Some examples of orthogonal matrix polynomials satisfying odd order differential equations

It is well known that if a finite order linear differential operator with polynomial coefficients has as eigenfunctions a sequence of orthogonal polynomials with respect to a positive measure (with support in the real line), then its order has to be even. This property no longer holds in the case of orthogonal matrix polynomials. The aim of this paper is to present examples of weight matrices such that the corresponding sequences of matrix orthogonal polynomials are eigenfunctions of certain linear differential operators of odd order. The weight matrices are of the form $$W(t)=t^{\alpha}e^{-t}e^{At}t^{B}t^{B^*}e^{A^* t},$$ where $A$ and $B$ are certain (nilpotent and diagonal, respectively) $N\times N$ matrices. These weight matrices are the first examples illustrating this new phenomenon which are not reducible to scalar weights.