Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Explicit construction of matrix-valued orthogonal polynomials of arbitrary size

Ignacio Bono Parisi

TL;DR

The paper addresses the explicit construction of matrix-valued orthogonal polynomials (MVOP) for weight matrices of the form $W(x)=T(x)\operatorname{diag}(w_{1},\dots,w_{N})T(x)^{*}$ with $T(x)=e^{Ax}$ and $A$ nilpotent, extending scalar theory to arbitrary size. It provides a closed-form expression for the MVOP sequence $Q_{n}(x)$ in terms of the scalar monic polynomials $p_{n}^{w_i}(x)$, and proves that these MVOPs are orthogonal with respect to $W$, with a controlled three-term recurrence and a second-order differential operator. The work establishes sufficient conditions for irreducibility of $W$ and for bispectrality, and develops several bispectral families arising from Laguerre, Hermite, Jacobi, and mixed Hermite–Laguerre weights, including explicit Darboux-transform connections. By delivering explicit $2\times2$ and $3\times3$ instances and a general construction for any size, the results substantially broaden the catalog of explicit, irreducible, bispectral MVOPs and facilitate applications in noncommutative spectral problems and time-band limiting on matrix-valued spaces.

Abstract

In this paper, we explicitly provide expressions for a sequence of orthogonal polynomials associated with a weight matrix of size $N$ constructed from a collection of scalar weights $w_{1}, \ldots, w_{N}$: $$W(x) = T(x)\operatorname{diag}(w_{1}(x), \ldots, w_{N}(x))T(x)^{\ast},$$ where $T(x)$ is a specific polynomial matrix. We provide sufficient conditions on the scalar weights to ensure that the weight matrix $W$ is irreducible. Furthermore, we give sufficient conditions on the scalar weights to ensure the constructed sequence of matrix orthogonal polynomials is an eigenfunction of a differential operator. We also study the Darboux transformations and bispectrality of the orthogonal polynomials in the particular case where the scalar weights are the classical weights of Jacobi, Hermite, and Laguerre. With these results, we construct a wide variety of bispectral matrix-valued orthogonal polynomials of arbitrary size, which satisfy a second-order differential equation.

Explicit construction of matrix-valued orthogonal polynomials of arbitrary size

TL;DR

The paper addresses the explicit construction of matrix-valued orthogonal polynomials (MVOP) for weight matrices of the form with and nilpotent, extending scalar theory to arbitrary size. It provides a closed-form expression for the MVOP sequence in terms of the scalar monic polynomials , and proves that these MVOPs are orthogonal with respect to , with a controlled three-term recurrence and a second-order differential operator. The work establishes sufficient conditions for irreducibility of and for bispectrality, and develops several bispectral families arising from Laguerre, Hermite, Jacobi, and mixed Hermite–Laguerre weights, including explicit Darboux-transform connections. By delivering explicit and instances and a general construction for any size, the results substantially broaden the catalog of explicit, irreducible, bispectral MVOPs and facilitate applications in noncommutative spectral problems and time-band limiting on matrix-valued spaces.

Abstract

In this paper, we explicitly provide expressions for a sequence of orthogonal polynomials associated with a weight matrix of size constructed from a collection of scalar weights : where is a specific polynomial matrix. We provide sufficient conditions on the scalar weights to ensure that the weight matrix is irreducible. Furthermore, we give sufficient conditions on the scalar weights to ensure the constructed sequence of matrix orthogonal polynomials is an eigenfunction of a differential operator. We also study the Darboux transformations and bispectrality of the orthogonal polynomials in the particular case where the scalar weights are the classical weights of Jacobi, Hermite, and Laguerre. With these results, we construct a wide variety of bispectral matrix-valued orthogonal polynomials of arbitrary size, which satisfy a second-order differential equation.

Paper Structure

This paper contains 16 sections, 11 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Orthogonal polynomials and the algebra $\mathcal{D}(W)$
  4. The Darboux Transformation
  5. Classical orthogonal polynomials
  6. Shifted Hermite Polynomials
  7. Laguerre Polynomials
  8. Jacobi Polynomials
  9. Construction of Matrix-Valued Orthogonal Polynomials
  10. Bispectrality and Darboux transformation
  11. Laguerre-type
  12. Hermite-type
  13. Jacobi-type
  14. Hermite-Laguerre-type
  15. Irreducibility
  16. ...and 1 more sections

Key Result

Proposition 2.7

Let $W$ and $\tilde{W}$ be weight matrices, and let $Q_{n}$ and $\tilde{Q}_{n}$ sequences of orthogonal polynomials for $W$ and $\tilde{W}$, respectively. If there exists a differential operator $\mathcal{D}_{1}$ such that with $A_{n} \in \operatorname{Mat}_{N}(\mathbb{C})$ nonsingular for all but finitely many $n\in \mathbb{N}_{0}$, then $\tilde{W}$ is a Darboux transformation of $W$.

Theorems & Definitions (28)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7: Theorem 3.3, BPZ24
  • Theorem 3.1
  • proof
  • Proposition 3.2
  • ...and 18 more