Chen Yang, Its Alexander R
Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI 53706, USA.
Philos Trans A Math Phys Eng Sci. 2008 Mar 28;366(1867):973-1003. doi: 10.1098/rsta.2007.2058.
In this paper, we study those polynomials, orthogonal with respect to a particular weight, over the union of disjoint intervals, first introduced by N. I. Akhiezer, via a reformulation as a matrix factorization or Riemann-Hilbert problem. This approach complements the method proposed in a previous paper, which involves the construction of a certain meromorphic function on a hyperelliptic Riemann surface. The method described here is based on the general Riemann-Hilbert scheme of the theory of integrable systems and will enable us to derive, in a very straightforward way, the relevant system of Fuchsian differential equations for the polynomials and the associated system of the Schlesinger deformation equations for certain quantities involving the corresponding recurrence coefficients. Both of these equations were obtained earlier by A. Magnus. In our approach, however, we are able to go beyond Magnus' results by actually solving the equations in terms of the Riemanni Theta-functions. We also show that the related Hankel determinant can be interpreted as the relevant tau-function.
在本文中，我们研究那些关于特定权重在不相交区间的并集上正交的多项式，这些多项式最初由N. I. 阿基耶泽尔引入，我们通过将其重新表述为矩阵分解或黎曼 - 希尔伯特问题来进行研究。这种方法补充了前一篇论文中提出的方法，前一篇论文中的方法涉及在超椭圆黎曼曲面上构造某个亚纯函数。这里描述的方法基于可积系统理论的一般黎曼 - 希尔伯特框架，并且将使我们能够以一种非常直接的方式推导出多项式的相关富克斯微分方程组以及涉及相应递推系数的某些量的施莱辛格变形方程组。这两个方程早些时候由A. 马格努斯得到。然而，在我们的方法中，我们能够超越马格努斯的结果，通过用黎曼θ函数实际求解这些方程。我们还表明相关的汉克尔行列式可以解释为相关的tau函数。
