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We introduce a family of weight matrices $W$ of the form $T(t)T^*(t)$,$T(t)=e^{\mathscr{A}t}e^{\mathscr{D}t^2}$, where $\mathscr{A}$ is certainnilpotent matrix and $\mathscr{D}$ is a diagonal matrix with negative realentries. The weight matrices $W$ have arbitrary size $N\times N$ and depend on$N$ parameters.  The orthogonal polynomials with respect to this family of weight matricessatisfy a second order differential equation with differential coefficientsthat are matrix polynomials $F_2$, $F_1$ and $F_0$ (independent of $n$) ofdegrees not bigger than 2, 1 and 0 respectively.  For size $2\times 2$, we find an explicit expression for a sequence oforthonormal polynomials with respect to $W$. In particular, we show that one ofthe recurrence coefficients for this sequence of orthonormal polynomials doesnot asymptotically behave as a scalar multiple of the identity, as it happensin the examples studied up to now in the literature.

Orthogonal matrix polynomials satisfying differential equations with recurrence coefficients having non-scalar limits