Research on systems of the biorthogonal functions, discrete andultradiscrete integrable systems, and their applications

利用正交多项式理论和Hirota的tau函数理论，借助经典的正交函数和非自治离散可积系统，研究了双正交函数系。在这项研究中，我们成功地导出了与斜正交多项式有关的离散可积系统，并用椭圆超几何函数阐明了Pade插值法与椭圆型Painleve方程的解之间的关系。从与经典双正交有理函数相关的非自治离散可积系统出发，给出了三对角矩阵束广义特征值问题的一种稳定的数值算法。

By using the theory of orthogonal polynomials and the theory of Hirota’s tau-functions, we study the system of bi-orthogonal functions with the help of the classical orthogonal functions and the nonautonomous discrete integrable systems. In this study, we have succeeded in deriving the discrete integrable systems associated with skew-orthogonal polynomials and making clear the relationship between the Padeinterpolations and the solutions of the elliptic Painleve equations in terms of elliptic hypergeometric functions. A stable numerical algorithm for the generalized eigenvalue problem of a tri-diagonal matrix pencil is introduced from the nonautonomous discrete integrable systems related to the classical biorthogonal rational functions.