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Session III.7 - Special Functions and Orthogonal Polynomials - Semi-plenary talk

Monday, June 19, 16:30 ~ 17:30

Orthogonal Polynomials and Symmetric Freud weights

Peter Clarkson

University of Kent, UK   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk I will discuss orthogonal polynomials associated with symmetric Freud weights, in particular the sextic weight \[ \omega(x;t,\tau,\rho)=|x|^\rho\exp(-x^6+\tau x^4+tx^2),\eqno(1)\] with $\tau$, $t$ and $\rho \gt -1$ parameters. I will describe properties of the recurrence coefficients in the three-term recurrence relation associated with these orthogonal polynomials. For the sextic weight (1) the recurrence coefficients satisfy a fourth-order discrete equation which is the second member of the first discrete Painlevé hierarchy, also known as the string equation, and also satisfiy a coupled system of second-order, nonlinear differential equations. When $\rho=0$, the weight (1) arises in the context of Hermitian matrix models and random symmetric matrix ensembles.

Joint work with Kerstin Jordaan (University of South Africa, South Africa) and Ana Loureiro (University of Kent, UK).

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