ABSTRACT

In this paper we present a connection between systems of differential equations for the recurrence coefficients of polynomials orthogonal with respect to the generalized Meixner and the deformed Laguerre weights. It is well-known that the recurrence coefficients of both generalized Meixner and deformed Laguerre orthogonal polynomials can be expressed in terms of solutions of the fifth Painlevé equation but no explicit relation between systems of differential equations for the recurrence coefficients was known. We also present certain limits in which the recurrence coefficients can be expressed in terms of solutions of the Painlevé XXXIV equation, which in the deformed Laguerre case extends previous studies and in the generalized Meixner case is a new result.

摘要

在本文中，我们介绍了多项式的递推系数的微分方程组之间的联系，这些多项式与广义 Meixner 和变形的 Laguerre 权重正交。众所周知，广义Meixner 和变形Laguerre 正交多项式的递推系数都可以用第五Painlevé 方程的解表示，但递推系数的微分方程组之间没有明确的关系。我们还提出了某些限制，其中可以根据 Painlevé XXXIV 方程的解来表达递推系数，这在变形的 Laguerre 案例中扩展了先前的研究，而在广义的 Meixner 案例中是一个新结果。