q-差分方程形式解及其在组合数学中的若干应用

With the rapid development of nonlinear mathematics and quantum mathematics, problems of q-integrals, generating functions for q-orthogonal polynomials and transformational identities in combinatorics become topics of current interest in research of the crossed mathematics, however, complex integral operations and finite summation formulae are the vital factors restricting the research progress. It is an effective way to solve some problems of combinatorics through the perspective of q-difference equation, with the aid of the formal solution method, the research will improve the computing efficiency of integrals and summations. The project is focused on the analytic function, constructing proper q-difference equation, and studying the following contents by using the method of formal solution of q-difference equations: 1. to generalize Andrews-Askey, Askey-Wilson type integrals and related identities of q-integrals; 2. to solve the open problems of higher dimensions q-Kibble-Slepian type formulae in paper [Journal of Functional Analysis, 2012, 262(1): 210-233.], and to generalize some related generating functions for q-orthogonal polynomials; 3. to generalize Rogers-Ramanujan type identities, integral type of Bailey's summation formula, and to discuss some related unilateral and bilateral transformational identities.

随着非线性数学和量子数学的快速发展，组合数学中q-积分、q-正交多项式生成函数及变换恒等式等问题成为当前数学交叉领域研究的热点，然而复杂的积分运算与有限的求和公式是制约研究进展的重要因素，引入q-差分方程并借助其形式解方法是解决组合数学若干问题的有效途径，研究将提高积分、求和运算效率。本项目针对解析函数，构造其所满足的q-差分方程，利用q-差分方程形式解方法研究：1.推广多变量Andrews-Askey型、Askey-Wilson型积分及相关q-积分恒等式; 2. 推广高维q-Kibble-Slepian型公式，解决文章[Journal of Functional Analysis, 2012, 262(1): 210-233.]中的公开问题，推广相关多变量q-正交多项式生成函数; 3. 推广Rogers-Ramanujan型恒等式、积分型Bailey求和公式，探讨相关单、双边变换恒等式.

本项目采用q-差分方程形式解及齐次q-偏差分方程形式解方法研究了q-Laguerre多项式关联的Cigler多项式和拓广多项式的系列生成函数问题、Askey-Wilson型积分问题、多线性生成函数、Ramanujan积分、Askey-Roy积分、Andrews-Askey积分、Ismail-Zhang型生成函数问题、分数阶q-积分及分数阶q-积分恒等式问题。探讨了q-差分方程形式解的结构、分数阶q-积分性质及齐次q-偏差分方程构造属性。.通过这些方法研究发现：1. q-Laguerre多项式关联的Cigler多项式和拓广多项式具有算子递推结构特征。2. q-差分方程形式解与Askey-Wilson型积分之间属性关联，系统推广了Askey-Wilson型积分。3. 以q-超几何多项式乘积展成的函数满足齐次q-偏差分方程，推广了多线性生成函数、Ramanujan积分、Askey-Roy积分、Andrews-Askey积分，Ismail-Zhang型生成函数等系列问题。4. q-差分方程与迭代分数阶q-积分之间关系，推广了二重分数阶q-积分及分数阶q-积分恒等式。5. q-差分方程与Ramanujan积分之间的内在关系，拓展了Ramanujan积分。.上述结果证实了q-差分方程形式解方法是研究q-Laguerre多项式关联的Cigler多项式生成函数、Askey-Wilson型积分、q-超几何多项式展开、分数阶q-积分、Ramanujan积分、Askey-Roy积分、Andrews-Askey积分，Ismail-Zhang型生成函数等系列问题一种重要手段，通过q-差分方程形式解方法揭示了q-正交多项式生成函数理论、q-积分理论及分数阶q-积分理论，并为进一步探索q-差分方程理论及应用提供了一个新的研究渠道。