Lectures on exceptional orthogonal polynomials and rational solutions to 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 Painlevé equation PIV and its higher order generalizations that belong to the $A_{2n}^{(1)}$-Painlevé hierarchy. The construction is based on dressing chains of Schrödinger 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.

49 pages, 7 figures. arXiv admin note: substantial text overlap with arXiv:1811.09274

References in corpus (4)

• Infinitely many shape invariant potentials and new orthogonal polynomials
• Solvable rational extensions of the isotonic oscillator
• Poles of Painlevé IV Rationals and their Distribution
• Complex SUSY Transformations and the Painlevé IV Equation

Cited by in corpus (1)

• Hidden symmetries and nonlinear (super)algebras