FoCM 2014 conference

Workshop C5 - Special Functions and Orthogonal Polynomials

December 19, 14:30 ~ 15:20 - Room A21

## Semi-classical orthogonal polynomials and the Painlev\'e equations

### Peter Clarkson

### University of Kent, UK - P.A.Clarkson@kent.ac.za

In this talk I shall discuss the relationship between the Painlev\'e equations and orthogonal polynomials with respect to semi-classical weights. 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 Wronskians that arise in the description of special function solutions of a Painlev\'e equation. The orthogonal polynomials discussed will include semi-classical Laguerre and Hermite weights, orthogonal polynomials with discontinuous weights and semi-classical generalizations of the Charlier and Meixner polynomials, which are discrete orthogonal polynomials.

Joint work with Kerstin Jordaan (University of Pretoria, South Africa) and J G Smith (University of Kent, United Kingdom).