Lie algebras of differential operators for matrix valued Laguerre type polynomials

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 $W^{(\nu )}_{\phi }(x) = x^{\nu }e^{-\phi (x)} W^{(\nu )}_\textrm{pol}(x)$, where $\nu >0$, $W^{(\nu )}_\textrm{pol}(x)$ is a certain matrix valued polynomial and $\phi$ is an analytic function. We introduce differential operators ${\mathcal {D}}$, ${\mathcal {D}}^{\dagger }$ which are mutually adjoint with respect to the matrix inner product induced by $W^{(\nu )}_{\phi }(x)$. We prove that the Lie algebra generated by ${\mathcal {D}}$ and ${\mathcal {D}}^{\dagger }$ is finite dimensional if and only if $\phi$ is a polynomial. For a polynomial $\phi$, we describe the structure of this Lie algebra. As a byproduct, we give a partial answer to a problem by Ismail about finite dimensional Lie algebras related to scalar Laguerre type polynomials. The case $\phi (x)=x$ is discussed in detail.