Some examples of orthogonal matrix polynomials satisfying odd order differential equations

Abstract

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 $$W(t)=t^{\alpha }{e}^{-t}{e}^{At}{t}^B{t}^{B^{\ast }}{e}^{A^{\ast }t},$$ 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.

arXiv comment28 pages

References in corpus (1)

• Matrix valued orthogonal polynomials related to SU$(N+1)$, their algebras of differential operators and the corresponding curves

Cited by in corpus (7)

• Reducibility of Matrix Weights
• The Algebra of Differential Operators for a Gegenbauer Weight Matrix
• Singular examples of the Matrix Bochner Problem
• The algebra $\mathcal{D}(W)$ via strong Darboux transformations
• Orthogonal matrix polynomials satisfying differential equations with recurrence coefficients having non-scalar limits
• Structural formulas for matrix-valued orthogonal polynomials related to $2\times 2$ hypergeometric operators
• A $3\times 3$ singular solution to the Matrix Bochner Problem with $\mathcal{D}(W)$ not of the form $\mathbb{C}[D]$