María Ángeles Garcia-Ferrero (Universidad
Autónoma de Madrid)
Exceptional orthogonal polynomials and Darboux transformations
Abstract:
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.