# Discrete Painleve equations and orthogonal polynomials

#### by Professor Walter Van Assche

Institution: Katholieke Universiteit Leuven, Belgium
Date: Wed 5th December 2007
Time: 11:00 AM
Location: Old Geology 1, The University of Melbourne

Abstract: We give four examples of families of orthogonal polynomials for which the coefficients in the recurrence relation satisfy a
discrete Painleve equation.

The first example deals with Freud weights $|x|^\rho \exp(-|x|^4)$ on the real line, for which the recurrence coefficients satisfy the
discrete Painleve I equation.

The second example deals with orthogonal polynomials on the unit circle
for the weight $\exp(\lambda \cos \theta)$. These orthogonal polynomials
are important in the theory of random unitary matrices. Periwal and
Shevitz have shown that the recurrence coefficients satisfy the discrete
Painleve II equation.

The third example deals with discrete orthogonal polynomials on the
positive integers. We show that the recurrence coefficients of
generalized Charlier polynomials can be obtained from a solution of the
discrete Painleve II equation.

The fourth example deals with orthogonal polynomials on $\{\pm q^n: n \in \mathbb{N}\}$, where $0 < q < 1$. We consider the discrete
q-Hermite I polynomials and some discrete q-Freud polynomials for which
the recurrence ceofficients satisfy a q-deformation of discrete
Painleve I.