### Electronic Research Archive

2020, Issue 2: 1031-1036. doi: 10.3934/era.2020055
Special Issues

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

• 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.

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

### Related Papers:

• 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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