2019/11/29 by Victor J. W. Guo, Guo, Victor J. W. · 1 citation
Mathematics · #Advanced Mathematical Identities #Advanced Combinatorial Mathematics #Analytic Number Theory Research
paper · pdf · doi:10.48550/arxiv.1912.00765
By applying Chinese remainder theorem for coprime polynomials and the\n"creative microscoping" method recently introduced by the author and Zudilin,\nwe establish parametric generalizations of three q-supercongruences modulo\nthe fourth power of a cyclotomic polynomial. The original q-supercongruences\nthen follow from these parametric generalizations by taking the limits as the\nparameter tends to 1 (l'H opital's rule is utilized here). In particular, we\nprove a complete q-analogue of the (J.2) supercongruence of Van Hamme and a\ncomplete q-analogue of a "divergent" Ramanujan-type supercongruence, thus\nconfirming two recent conjectures of the author. We also put forward some\nrelated conjectures, including a q-supercongruence modulo the fifth power of\na cyclotomic polynomial.\n