Orthogonal polynomials, Toda lattices and Painlev\'e equations

TL;DR

This paper surveys the relationships between orthogonal polynomials, Toda lattices, and Painlevé equations, highlighting their interconnected roles in mathematical physics and integrable systems.

Contribution

It provides a comprehensive overview of how orthogonal polynomials relate to Toda lattices and Painlevé equations, emphasizing their interconnected structures.

Findings

Unified view of orthogonal polynomials and integrable systems

Connections between Toda lattices and Painlevé equations

Insights into discrete and continuous Painlevé equations

Abstract

We give a survey of the connection between orthogonal polynomials, Toda lattices and related lattices, and Painlev\'e equations (discrete and continuous).