Differential Systems for Biorthogonal Polynomials Appearing in 2-Matrix Models and the Associated Riemann–Hilbert Problem
Differential Systems for Biorthogonal Polynomials Appearing in 2-Matrix Models and the Associated Riemann–Hilbert Problem
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2 矩阵模型中出现的双正交多项式的微分系统及相关黎曼-希尔伯特问题
DOI：
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
J. Harnad
中科院分区：
文献类型：
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作者：
M. Bertola;B. Eynard;J. Harnad
本文研究了双正交多项式，它是在研究具有两个任意阶多项式势V1（x），V2（y）的任意复系数的双矩阵Hermitian模型的推广中出现的。长度等于势V1和V2的次数的双正交多项式（“窗口”）的有限连续连续性满足具有多项式系数的ODE系统以及关于势的系数的PDE系统（变形方程）和连接连续窗口的递归关系。构造了这些方程的基本解系的相容序列。通过鞍点积分导出了这些基本系统的（Stokes）扇形渐近性，并推导出表征微分方程的Riemann-Hilbert问题。
AbstractWe consider biorthogonal polynomials that arise in the study of a generalization of two–matrix Hermitian models with two polynomial potentials V1(x), V2(y) of any degree, with arbitrary complex coefficients. Finite consecutive subsequences of biorthogonal polynomials (‘‘windows’’), of lengths equal to the degrees of the potentials V1 and V2, satisfy systems of ODE’s with polynomial coefficients as well as PDE’s (deformation equations) with respect to the coefficients of the potentials and recursion relations connecting consecutive windows. A compatible sequence of fundamental systems of solutions is constructed for these equations. The (Stokes) sectorial asymptotics of these fundamental systems are derived through saddle-point integration and the Riemann-Hilbert problem characterizing the differential equations is deduced.