Ramanujan theta函数中的代数组合方法

Ramanujan theta函数是组合数学和特殊函数领域中备受关注的一个重要研究对象。本项目旨在围绕Ramanujan theta函数、偏theta函数和仿theta函数，利用代数组合学中的分析方法和计算技巧，系统研究这三类theta函数恒等式的推导和拓展及其相关的组合性质和同余性质，主要包括以下几个方面的内容：利用基本超几何级数理论、Bailey对理论和q正交多项式理论，研究这三类theta函数恒等式的参数的扩充和形式的推广；以Warnaar提出的寻找Jacobi三重积在偏theta函数上推广形式的组合证明的问题为切入点，研究特殊theta函数恒等式的组合性质；以整数分拆同余式和模形式理论的研究为基础，以三阶和五阶仿theta函数为初始研究对象，逐步将Ramanujan theta函数理论应用于分拆同余式的研究中。

Ramanujan theta functions have attracted a lot of attention in the field of combinatorics and special functions. This project is mainly concerned with the algebraic combinatorial methods on Ramanujan's theta functions, partial theta functions and mock theta functions. We aim to study the derivation and the extension methods for Ramanujan theta function identities and the related combinatorial properties and congruences. We shall focus on the following: applying the theory of basic hypergeometric series, Bailey pairs and q-orthogonal polynomials to extend identities on Ramanujan's theta functions; studying the combinatorial properties of special theta functions by starting with Warnaar's extension of Jacobi's triple product identity on partial theta functions; employing Ramanujan theta function to study the partition congruences by using the theory of partitions and modular forms, especially for mock theta functions of order 3 and 5.