Berndt-Type Integrals and Series Associated with Ramanujan and Jacobi Elliptic Functions

：徐策（安徽师范大学）
：2023-06-02 09:30
：海韵园实验楼104报告厅

报告人：徐策（安徽师范大学）

时  间：2023年6月2日9:30

地  点：海韵园实验楼104报告厅

内容摘要:

In this talk, we will introduce how to evaluate in closed forms two families of infinite integrals containing hyperbolic and trigonometric functions in their integrands. We call them Berndt-type integrals since he initiated the study of similar integrals. We first establish explicit evaluations of four classes of hyperbolic sums by special values of the Gamma function, by two completely different approaches, which extend those sums considered by Ramanujan and Zucker previously. We discover the first by refining two results of Ramanujan concerning some q-series. For the second we compare both the Fourier series expansions and the Maclaurin series expansions of a few Jacobi elliptic functions. Next, by contour integrations we convert two families of Berndt-type integrals to the above hyperbolic sums, all of which can be evaluated in  closed forms. We then discover explicit formulas for one of the two families. Throughout the paper we present many examples which enable us to formulate a conjectural explicit formula for the other family of the Berndt-type integrals at the end.

个人简介：

徐策，硕士生导师，安徽师范大学数学与统计学院副教授。2020年博士毕业于厦门大学，同年加盟安徽师范大学数学与统计学院。曾在日本九州大学访学一年，师从Masanobu Kaneko教授，长期从事多重zeta值（Multiple zeta values, MZVs）及其相关变形的研究。主持国家自然科学基金（青年项目），安徽省自然科学基金（青年项目）和安徽省教育厅高校项目各1项。在《Mathematische Zeitschrift》、《Journal of Algebra》、《Forum Mathematicum》、《Advances in Applied Mathematics》、《Journal of Number Theory》、《European Journal of Combinatorics》等期刊发表论文40余篇。

 

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