Abstract
AbstractThese 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.KeywordsSturm-Liouville problemsClassical polynomialsDarboux transformationsExceptional polynomialsPainlevé equationsRational solutionsDarboux dressing chainsMaya diagramsWronskian determinantsMathematics Subject Classification (2000)Primary 33C45; Secondary 34M55
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