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

TL;DR

This paper constructs a new family of matrix-valued orthogonal polynomials associated with specific weight matrices, demonstrating their differential equations and revealing non-scalar asymptotic behavior of recurrence coefficients.

Contribution

It introduces a novel class of weight matrices and orthogonal polynomials with explicit differential equations and non-scalar asymptotic recurrence coefficients.

Findings

Explicit 2x2 orthogonal polynomial sequence derived

Differential equations with matrix polynomial coefficients established

Recurrence coefficients exhibit non-scalar asymptotic behavior

Abstract

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 certain nilpotent matrix and $\mathscr{D}$ 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.