A $q$-supercongruence modulo the fourth power of a cyclotomic polynomial

Xiaoxia Wang; Chang Xu

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arXiv · 2201.05785

A $q$-supercongruence modulo the fourth power of a cyclotomic polynomial

Xiaoxia Wang·Chang Xu

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Abstract

In this paper, a new $q$-supercongruence with two free parameters modulo the fourth power of a cyclotomic polynomial is obtained. Our main auxiliary tools are Watson's $_8\phi_7$ transformation formula for basic hypergeometric series, the `creative microscoping' method recently introduced by Guo and Zudilin and the Chinese remainder theorem for coprime polynomials. By taking suitable parameter substitutions in the established $q$-supercongruence, some nice congruences involving the Bernoulli numbers are derived.

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Xiaoxia Wang, Chang Xu. 2022-01-15. A $q$-supercongruence modulo the fourth power of a cyclotomic polynomial. https://arxiv.org/abs/2201.05785

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