（组合恒等式）超几何级数、基本超几何级数和theta-椭圆超几何级数是特殊函数理论重要的组成部分.模形式是theta函数与椭圆函数的本质推广.它们都是研究解析数论不可或缺的理论基础与工具.本项目将围绕作者近期提出的t-系数法来展开对特殊函数理论与模形式的研究.主要内容包括:利用组合反演方法建立theta-椭圆超几何级数的变换与求和公式;有限个theta 函数之积的傅里叶级数展开;Ramanujan关于theta函数的循环和式一般显式表达式的建立;研究Abelian椭圆函数与Jacobian椭圆函数理论与方法之间的相互关系;并探索相关结论在自然数平方和表示与整数分拆等组合与解析数论问题上的应用.

(Combinatorial identities)Hypergeometric series,basic hypergeometric series and theta-elliptic hypergeometric series are fundamental parts of special functions. Modular forms are essential generalizations of theta and elliptic functions.They are necessarily theoretical tools for the study of analytic number theory. This proposal is mainly concerned with potential applications of the so-called t-coefficient method very recently discovered by the applicant to Special functions and Modular forms. The main content is planned as follows: to find any possibly summation and transformation formulas of elliptic hypergeometric series via combinatorial inverse technique；to study series expansions for the product of arbitrary finite theta functions;to find general explicit expressions of Ramanujan's circulant summation for theta function;to track relationship between Abelian elliptic functions and Jacobian elliptic functions; ultimately to apply all obtained results to Combinatorial-Analytic Number Theory such as quadratic sum of positive integer representations and integer partitions and so forth.