Two-step Darboux transformations and exceptional Laguerre polynomials

TL;DR

This paper introduces a method to generate new exceptional Laguerre polynomials using multi-step Darboux transformations, expanding the family of orthogonal polynomials with potential applications in mathematical physics.

Contribution

It presents a novel algebraic approach to construct exceptional orthogonal polynomials via multiple-step Darboux transformations, including explicit examples with Laguerre polynomials.

Findings

New orthogonal polynomial systems parameterized by two integers.

Recovery of classical Laguerre and $X_$-Laguerre polynomials for specific parameters.

Demonstration of the construction method through a 2-step Darboux transformation.

Abstract

It has been recently discovered that exceptional families of Sturm-Liouville orthogonal polynomials exist, that generalize in some sense the classical polynomials of Hermite, Laguerre and Jacobi. In this paper we show how new families of exceptional orthogonal polynomials can be constructed by means of multiple-step algebraic Darboux transformations. The construction is illustrated with an example of a 2-step Darboux transformation of the classical Laguerre polynomials, which gives rise to a new orthogonal polynomial system indexed by two integer parameters. For particular values of these parameters, the classical Laguerre and the type II $X_\ell$-Laguerre polynomials are recovered.