q-Supercongruences modulo the fourth power of a cyclotomic polynomial via creative microscoping
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-supercongruen...
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| Published in | Advances in applied mathematics Vol. 120; p. 102078 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
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Elsevier Inc
01.09.2020
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| ISSN | 0196-8858 |
| DOI | 10.1016/j.aam.2020.102078 |
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| Abstract | 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. |
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| AbstractList | 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. |
| ArticleNumber | 102078 |
| Author | Guo, Victor J.W. |
| Author_xml | – sequence: 1 givenname: Victor J.W. surname: Guo fullname: Guo, Victor J.W. email: jwguo@hytc.edu.cn organization: School of Mathematics and Statistics, Huaiyin Normal University, Huai'an 223300, Jiangsu, People's Republic of China |
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| Snippet | By applying the Chinese remainder theorem for coprime polynomials and the “creative microscoping” method recently introduced by the author and Zudilin, we... |
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| SubjectTerms | Creative microscoping Cyclotomic polynomial q-Congruence The Chinese remainder theorem |
| Title | q-Supercongruences modulo the fourth power of a cyclotomic polynomial via creative microscoping |
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