q-Supercongruences modulo the fourth power of a cyclotomic polynomial via creative microscoping
q-Supercongruences modulo the fourth power of a cyclotomic polynomial via creative microscoping
复制标题
q-超同余通过创造性显微镜对分圆多项式的四次方取模
DOI：
10.1016/j.aam.2020.102078
复制
发表时间：
2020-09-01
影响因子：
1.1
通讯作者：
Guo, Victor J. W.
中科院分区：
文献类型：
--
作者：
Guo, Victor J. W.
利用互素多项式的中国剩余定理和作者与Zudilin最近提出的“创造性显微”方法，建立了分圆多项式模四次幂的三个q-超同余的参数推广.原始的q-超同余则通过取极限作为参数趋于1（这里使用l 'Hopitars规则）从这些参数推广而来。特别地，我们证明了货车Hamme的（J.2）超同余的一个完全q-模拟和一个“发散的”Ramanujan型超同余的一个完全q-模拟，从而证实了作者最近的两个猜想.我们也给出了一些相关的证明，包括分圆多项式模五次幂的q-超同余。(C)2020爱思唯尔公司All rights reserved.
By applying the 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'Hopitars 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. (C) 2020 Elsevier Inc. All rights reserved.