Recurrence relations for the generalized Laguerre and Charlier orthogonal polynomials and discrete Painlevé equations on the $D_{6}^{(1)}$ Sakai surface
Xing Li, Anton Dzhamay, Galina Filipuk, Da-jun Zhang
备注：
30 pages
发表信息：
Mathematical Physics, Analysis and Geometry 28, 5 (2025)
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.
Generalized Charlier Recurrence Coefficients: Hankel Formulas, Laguerre--Freud Dynamics, and Painlevé V Connections
International Chair of Mathematical Physics and Applications (ICMPA–UNESCO Chair)（国际数学物理与应用 chair（ICMPA-UNESCO 讲席））；University of Abomey-Calavi（阿波美卡拉维大学）
On the discrete Painlevé equivalence problem, non-conjugate translations and nodal curves
Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京数学与应用研究所）；Institute of Mathematics, University of Warsaw（华沙大学数学研究所）；Waseda Institute for Advanced Study, Waseda University（早稻田大学高等研究院）
