Discrete Painlevé equations

Summary

Discrete Painlevé equations are integrable nonlinear second-order difference equations that play the role of the Painlevé equations for recurrences. They are non-autonomous, and in a suitable continuum limit they tend to one of the six Painlevé differential equations. Their integrability shows up as singularity confinement, as a Lax pair of linear problems, and as a description by birational maps of rational surfaces. They arose in the theory of orthogonal polynomials and random matrix models, where recurrence coefficients satisfy them.