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Exceptional Legendre polynomials and confluent Darboux transformations

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Garcia Ferrero, María Ángeles and Gómez-Ullate Oteiza, David and Milson, Robert (2021) Exceptional Legendre polynomials and confluent Darboux transformations. Symmetry integrability and geometry: methods and applications (SIGMA), 17 . ISSN 1815-0659

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Official URL: http://dx.doi.org/10.3842/SIGMA.2021.016




Abstract

Exceptional orthogonal polynomials are families of orthogonal polynomials that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing for polynomial sequences that miss a finite number of "exceptional" degrees. In this paper we introduce a new construction of multi-parameter exceptional Legendre polynomials by considering the isospectral deformation of the classical Legendre operator. Using confluent Darboux transformations and a technique from inverse scattering theory, we obtain a fully explicit description of the operators and polynomials in question. The main novelty of the paper is the novel construction that allows for exceptional polynomial families with an arbitrary number of real parameters.


Item Type:Article
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© 2021 Natl Acad Sci Ukraine, Inst Math.
MAGF would like to thank the Max-Planck-Institute for Mathematics in the Sciences, Leipzig (Germany), where some of her work took place. DGU acknowledges support from the Spanish MICINN under grants PGC2018-096504-B-C33 and RTI2018-100754-B-I00 and the European Union under the 2014-2020 ERDF Operational Programme and by the Department of Economy, Knowledge, Business and University of the Regional Government of Andalusia (project FEDER-UCA18-108393).

Uncontrolled Keywords:Potentials; Hermite.
Subjects:Sciences > Physics > Physics-Mathematical models
Sciences > Physics > Mathematical physics
ID Code:64748
Deposited On:12 Apr 2021 17:41
Last Modified:13 Apr 2021 06:47

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