Xiaoxia Wang, Chang Xu: A $q$-supercongruence modulo the fourth power of a cyclotomic polynomial, 345-353

Abstract:

Recently, Liu provided several nice supercongruences. Inspired by his work, in this paper, we establish a new $q$-supercongruence with two free parameters modulo the fourth power of a cyclotomic polynomial. By taking suitable parameter substitutions in this $q$-supercongruence, we derive some new results including a partial $q$-analogue of Liu's supercongruence. Our main auxiliary tools are Watson's $_8\phi_7$ transformation formula for basic hypergeometric series, the `creative microscoping' method introduced by Guo and Zudilin and the Chinese remainder theorem for coprime polynomials.

Key Words: Basic hypergeometric series, Watson's $_8\phi_7$ transformation, creative microscoping, Chinese remainder theorem.

2020 Mathematics Subject Classification: Primary 33D15; Secondary 11A07, 11B65.

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