本项目旨在研究组合分析、正交多项式和特殊函数理论中的有关组合反演, q-函数展开与基本超几何级数的变换, 以及部分theta函数恒等式等问题. 为此本项目将发展目前初见成效的参数算子法、Bailey算子法、t-系数法等方法. 具体内容包括:(1)研究沿Dyck路加权的和Gessel形式Lagrange反演公式与组合反演的联系;利用特征函数建立带限制条件的组合反演和(f,g)反演的多截断形式;研究这些反演在正交多项式与特殊函数领域的应用;(2)研究(1-xy,y-x)-展开公式和其它类似的q-展开公式及应用;(3)研究由Bailey对与Bailey引理抽象出的Bailey算子及应用;(4)研究"视参数为变量"观点下的参数代换算子的理论与应用;(5)推广Andrews-Warnaar关于部分theta 函数和、积恒等式,部分theta 函数的二元表示与应用.

The theme of this proposal is to study some problems in Combination Analysis, Orthogonal Polynomials and Special Functions which cover combinatorial inversions, q-function expansions, transformations of q-series and the partial theta functions. For this purpose, this project will further develop some useful methods such as the substitutions of parameter operators, the Bailey operators and the t-coefficient method initially proposed by the applicant himself. The detailed content is as follows. (i) To study any applications of the Lagrange inversion formula on certain weighted Dyck path and Gessel's formal Lagrange inversion formula to linear or non-linear combinatorial inversions; using the characteristic function to find combinatorial inversions with various restricted conditions and multi-dissected (f,g)-inversion. Applications of these inversions in Orthogonal polynomials and Special Functions will be investigated simultaneously; (ii) To investigate the (1-xy,y-x)-expansion formula and so on; (iii) To study the so-called Bailey operators steming from the Bailey pairs and the Bailey lemma and its applications to basic hypergeometric series. (iv) To develop the method of the substitutions of parameters operators saying "parameters being treated as variables" and its applications to basic hypergeometric series transformation;(v) To find any extension of the sum and product formula of the partial theta function due to Andrews and Warnaar, as well as its binary representation.