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SUPERCONGRUENCES INVOLVING $p$-ADIC GAMMA FUNCTIONS

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

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. 

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