Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 016, 19 pages      arXiv:2008.02822      https://doi.org/10.3842/SIGMA.2021.016

Exceptional Legendre Polynomials and Confluent Darboux Transformations

María Ángeles García-Ferrero a, David Gómez-Ullate bc and Robert Milson d
a) Institut für Angewandte Mathematik, Ruprecht-Karls-Universität Heidelberg, Im Neunheimer Feld 205, 69120 Heidelberg, Germany
b) Departamento de Ingeniería Informática, Escuela Superior de Ingeniería, Universidad de Cádiz, 11519 Puerto Real, Spain
c) Departamento de Física Teórica, Universidad Complutense de Madrid, 28040 Madrid, Spain
d) Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, B3H 3J5, Canada

Received September 22, 2020, in final form February 03, 2021; Published online February 20, 2021

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.

Key words: exceptional orthogonal polynomials; Darboux transformations; isospectral deformations.

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