Recurrence Relations for the Generalized Laguerre and Charlier Orthogonal Polynomials and Discrete Painlevé Equations on the \(D_{6}^{(1)}\) Sakai Surface

IF 0.9 3区 数学 Q3 MATHEMATICS, APPLIED Mathematical Physics, Analysis and Geometry Pub Date : 2025-03-15 DOI:10.1007/s11040-025-09502-6
Xing Li, Anton Dzhamay, Galina Filipuk, Da-jun Zhang
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Abstract

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.

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来源期刊
Mathematical Physics, Analysis and Geometry
Mathematical Physics, Analysis and Geometry 数学-物理:数学物理
CiteScore
2.10
自引率
0.00%
发文量
26
审稿时长
>12 weeks
期刊介绍: MPAG is a peer-reviewed journal organized in sections. Each section is editorially independent and provides a high forum for research articles in the respective areas. The entire editorial board commits itself to combine the requirements of an accurate and fast refereeing process. The section on Probability and Statistical Physics focuses on probabilistic models and spatial stochastic processes arising in statistical physics. Examples include: interacting particle systems, non-equilibrium statistical mechanics, integrable probability, random graphs and percolation, critical phenomena and conformal theories. Applications of probability theory and statistical physics to other areas of mathematics, such as analysis (stochastic pde''s), random geometry, combinatorial aspects are also addressed. The section on Quantum Theory publishes research papers on developments in geometry, probability and analysis that are relevant to quantum theory. Topics that are covered in this section include: classical and algebraic quantum field theories, deformation and geometric quantisation, index theory, Lie algebras and Hopf algebras, non-commutative geometry, spectral theory for quantum systems, disordered quantum systems (Anderson localization, quantum diffusion), many-body quantum physics with applications to condensed matter theory, partial differential equations emerging from quantum theory, quantum lattice systems, topological phases of matter, equilibrium and non-equilibrium quantum statistical mechanics, multiscale analysis, rigorous renormalisation group.
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Recurrence Relations for the Generalized Laguerre and Charlier Orthogonal Polynomials and Discrete Painlevé Equations on the \(D_{6}^{(1)}\) Sakai Surface Propagation of Chaos and Residual Dependence in Gibbs Measures on Finite Sets Matrix Solutions of the Cubic Szegő Equation on the Real Line A Novel Discrete Integrable System Related to Hyper-Elliptic Curves of Genus Two Umbilicity and the First Stability Eigenvalue of a Subclass of CMC Hypersurfaces Immersed in Certain Einstein Manifolds
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