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Published in:
An Introduction to Orthogonal Polynomials Read chapter Read first chapter

2020 | OriginalPaper | Chapter

Exceptional Orthogonal Polynomials and Rational Solutions to Painlevé Equations

Authors : David Gómez-Ullate, Robert Milson

Published in: Orthogonal Polynomials

Publisher: Springer International Publishing

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Abstract

These are the lecture notes for a course on exceptional polynomials taught at the 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 10 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.

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previous chapter Two Variable Orthogonal Polynomials and Fejér-Riesz Factorization
next chapter -Rogers–Szegö and Hermite Polynomials, and Induced Deformed Quantum Algebras
Footnotes
1
Solving this question is equivalent to classifying exceptional polynomials and operators, a question that we shall mention below.
 
2
In the case of an unbounded interval with a = −∞ and/or b = +∞, or if solutions y(z) of (3.1) have no defined value at the endpoints, one has to consider the asymptotics of the corresponding solutions and impose boundary conditions of a more general form:
$$\displaystyle \begin{aligned} \begin{aligned} \alpha_0(z) y(z) + \alpha_1(z) y'(z) \to 0& \quad \text{as } z\to a^{-}\\ \beta_0(z) y(z) + \beta_1(z) y'(z) \to 0& \quad \text{as } z\to b^{+} \end{aligned} \end{aligned}$$
where α 0(z), α 1(z), β 0(z), β 1(z) are continuous functions defined on I.
 
3
A multi-set is generalization of the concept of a set that allows for multiple instances for each of its elements.
 
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Metadata
Title
Exceptional Orthogonal Polynomials and Rational Solutions to Painlevé Equations
Authors
David Gómez-Ullate
Robert Milson
Copyright Year
2020
Book
Orthogonal Polynomials

Print ISBN: 978-3-030-36743-5

Electronic ISBN: 978-3-030-36744-2

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-3-030-36744-2_15

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