距量积分与Carlitz型q-Hardy-Hille公式的研究

近些年随着量子数学中q-正交多项式理论的飞速发展，q-Laguerre多项式重要类型的生成函数问题成为当前数学研究的热点课题。但是，通过q-模拟的经典方法研究相应的q-Laguerre多项式的生成函数愈来愈困难，距量积分方法是解决该问题的有效手段。本项目以Carlitz型q-Hardy-Hille公式为研究对象，利用距量积分方法系统地研究该类型公式的单重、二重、三重及多重情况，并结合指数算子分解、q-指数算子等方法刻画相应成函数的结构特性，并且研究它们所对应的q-差分方程、距量积分恒等式的性质，进而获得方程解的特征、q-级数变换公式。本项目的研究方法和成果揭示了一类重要q-正交多项式生成函数的特性，提供了研究其它q-正交多项式生成函数的途径，丰富了q-正交多项式生成函数的理论。

With the rapid development of the q-orthogonal polynomial theory in quantum mathematics, problems from important types of generating functions for q-Laguerre polynomials become a focus of mathematical research. However, it become more and more difficult to study generating functions for q-Laguerre polynomials by classical q-analogues methods respectively, a method of moment integral is an effective way to solve this problem. This project aims at Carlitz-type q-Hardy-Hille formulas, whose cases of single, double, triple and multiple are studied by utilizing moment integral systematically, the structural features of these generating functions are revealed by using techniques of exponential operator decomposition and q-exponential operator etc, the properties of their corresponding q-difference equations and moment integral identities are discussed, further more, the characters of solutions for the equations and q-series transformations are obtained. The method and results in this project reveal the characters of a class of generating functions for q-orthogonal polynomials，its also provide an approach to investigate other q-orthogonal polynomials，and consequently enrich the theory of q-orthogonal polynomials.