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Exceptional Orthogonal Polynomials and Rational Solutions to Painlevé Equations

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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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Notes

  1. 1.

    Solving this question is equivalent to classifying exceptional polynomials and operators, a question that we shall mention below.

  2. 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. 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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Acknowledgements

The research of DGU has been supported in part by Spanish MINECO-FEDER Grants MTM2015-65888-C4-3 and MTM2015-72907-EXP, and by the ICMAT-Severo Ochoa project SEV-2015-0554. The research of RM was supported in part by NSERC grant RGPIN-228057-2009. DGU would like to thank the Volkswagen Stiftung and the African Institute of Mathematical Sciences for their hospitality during the Workshop on Introduction to Orthogonal Polynomials and Applications, Duala (Cameroon), where these lectures were first taught.

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Authors and Affiliations

  1. Escuela Superior de Ingeniería, Universidad de Cádiz, Puerto Real, Spain

    David Gómez-Ullate

  2. Departamento de Física Teórica, Universidad Complutense de Madrid, Madrid, Spain

    David Gómez-Ullate

  3. Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, Canada

    Robert Milson

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  1. David Gómez-Ullate
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  2. Robert Milson
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Correspondence to David Gómez-Ullate .

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Editors and Affiliations

  1. University of Yaoundé I, Yaoundé, Cameroon; African Institute for Mathematical Sciences, Limbe, Cameroon

    Mama Foupouagnigni

  2. Institute for Mathematics, University of Kassel, Kassel, Germany

    Wolfram Koepf

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Gómez-Ullate, D., Milson, R. (2020). Exceptional Orthogonal Polynomials and Rational Solutions to Painlevé Equations. In: Foupouagnigni, M., Koepf, W. (eds) Orthogonal Polynomials. AIMSVSW 2018. Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-36744-2_15

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