Exceptional orthogonal polynomials and the Darboux transformation

David Gomez-Ullate, Niky Kamran, Robert Milson

Published 2010-02-13, updated 2010-03-04Version 2

We adapt the notion of the Darboux transformation to the context of polynomial Sturm-Liouville problems. As an application, we characterize the recently described $X_m$ Laguerre polynomials in terms of an isospectral Darboux transformation. We also show that the shape-invariance of these new polynomial families is a direct consequence of the permutability property of the Darboux-Crum transformation.