Lectures on exceptional orthogonal polynomials and rational solutions to Painlev\'e equations

TL;DR

This paper reviews exceptional orthogonal polynomials, their construction, and introduces new methods for generating rational solutions to Painlevé IV and its hierarchy using dressing chains and rational extensions of the harmonic oscillator.

Contribution

It provides a comprehensive summary of exceptional polynomials and introduces novel results on rational solutions to Painlevé equations via dressing chains and rational extensions.

Findings

New results on genus, interlacing, and cyclic Maya diagrams.

Construction of rational solutions to Painlevé IV and higher order hierarchies.

Integration of classical and novel techniques in orthogonal polynomials and Painlevé equations.

Abstract

These are the lecture notes for a course on exceptional polynomials taught at the \textit{AIMS-Volkswagen Stiftung Workshop on Introduction to Orthogonal Polynomials and Applications} that took place in Douala (Cameroon) from October 5-12, 2018. They summarize the basic results and construction of exceptional poynomials, developed over the past ten years. In addition, some new results are presented on the construction of rational solutions to Painlev\'e equation PIV and its higher order generalizations that belong to the $A_{2n}^{(1)}$-Painlev\'e hierarchy. The construction is based on dressing chains of Schr\"odinger operators with potentials that are rational extensions of the harmonic oscillator. Some of the material presented here (Sturm-Liouville operators, classical orthogonal polynomials, Darboux-Crum transformations, etc.) are classical and can be found in many textbooks, while some results (genus, interlacing and cyclic Maya diagrams) are new and presented for the first time in this set of lecture notes.