标题： 可积微分系统用于变形的拉盖尔-哈恩正交多项式 显示英文标题

标题： Integrable Differential Systems for Deformed Laguerre-Hahn Orthogonal Polynomials

摘要： 我们的工作研究了拉盖尔-哈恩类的正交多项式序列$ \{P_{n}(x)\}_{n=0}^{\infty} $，其斯特尔蒂斯函数满足具有多项式系数的类型微分方程，受变形参数$t$的影响。 我们推导了微分方程组并给出了Lax对，从而得到了正交多项式的递推关系系数和Lax矩阵在$t$中的非线性微分方程。 对通过与修改后的雅可比权相关的斯特尔蒂斯函数的莫比乌斯变换得到的非半经典情况进行了详细研究，表明该系统由Painlevé类型的微分方程P$_\textrm{VI}$控制。 这里出现的P$_\textrm{VI}$的特殊情况与Magnus [A.P. Magnus, Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials, J. Comput. Appl. Math., 57:215-237, 1995] 找到的解具有相同的四个参数，但在边界条件上有所不同。 显示英文摘要

摘要： Our work studies sequences of orthogonal polynomials $ \{P_{n}(x)\}_{n=0}^{\infty} $ of the Laguerre-Hahn class, whose Stieltjes functions satisfy a Riccati type differential equation with polynomial coefficients, are subject to a deformation parameter $t$. We derive systems of differential equations and give Lax pairs, yielding non-linear differential equations in $t$ for the recurrence relation coefficients and Lax matrices of the orthogonal polynomials. A specialisation to a non semi-classical case obtained via a M\"{o}bius transformation of a Stieltjes function related to a modified Jacobi weight is studied in detail, showing this system is governed by a differential equation of the Painlev\'e type P$_\textrm{VI}$. The particular case of P$_\textrm{VI}$ arising here has the same four parameters as the solution found by Magnus [A.P. Magnus, Painlev\'e-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials, J. Comput. Appl. Math., 57:215-237, 1995] but differs in the boundary conditions.