### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 088, 19 pages      arXiv:1804.02856      https://doi.org/10.3842/SIGMA.2018.088
Contribution to the Special Issue on Painlevé Equations and Applications in Memory of Andrei Kapaev

### Discrete Orthogonal Polynomials with Hypergeometric Weights and Painlevé VI

Galina Filipuk a and Walter Van Assche b
a) Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, Warsaw, 02-097, Poland
b) Department of Mathematics, KU Leuven, Celestijnenlaan 200B box 2400, BE-3001 Leuven, Belgium

Received April 10, 2018, in final form August 20, 2018; Published online August 24, 2018

Abstract
We investigate the recurrence coefficients of discrete orthogonal polynomials on the non-negative integers with hypergeometric weights and show that they satisfy a system of non-linear difference equations and a non-linear second order differential equation in one of the parameters of the weights. The non-linear difference equations form a pair of discrete Painlevé equations and the differential equation is the $\sigma$-form of the sixth Painlevé equation. We briefly investigate the asymptotic behavior of the recurrence coefficients as $n\to \infty$ using the discrete Painlevé equations.

Key words: discrete orthogonal polynomials; hypergeometric weights; discrete Painlevé equations; Painlevé VI.

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