### Discrete Painlevé equations and orthogonal polynomials

van Assche, W *(Katholieke Universiteit Leuven)*

Friday 12 July 2013, 14:00-14:30

Seminar Room 1, Newton Institute

#### Abstract

It is now well known that the recurrence coefficients of many semi-classical orthogonal polynomials satisfy discrete and continuous Painlevé equations. In the talk we show how to find the discrete Painlevé equations by using compatibility relations or ladder operators. A combination with Toda type equations allows to find continuous Painlevé equations. We have made an attempt to go through the literature and to collect all known examples of discrete (and continuous) Painlevé equations for the recurrence coefficients of orthogonal polynomials. All input in completing the list is welcome. The required solutions usually satisfy some positivity constraint, leading to a unique solution with one boundary condition. Often the solution corresponds to solutions of the (continuous) Painlevé equation in terms of special functions.

#### Presentation

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