Discrete Painlev\'e equations for recurrence coefficients of orthogonal polynomials

Abstract

We give four examples of families of orthogonal polynomials for which the coefficients in the recurrence relation satisfy a discrete Painlev\'e equation. The first example deals with Freud weights $|x|^\rho \exp(-|x|^m)$ on the real line, and we repeat Freud's derivation and analysis for the cases $m=2,4,6$. The Freud equations for the recurrence coefficients when $m=4$ corresponds to the discrete Painlev\'e 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 Painlev\'e 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 Painlev\'e II equation. The fourth example deals with orthogonal polynomials on $\pm q^n$. 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 Painlev\'e I.