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NEW GENERALISATIONS OF VAN HAMME’S (G.2) SUPERCONGRUENCE

Part of: Sequences and sets Basic hypergeometric functions Elementary number theory

Published online by Cambridge University Press:  18 May 2022

NA TANG Open the ORCID record for NA TANG [Opens in a new window]
NA TANG*
Affiliation:
School of Mathematics and Statistics, Huaiyin Normal University, Huai’an 223300, Jiangsu, PR China
*
e-mail: hytn999@126.com
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Abstract

Swisher [‘On the supercongruence conjectures of van Hamme’, Res. Math. Sci. 2 (2015), Article no. 18] and He [‘Supercongruences on truncated hypergeometric series’, Results Math. 72 (2017), 303–317] independently proved that Van Hamme’s (G.2) supercongruence holds modulo $p^4$ for any prime $p\equiv 1\pmod {4}$ . Swisher also obtained an extension of Van Hamme’s (G.2) supercongruence for $p\equiv 3 \pmod 4$ and $p>3$ . In this note, we give new one-parameter generalisations of Van Hamme’s (G.2) supercongruence modulo $p^3$ for any odd prime p. Our proof uses the method of ‘creative microscoping’ introduced by Guo and Zudilin [‘A q-microscope for supercongruences’, Adv. Math. 346 (2019), 329–358].

Keywords

q-supercongruencesupercongruenceJackson’s 6 ϕ 5 summationcreative microscoping

MSC classification

Primary: 11B65: Binomial coefficients; factorials; $q$-identities
Secondary: 11A07: Congruences; primitive roots; residue systems 33D15: Basic hypergeometric functions in one variable, $_rphi_s$
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 107 , Issue 2 , April 2023 , pp. 177 - 183
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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