Scholar
Tauraso, R., ‘An elementary proof of a Rodriguez-Villegas supercongruence’, Preprint, 2009, arXiv:0911.4261.Google Scholar
Tauraso, R., ‘Supercongruences for a truncated hypergeometric series’, Integers 12 (2012), A45, 12 pages.Google Scholar