Riemann-Hilbert方法及若干相关问题的研究

项目名称： Riemann-Hilbert方法及若干相关问题的研究

项目编号： No.11201070

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 周建荣

作者单位： 佛山科学技术学院

项目金额： 22万元

中文摘要： Riemann-Hilbert方法对研究当前许多重要的渐近性问题非常有效。该方法在正交多项式、特殊函数、随机矩阵论等领域内的深化研究已日渐成为目前应用分析的一个重要前沿课题。 本项研究拟考虑以下问题：(i) 融合Riemann-Hilbert方法与M. Berry - C. Howls的想法，探讨权函数的支集由无穷个离散点与一连续区间所构成的非Szego类广义Pollaczek正交多项式在全复平面内的一致渐近。(ii) 从超几何函数的微分方程入手，先借助于位势理论方法求出其零点的渐近分布曲线，再结合Riemann-Hilbert方法，探讨超几何函数在全复平面内的一致渐近及其渐近零点分布，着重考虑Gauss超几何函数的渐近性质及其零点的精确分布。 本课题的提出和解决，不仅有助于Riemann-Hilbert方法自身的发展，而且有助于深化认识超几何函数的渐近性质，尤其是零点的渐近分布问题。

中文关键词： Riemann-Hilbert方法；零点的渐近分布；一致渐近；正交多项式；超几何函数

英文摘要： Riemann-Hilbert method is an effective tool in the study of some important current asymptotic problems. Deepening studies of Riemann-Hilbert method in orthogonal polynomials, special functions, random matrix theory and other areas have increasingly become an important current topic in applied analysis. This study intends to consider the following questions：(i)Combining the Riemann-Hilbert method and thoughts of M. Berry - C. Howls, we investigate the uniform asymptotic of the non-Szego class generalized Pollaczek orthogonal polynomials, the support of whose weight function is composed of infinite discrete points and a continuous interval.(ii) Starting from differential equation of hypergeometric functions, we first lead to the asymptotic curve of zeros of hypergeometric functions by using potential theoretical methods, then obtain uniform asymptotic and asymptotic zero distribution of hypergeometric functions by combing Riemann-Hilbert method. We will focus on the asymptotic properties and precise distribution of zeros of Gauss hypergeometric functions. The present research is very useful not only in the self-development of Riemann-Hilbert method but also in deepening the cognition of asymptotic properties of hypergeometric functions,especially the asymptotic distribution of the zeros.

英文关键词： Riemann-Hilbert method；distribution of zeros；uniform asymptotic；orthogonal polynomials；hypergeometric functions