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q-Supercongruences modulo the fourth power of a cyclotomicpolynomial via creative microscoping
Advances in Applied Mathematics ( IF 1.1 ) Pub Date : 2020-09-01 , DOI: 10.1016/j.aam.2020.102078 Victor J.W. Guo
Advances in Applied Mathematics ( IF 1.1 ) Pub Date : 2020-09-01 , DOI: 10.1016/j.aam.2020.102078 Victor J.W. Guo
By applying Chinese remainder theorem for coprime polynomials and the "creative microscoping" method recently introduced by the author and Zudilin, we establish parametric generalizations of three $q$-supercongruences modulo the fourth power of a cyclotomic polynomial. The original $q$-supercongruences then follow from these parametric generalizations by taking the limits as the parameter tends to $1$ (l'Hopital's rule is utilized here). In particular, we prove a complete $q$-analogue of the (J.2) supercongruence of Van Hamme and a complete $q$-analogue of a "divergent" Ramanujan-type supercongruence, thus confirming two recent conjectures of the author. We also put forward some related conjectures, including a $q$-supercongruence modulo the fifth power of a cyclotomic polynomial.
更新日期：2020-09-01
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