On linear systems and <em>τ</em> functions associated with Lamé's equation and Painlevé's equation VI

Painlevé's transcendental differential equation may be expressed as the consistency condition for a pair of linear differential equations with matrix coefficients with rational entries. By a construction due to Tracy and Widom, this linear system is associated with certain kernels which give trace class operators on Hilbert space. This paper expresses such operators in terms of Hankel operators of linear systems which are realised in terms of the Laurent coefficients of the solutions of the differential equations. Let be the orthogonal projection; then the Fredholm determinant defines the function, which is here expressed in terms of the solution of a matrix Gelfand–Levitan equation. For suitable values of the parameters, solutions of the hypergeometric equation give a linear system with similar properties. For meromorphic transfer functions that have poles on an arithmetic progression, the corresponding Hankel operator has a simple form with respect to an exponential basis in ; so can be expressed as a series of finite determinants. This applies to elliptic functions of the second kind, such as satisfy Lamé's equation with .

潘勒韦超越微分方程可表述为一对具有有理项矩阵系数的线性微分方程的相容性条件。通过特雷西和威多姆提出的一种构造，该线性系统与某些核相关联，这些核在希尔伯特空间上给出迹类算子。本文根据线性系统的汉克尔算子来表示此类算子，这些线性系统是通过微分方程解的洛朗系数实现的。设\(P\)为正交投影；那么弗雷德霍姆行列式\(\det(I - zP KP)\)定义了\(\tau\)函数，在此它根据一个矩阵盖尔范德 - 列维坦方程的解来表示。对于参数的适当取值，超几何方程的解给出一个具有类似性质的线性系统。对于在算术级数上有极点的亚纯传递函数\(K\)，相应的汉克尔算子相对于\(L^{2}(0, \infty)\)中的一个指数基具有简单形式；因此\(\det(I - zP KP)\)可表示为一系列有限行列式。这适用于第二类椭圆函数，例如满足\(n = 1\)时的拉梅方程的函数。