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A NEW $\boldsymbol {q}$-ANALOGUE OF VAN HAMME’S (A.2) SUPERCONGRUENCE

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

Published online by Cambridge University Press:  20 May 2022

VICTOR J. W. GUO Open the ORCID record for VICTOR J. W. GUO [Opens in a new window]
VICTOR J. W. GUO*
Affiliation:
School of Mathematics and Statistics, Huaiyin Normal University, Huai’an 223300, Jiangsu, PR China
*
e-mail: jwguo@math.ecnu.edu.cn
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Abstract

We give a new q-analogue of the (A.2) supercongruence of Van Hamme. Our proof employs Andrews’ multiseries generalisation of Watson’s $_{8}\phi _{7}$ transformation, Andrews’ terminating q-analogue of Watson’s $_{3}F_{2}$ summation, a q-Watson-type summation due to Wei–Gong–Li and the creative microscoping method, developed by the author and Zudilin [‘A q-microscope for supercongruences’, Adv. Math. 346 (2019), 329–358]. As a conclusion, we confirm a weaker form of Conjecture 4.5 by the author [‘Some generalizations of a supercongruence of van Hamme’, Integral Transforms Spec. Funct. 28 (2017), 888–899].

Keywords

basic hypergeometric seriesq-congruencesupercongruencecreative 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 1 , February 2023 , pp. 22 - 30
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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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References

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