An Isaac Newton Institute Workshop

Painleve Equations and Monodromy Problems: Recent Developments

Discrete Painlevé equations for recurrence coefficients of orthogonal polynomials

18th September 2006

Author: Walter Van Assche (Katholieke Universiteit Leuven, Belgium)

Abstract

The recurrence coefficients of certain semi-classical orthogonal polynomials satisfy discrete Painlevé equations. The Freud equation for the recurrence coefficients of the orthogonal polynomials for the weight exp(-x^4+ t x^2) is in fact a special case of discrete Painlevé I, the Verblunsky coefficients of orthogonal polynomials on the unit circle with weight exp(K cos t) satisfy discrete Painlevé II, the recurrence coefficients of generalized Charlier polynomials can be written in terms of a solution of discrete Painlevé II, and a q-deformation of the Freud polynomials on the exponential lattice has recurrence coefficients that satisfy a q-discrete Painlevé I equation. Unfortunately, these non-linear recurrence relations are not suited for computing the recurrence coefficients starting from two initial conditions, since minor deviations from the correct initial values quickly leads to major deviations from the correct value. For the Freud equations for the weight exp(-x^4) Lew and Quarles showed that there is a unique solution of the discrete Painlevé I equation which starts at 0 and remains positive for all n. This positive solution is in fact a fixed point in a metric space of sequences, and it can be found by successive iterations of a contractive mapping. This procedure give a numerically stable way to compute the recurrence coefficients. We will show that a similar result is also true for the discrete Painlevé II equation and for the q-discrete Painlevé I equation. In both cases the fixed point solution is precisely the solution that gives the recurrence coefficients of the corresponding orthogonal polynomals.

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