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

Dzhamay, A; Filipuk, G; Stokes, A; (2020) Recurrence coefficients for discrete orthogonal polynomials with hypergeometric weight and discrete Painlevé equations. Journal of Physics A: Mathematical and Theoretical 10.1088/1751-8121/abbd54. (In press). Green open access

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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.

Type: Article
Title: Recurrence coefficients for discrete orthogonal polynomials with hypergeometric weight and discrete Painlevé equations
Open access status: An open access version is available from UCL Discovery
DOI: 10.1088/1751-8121/abbd54
Publisher version: https://doi.org/10.1088/1751-8121/abbd54
Language: English
Additional information: As the Version of Record of this article is going to be/has been published on a subscription basis, this Accepted Manuscript will be available for reuse under a CC BY-NC-ND 3.0 licence after a 12 month embargo period.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10112239
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