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arXiv · 1804.02856

Discrete Orthogonal Polynomials with Hypergeometric Weights and Painlev\'e VI

Galina Filipuk·Walter Van Assche
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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 Painlev\'e equations and the differential equation is the $\sigma$-form of the sixth Painlev\'e equation. We briefly investigate the asymptotic behavior of the recurrence coefficients as $n\to \infty$ using the discrete Painlev\'e equations.

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Galina Filipuk, Walter Van Assche. 2018-04-09. Discrete Orthogonal Polynomials with Hypergeometric Weights and Painlev\'e VI. https://doi.org/10.3842/sigma.2018.088

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