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Colloquium on Semi-classical Orthogonal Polynomials and Integrable Systems

On Monday 23rd May a short MIGSAA Colloquium will take place in the Newhaven Lecture Theatre, International Centre for Mathematical Sciences, 15 South College Street, Edinburgh, from 16.10 pm till 17.00 pm.  Professor Peter Clarkson (University of Kent) will preside.

Abstract: 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 Painleve ́ equations and discrete Painleve ́ 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 a Painleve ́ equation. 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 semi-classical Hermite weight, a generalization of the Freud weight and an Airy weight.

The event will begin with light refreshments from 15.30 pm outside the Newhaven Lecture Theatre.

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