Mock theta函数的各种表示形式与多变量拓广的研究

起源于Ramanujan的Mock theta函数是q-级数理论研究的重要内容，在当代数论、物理学、代数学等领域均有非常广泛的应用，模形式与q-级数的重叠问题是国际著名数论专家、美国科学院院士Andrews在2000年国际数论大会上提出的挑战问题。本项目旨在进一步研究Mock theta函数的各种表示及其相关问题，回答或部分回答此挑战问题。具体表现为创新新思想和新方法，改造和提升旧方法，建立新的双边Mock theta函数表示及Mock theta函数的双边半有限形式与其相关的Ramnujan的径向极限以及第二类对偶的对偶性等性质，得到Mock theta函数和经典Mock theta函数双重和与多重和的新表示；构造Mock theta函数的多变量U(n+1)、Cn与Dn等拓广，从新的视角，更深层次刻画模形式与q-级数之间的内在本质联系。

Mock theta function, which originated from Ramanujan, is an important part of q-series theory, and has been widely used in contemporary number theory, physics, algebra and other fields. The overlap between modular form and q-series was a challenge posed by Andrews, an internationally famous expert in number theory and an academician of the American Academy of Sciences, at the International Conference on Number Theory in 2000. The purpose of this project is to further explore various representations of the Mock theta functions and their related questions to answer or partially answer this challenge question. The concrete performance is innovating new ideas and methods, reforming and upgrading the old methods, establishing new bilateral representations and the bilateral semi finite forms of Mock theta functions, the related ramnujan radial limit and the duality of the second kind of duality, etc., obtaining the new representations of double sums and multiple sums of Mock theta function and classical Mock theta functions, and constructing the extensions of multivariate U(n+1), Cn and Dn of Mock theta functions. From a new perspective, the intrinsic relationship between modular form and q-series is described in a deeper level.