Some q-supercongruences modulo the fourth power of a cyclotomic polynomial

With the help of the creative microscoping method recently introduced by Guo and Zudilin and the Chinese remainder theorem for coprime polynomials, we establish a q-supercongruence with two parameters modulo [n]Phi(n)(q)(3). Here [n] = (1 - q(n))/(1 - q) and Phi(n)(q) is the n-th cyclotomic polynomial in q. In particular, we confirm a recent conjecture of Guo and give a complete q-analogue of Long's supercongruence. The latter is also a generalization of a recent q-supercongruence obtained by Guo and Schlosser. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

With the help of the creative microscoping method recently introduced by Guo and Zudilin and the Chinese remainder theorem for coprime polynomials, a q-supercongruence with two parameters modulo [n]Phi(n)(q)(3) is established. This confirms a recent conjecture of Guo and gives a complete q-analogue of Long's supercongruence, which is also a generalization of a recent q-supercongruence obtained by Guo and Schlosser.