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Discrete Painlev equations and orthogonal polynomials

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It is now well known that the recurrence coefficients of many semi-classical orthogonal polynomials satisfy discrete and continuous Painlev equations. In the talk we show how to find the discrete Painlev equations by using compatibility relations or ladder operators. A combination with Toda type equations allows to find continuous Painlev equations. We have made an attempt to go through the literature and to collect all known examples of discrete (and continuous) Painlev 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) Painlev equation in terms of special functions.

This talk is part of the Isaac Newton Institute Seminar Series series.

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