Using quasi-Darboux transformations to construct exceptional matrix polynomials

Abstract

We introduce a couple of methods to construct exceptional matrix polynomials. One of them uses what we have called quasi-Darboux transformations. This seems to be a more powerful method to deal with the non-commutativity problems that appear when matrix-valued polynomials are considered. The other method does not use any transformation of Darboux type. Using both methods, we construct a collection of five illustrative examples that show how powerful our two methods are. The examples include exceptional matrix polynomials of Hermite, Laguerre, and Gegenbauer type, as well as an example with a weight matrix having a Dirac delta.