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Bull. Korean Math. Soc. 2023; 60(3): 775-783

Online first article May 11, 2023      Printed May 31, 2023

https://doi.org/10.4134/BKMS.b220369

Copyright © The Korean Mathematical Society.

A new $q$-analogue of Van Hamme's $($G.2$)$ supercongruence for primes $p\equiv 3\pmod{4}$

Victor J. W. Guo , Xiuguo Lian

Huaiyin Normal University; Huaiyin Normal University

Abstract

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Van Hamme's (G.2) supercongruence modulo $p^4$ for primes $p\equiv 3 \pmod 4$ and $p>3$ was first established by Swisher. A $q$-analogue of this supercognruence was implicitly given by the first author and Schlosser. In this paper, we present a new $q$-analogue of Van Hamme's (G.2) supercongruence for $p\equiv 3\pmod{4}$.

Keywords: Cyclotomic polynomials, $q$-supercongruences, supercongruences, Jackson's ${}_6\phi_5$ summation, creative microscoping

MSC numbers: Primary 33D15, 11A07, 11B65

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