Exceptional orthogonal polynomials and rational solutions to Painlevé equations. (English) Zbl 1443.33014
Foupouagnigni, Mama (ed.) et al., Orthogonal polynomials. Proceedings of the 2nd AIMS-Volkswagen Stiftung workshop on introduction to orthogonal polynomials and applications, Douala, Cameroon, October 5–12, 2018. Cham: Birkhäuser. Tutor. Sch. Workshops Math. Sci., 335-386 (2020).
Summary: 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 \(\mathrm{P}_{\mathrm{IV}}\) 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.
For the entire collection see [Zbl 1442.33005].
For the entire collection see [Zbl 1442.33005].
MSC:
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 34M55 | Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies |
Keywords:
Sturm-Liouville problems; classical polynomials; Darboux transformations; exceptional polynomials; Painlevé equations; rational solutions; Darboux dressing chains; Maya diagrams; Wronskian determinantsReferences:
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