María Ángeles Garcia-Ferrero (Universidad Autónoma de Madrid)

Exceptional orthogonal polynomials and Darboux transformations


Exceptional orthogonal polynomials arise as eigenfunctions of Sturm-Liouville problems and form complete bases in spaces of square integrable functions with weights. Nevertheless, contrary to the classical polynomials of Hermite, Laguerre and Jacobi, their sets of degrees miss finitely many natural numbers. In this talk, we will see how we can construct (all) exceptional orthogonal polynomials from the classical polynomials by applying Darboux transformations. In the case of starting with Jacobi polynomials with integer parameters α, β, families of exceptional polynomials which depend on an arbitrary number of continuous parameters appear. This is based on joint work with David Gómez-Ullate and Robert Milson.