A family of $ q $-congruences modulo the square of a cyclotomic polynomial

Victor J. W. Guo; Victor J. W. Guo

doi:10.3934/era.2020055

Electronic Research Archive

2020, Volume 28, Issue 2: 1031-1036. doi: 10.3934/era.2020055

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A family of $ q $-congruences modulo the square of a cyclotomic polynomial

Victor J. W. Guo

School of Mathematics and Statistics, Huaiyin Normal University, Huai'an 223300, Jiangsu, China

Received: 01 January 2020 Revised: 01 April 2020
Primary: 33D15; Secondary: 11A07, 11B65

Using Watson's terminating $ _8\phi_7 $ transformation formula, we prove a family of $ q $-congruences modulo the square of a cyclotomic polynomial, which were originally conjectured by the author and Zudilin [J. Math. Anal. Appl. 475 (2019), 1636-646]. As an application, we deduce two supercongruences modulo $ p^4 $ ($ p $ is an odd prime) and their $ q $-analogues. This also partially confirms a special case of Swisher's (H.3) conjecture.
- Cyclotomic polynomial,
- $ q $-congruence,
- supercongruence,
- Swisher's conjecture,
- Watson's transformation
Citation: Victor J. W. Guo. A family of $ q $-congruences modulo the square of a cyclotomic polynomial[J]. Electronic Research Archive, 2020, 28(2): 1031-1036. doi: 10.3934/era.2020055

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Abstract

Using Watson's terminating $ _8\phi_7 $ transformation formula, we prove a family of $ q $-congruences modulo the square of a cyclotomic polynomial, which were originally conjectured by the author and Zudilin [J. Math. Anal. Appl. 475 (2019), 1636-646]. As an application, we deduce two supercongruences modulo $ p^4 $ ($ p $ is an odd prime) and their $ q $-analogues. This also partially confirms a special case of Swisher's (H.3) conjecture.

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