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

Abstract

We introduce a family of weight matrices $W$ of the form $T(t)T^{\ast }(t)$, $T(t)=e^{\mathcal{A}t}{e}^{\mathcal{D}{t}^2}$, where $\mathcal{A}$ is certain nilpotent matrix and $\mathcal{D}$ is a diagonal matrix with negative real entries. 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 matrices satisfy a second order differential equation with differential coefficients that are matrix polynomials $F_2$, $F_1$ and $F_0$ (independent of $n$) of degrees not bigger than 2, 1 and 0 respectively. For size $2\times 2$, we find an explicit expression for a sequence of orthonormal polynomials with respect to $W$. In particular, we show that one of the recurrence coefficients for this sequence of orthonormal polynomials does not asymptotically behave as a scalar multiple of the identity, as it happens in the examples studied up to now in the literature.

17 pages