The matrix Bochner problem

Matrix exceptional Laguerre polynomials

“…Now by Refs. [10] and [39, Lemma 1], we get M(Γ 𝑛 − 𝜆) = 0 for all 𝑛 ∈ ℕ 0 , and since (Γ 𝑛 − 𝜆) is invertible for all 𝑛 ∈ ℕ 0 we obtain M = 0. □…”

“…Next, we adapt the construction of the Fourier algebras given in Ref. [10, Definition 2.20] to our setting. Definition Given a sequence of matrix‐valued polynomials $$(Q_n(x))_{n\in \mathbb {N}_0}$$, we define $$$$\begin{equation} \begin{split} \mathcal {F}_L(Q)&=\lbrace M\in \mathcal {N}_N: \exists \mathcal {D}\in \mathcal {M}_N,\, M\cdot Q = Q\cdot \mathcal {D}\rbrace \subset \mathcal {N}_{N},\\ \mathcal {F}_R(Q)&=\lbrace \mathcal {D}\in \mathcal {M}_N: \exists M\in \mathcal {N}_N,\, M\cdot Q = Q\cdot \mathcal {D}\rbrace \subset \mathcal {M}_{N}.$$…”

“…where 𝑇 1 is given in (10) and 𝑆 1 is a self-adjoint extension of 𝑇 1 given in (21). Let Ŵ be as in Definition 1, then…”

“…Next, we adapt the construction of the Fourier algebras given in Ref. [10,Definition 2.20] to our setting.…”

“…In particular, the families of MVOPs, which are eigenfunctions of a second-order differential operator with matrix-valued coefficients, have been basically classified in a recent paper by Casper and Yakimov. 10 The classification relies on the study of a pair of isomorphic algebras of differential and difference operators, called the Fourier algebras. However, the construction of explicit examples of MVOPs of arbitrary size with the property of being eigenfunctions of a second-order differential operator is not an easy task.…”