本项目主要研究Painleve方程与正交多项式、随机矩阵以及Heun方程的相关问题,主要研究内容包括：Painleve方程有理解的Hankel行列式表示和相关生成函数、正交多项式、渐近分析及isomonodromy问题；双合流Heun方程的特殊函数展开解以及与Painleve IV方程的联系；正交多项式、随机矩阵系综与Painleve方程之间的联系；Heun方程的(semi)finite-gap问题。

The project focuses on the studies of related problems among Painleve equations, orthogonal polynomials, random matrix theory and Heun equations. The main contents include the following aspects. We will give the Hankel determinant representation of rational solutions to Painleve III and corresponding generating functions,orthogonal polynomials, asymptotical behavior and isomonodromy problem. We will present series solutions of biconfluent Heun equation with special functions as expansion basis and explain the relation between Heun and Painleve equations from the point of view of solutions. We will investigate links of orthogonal polynomials, random matrix ensembles and Painleve equations.. Finally, we will consider (semi)finite-gap problems of Heun equations.