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 one-parametric families of monic 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 to a weight matrix at a certain point. In this article it is described how to choose a point t(0), a discrete mass M(t(0)), and the weight matrix W, so that the new weight matrix W + delta(t0) M(t(0)) 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. For some of these examples, we characterize the convex cone of weight matrices for which the differential operator is symmetric.