Second order differential operators having several families of orthogonal matrix polynomials as eigenfunctions

The aim of this paper is to bring into the picture a new phenomenon in the theory of orthogonal matrix polynomials satisfying second order differential equations. The last few years have witnessed some examples of a (fixed) family of orthogonal matrix polynomials whose elements are common eigenfunctions of several linearly independent second order differential operators. We show that the dual situation is also possible: there are examples of different families of matrix polynomials, each family orthogonal with respect to a different weight matrix, whose elements are eigenfunctions of a common second order differential operator. These examples are constructed by adding a discrete mass at certain point to a weight matrix: W̃ = W + δt0M(t0). Our method consists in showing how to choose the discrete mass M(t0), the point t0 where the mass lives and the weight matrix W so that the new weight matrix W̃ inherits some of the symmetric second order differential operators associated with W . It is well known that this situation is not possible for the classical scalar families of Hermite, Laguerre and Jacobi.