Discrete Painlev Equations and Orthogonal Polynomials

Abstract:

The discrete Painlevé equations (dP) are discrete analogues of the celebrated continuous Painlevé equations. They all satisfy the property of singularity confinement, which was first described by Grammaticos et al. in 1991, who also discovered the first dP's. The list of dP's is still expanding and contains definitely more equations than the six classical ones.

It is well known that orthogonal polynomials satisfy a three-term recurrence relation. We show that the recurrence coeficients satisfy, for well-chosen weights, a discrete Painlevé equation, using compatibility relations for the associated orthogonal polynomials.