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

日期:2020-12-01 来源:

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

主讲: 郭军伟 教授

时间: 12月2日(周三) 15:30

地点: 文理楼B302

主办: 数理部、数学与信息安全研究所


专家简介: 郭军伟,淮阴师范学院教授,翔宇学者。曾任华东师范大学数学系教授,博士生导师。主要从事组合数学,q-级数和数论的研究,是中国组合数学界杰出的青年数学家。郭军伟教授共发表SCI论文一百余篇,最近他利用单位根来证明q-同余式的新方法,是q-同余式方向的一个重大突破,其研究成果已经被国际权威期刊《Advances in Mathematics》发表。郭军伟教授先后主持三项国家自然科学基金,以及上海市教育发展基金会晨光计划,上海市科委青年科技启明星计划,江苏省自然科学基金等项目,并入选江苏省教育厅“青蓝工程”中青年学术带头人等。


报告主要内容:

 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. 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.