随机矩阵在物理学中有着广泛的应用。Hankel行列式的渐近展开与随机矩阵中的许多问题相关，并且受到很多研究者的关注。Riemann-Hilbert方法是研究Hankel行列式渐近展开的有效途径。近年来，Riemann-Hilbert方法得到国内外学者的大量发展。申请人已经通过Riemann-Hilbert方法，研究了几类正交多项式的一致渐近展开以及一类Hankel行列式的渐近展开。本项目以具有跳跃奇点的非Szegö类权函数以及具有多个奇点重合的权函数为背景，通过Riemann-Hilbert方法研究Hankel行列式的渐近展开及差分方程系数的渐近展开。本项目的研究对Riemann-Hilbert方法的深入发展有着重要的理论意义。

Random matrix theory has a great many applications in physics. Asymptotic expansions of the Hankel determinant, with a lot of researchers' attentions, are connected with plenty of problems in random matrix theory. Riemann-Hilbert approach is a powful method to obtain the asymptotic expansions of the Hankel determinant. Recently, Riemann- Hilbert approach has been made lots of developments.The proposer has considered the uniform asymptotic expansions of several classes of orthogonal polynomials, and asymptotic expansion of a class of Hankel determinant. In this project, we aim to investigate the asymptotic expansions of the Hankel determinant and the recurrence coefficients, related to the non-Szegö class weight with a jump and the weight with coalescing critical points. This research has important theoretical significance to the deep development of the Riemann-Hilbert approach.