# 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.

*For More Information:* Dr Owen Jones O.D.Jones@ms.unimelb.edu.au