Exceptional Orthogonal Polynomials and Rational Solutions to Painlevé Equations

  • David Gómez-UllateEmail author
  • Robert Milson
Conference paper
Part of the Tutorials, Schools, and Workshops in the Mathematical Sciences book series (TSWMS)


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.


Sturm-Liouville problems Classical polynomials Darboux transformations Exceptional polynomials Painlevé equations Rational solutions Darboux dressing chains Maya diagrams Wronskian determinants 

Mathematics Subject Classification (2000)

Primary 33C45; Secondary 34M55 



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. 1.Escuela Superior de IngenieríaUniversidad de CádizPuerto RealSpain
  2. 2.Departamento de Física TeóricaUniversidad Complutense de MadridMadridSpain
  3. 3.Department of Mathematics and StatisticsDalhousie UniversityHalifaxCanada

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