论文标题
用于变形的laguerre-hahn正交多项式的可集成差分系统
Integrable Differential Systems for Deformed Laguerre-Hahn Orthogonal Polynomials
论文作者
论文摘要
我们的工作研究序列的正交多项式$ \ {p_ {n}（x）\} _ {n = 0}^{\ infty} $的laguerre-hahn类$ stieltjes的stieltjes的功能可满足riccati类型的差异方程，并具有多元元素的参数，$ ns obseff us $ $ ns obseft $ ns obsort $ ns obsorts $ ns $我们得出了微分方程的系统并给出松弛对，对于正交多项式的复发关系系数和LAX矩阵，在$ t $中产生非线性微分方程。详细研究了通过与改良的雅各比重量相关的stieltjes函数的Möbius变换获得的非半古典案例的专业化，并详细研究了该系统，显示该系统受Painlevé类型P $ _ \ textrm {VI} $的差分方程。 P $ _ \ textrm {Vi} $的特定情况与Magnus [A.P. Magnus，半古典正交多项式复发系数的Painlevé-Type微分方程，J。Comput。应用。数学，57：215-237，1995]，但在边界条件下有所不同。
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 Möbius 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 Painlevé 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, Painlevé-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.
