Lie algebras of differential operators for Matrix valued Laguerre type polynomials

2023/03/13 by Andrea L. Gallo, Gallo, Andrea L., Pablo Román +1
Mathematics · Physics and Astronomy · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics

paper · pdf · doi:10.48550/arxiv.2303.06805

openalex publication_date 2023/03/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study algebras of differential and difference operators acting on matrix valued orthogonal polynomials (MVOPs) with respect to a weight matrix of the form W^(ν)_ϕ(x) = x^νe^-ϕ(x) W^(ν)_pol(x), where ν>0, W^(ν)_pol(x) is certain matrix valued polynomial and ϕ an entire function. We introduce a pair differential operators D, D^† which are mutually adjoint with respect to the matrix inner product induced by W^(ν)_ϕ(x). We prove that the Lie algebra generated by D and D^† is finite dimensional if and only if ϕ is a polynomial, giving a partial answer to a problem by M. Ismail. In the case ϕ polynomial, we describe the structure of this Lie algebra. The case ϕ(x)=x, is discussed in detail. We derive difference and differential relations for the MVOPs. We give explicit expressions for the entries of the MVOPs in terms of classical Laguerre and Dual Hahn polynomials.

Related

• Real-Root Preserving Differential Operator Representations of Orthogonal Polynomials
• Structure of operator algebras for matrix orthogonal polynomials
• Matrix-Valued Hermite and Laguerre polynomials via Quadratic Transformation
• Differential Recursion Relations for Laguerre Functions on Hermitian Matrices
• Second order differential operators having several families of orthogonal matrix polynomials as eigenfunctions