论文标题
$ d_ {6}^{（1）} $ sakai Surface上的广义拉瓜和查理尔和charlier正交多项式的重复关系
Recurrence relations for the generalized Laguerre and Charlier orthogonal polynomials and discrete Painlevé equations on the $D_{6}^{(1)}$ Sakai surface
论文作者
论文摘要
本文涉及Painlevé识别问题的离散版本，即如何识别某种复发关系为离散的Painlevé方程。通常可以从问题的设置中看到一些线索，例如，当复发与某些差分painlevé方程相连，或者从方程式不确定点的配置的几何形状连接时。我们论文的主要信息是，实际上，这仅允许我们识别动态系统的配置空间，而不是动态本身。识别问题的精致版本在于确定动力学的翻译方向，这反过来又需要Painlevé方程的几何理论的全部力量。 为了说明这一点，在本文中，我们考虑了正交多项式理论中出现的这类复发的两个例子。我们之所以选择这些示例，是因为它们在同一sakai表面上进行了正规化，但同时却不等效，因为它们导致了非等效的翻译方向。此外，我们还使用Sakai的几何方法来回答此类问题的最近提议的识别程序，用于离散的Painlevé方程的有效性。特别是，这种方法不需要对可能类型的方程式的先验知识。
This paper concerns the discrete version of the Painlevé identification problem, i.e., how to recognize a certain recurrence relation as a discrete Painlevé equation. Often some clues can be seen from the setting of the problem, e.g., when the recurrence is connected with some differential Painlevé equation, or from the geometry of the configuration of indeterminate points of the equation. The main message of our paper is that, in fact, this only allows us to identify the configuration space of the dynamic system, but not the dynamics themselves. The refined version of the identification problem lies in determining, up to the conjugation, the translation direction of the dynamics, which in turn requires the full power of the geometric theory of Painlevé equations. To illustrate this point, in this paper we consider two examples of such recurrences that appear in the theory of orthogonal polynomials. We choose these examples because they get regularized on the same family of Sakai surfaces, but at the same time are not equivalent, since they result in non-equivalent translation directions. In addition, we show the effectiveness of a recently proposed identification procedure for discrete Painlevé equations using Sakai's geometric approach for answering such questions. In particular, this approach requires no a priori knowledge of a possible type of the equation.
