q-Supercongruences modulo the fourth power of a cyclotomic polynomial via creative microscoping

• Published in: Advances in applied mathematics Vol. 120; p. 102078
• Main Author: Guo, Victor J.W.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.09.2020
• Subjects: Creative microscoping; Cyclotomic polynomial; q-Congruence; The Chinese remainder theorem; 11A07; q-Congruence; Creative microscoping; 11B65; Cyclotomic polynomial; The Chinese remainder theorem; 33D15
• ISSN: 0196-8858
• DOI: 10.1016/j.aam.2020.102078

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'Hôpital'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.