RAMANUJAN’S MOCK THETA FUNCTIONS AND SINGULAR q-DIFFERENCE EQUATIONS

：张长贵（法国里尔大学）
：2025-05-09 14:30
：海韵园行政楼C503

报告人：张长贵（法国里尔大学）

时  间：2025年5月9日14:30

地  点：海韵园行政楼C503

内容摘要:

In his final letter to Hardy in 1920, Ramanujan introduced functions he termed mock theta—families of analytic functions defined in the unit disk |q| < 1, but possessing infinitely many singularities on its boundary. They are expected to possess certain characteristic properties of a theta function in terms of their behavior near roots of unity.

In this talk, after reviewing connections between partitions, rank generating functions, and Appell-Lerch series, we will examine Ramanujan’s series as special values of solutions to linear q-difference equations that have a non-Fuchsian singularity at zero or infinity in the complex plane. The analysis of these singularities helps us understand the analytic structure of the governing equation and will lead us to different possible representations of a Ramanujan mock theta function. In particular, we will obtain certain modular-type relations in terms of the Stokes phenomenon, known for divergent series.

个人简介：

张长贵，法国里尔大学教授。研究微分方程，差分方程和相关的特殊函数及其在解析数论方面的应用。其工作涉及带非正则奇点的微分方程和 q-差分方程发散级数解的求和，q-差分方程解析分类， Ramanujan 整函数和拟模函数等特殊函数的研究. 发表专著一册（Local analytic classification of qdifference equations.），在 Adv. Math., Ann. Inst. Fourier, Ann. Fac. Sci. Toulouse Math., Asymptot. Anal., J. Approx. Theory, J. Math. Anal. Appl., J. Math. Sci. Univ. Tokyo, Ramanujan J.Results Math 等杂志上发表论文三十八篇。

 

联系人：朱玉峻