Recurrence coefficients for discrete orthogonal polynomials with hypergeometric weight and discrete Painlevé equations

Abstract

Over the last decade it has become clear that discrete Painlevé equations appear in a wide range of important mathematical and physical problems. Thus, the question of recognizing a given non-autonomous recurrence as a discrete Painlevé equation and determining its type according to Sakai's classification scheme, understanding whether it is equivalent to some known (model) example, and especially finding an explicit change of coordinates transforming it to such an example, becomes one of the central ones. Fortunately, Sakai's geometric theory provides an almost algorithmic procedure for answering this question. In this paper we illustrate this procedure by studying an example coming from the theory of discrete orthogonal polynomials. There are many connections between orthogonal polynomials and Painlevé equations, both differential and discrete. In particular, often the coefficients of three-term recurrence relations for discrete orthogonal polynomials can be expressed in terms of solutions of discrete Painlevé equations. In this work we study discrete orthogonal polynomials with general hypergeometric weight and show that their recurrence coefficients satisfy, after some change of variables, the standard discrete Painlevé-V equation. We also provide an explicit change of variables transforming this equation to the standard form.

24 pages

References in corpus (7)

• Affine Weyl groups, discrete dynamical systems and Painleve equations
• Geometric Aspects of Painlevé Equations
• Fiber-dependent deautonomization of integrable 2D mappings and discrete Painlevé equations
• Discrete Schlesinger Transformations, their Hamiltonian Formulation, and Difference Painlevé Equations
• On Some Applications of Sakai's Geometric Theory of Discrete Painlevé Equations
• Discrete Orthogonal Polynomials with Hypergeometric Weights and Painlevé VI
• Polynomial Sequences Associated with the Moments of Hypergeometric Weights

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• Special solutions of a discrete Painlevé equation for quantum minimal surfaces
• Recurrence relations for the generalized Laguerre and Charlier orthogonal polynomials and discrete Painlevé equations on the $D_6^{(1)}$ Sakai surface
• On the recurrence coefficients for the $q$-Laguerre weight and discrete Painlevé equations
• Orthogonal Polynomials for the Gaussian Weight with a Jump and Discrete Painlevé Equations
• Fluctuations of Young diagrams for symplectic groups and semiclassical orthogonal polynomials