BIRS Workshop Lecture Videos
Orthogonal Polynomials and Integrable Systems Clarkson, Peter
In this talk I shall discuss the relationship between orthogonal polynomials with respect to semi-classical weights, which are generalisations of the classical weights and arise in applications such as random matrices, and integrable systems, in particular the Painlev\'e equations and discrete Painlev\'e equations. It is well-known that orthogonal polynomials satisfy a three-term recurrence relation. I will show that for some semi-classical weights the coefficients in the recurrence relation can be expressed in terms of Hankel determinants, which are Wronskians, that also arise in the description of special function solutions of Painleve equations. The determinants arise as partition functions in random matrix models and the recurrence coefficients satisfy a discrete Painleve equation. The semi-classical orthogonal polynomials discussed will include a generalization of the Freud weight and an Airy weight.
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