Coherent pairs and Sobolev-type orthogonal polynomials on the real line: An extension to the matrix case

Abstract

In this contribution, we extend the concept of coherent pair for two quasi-definite matrix linear functionals $u_{0}$ and $u_{1}$. Necessary and sufficient conditions for these functionals to constitute a coherent pair are determined, when one of them satisfies a matrix Pearson-type equation. Moreover, we deduce algebraic properties of the matrix orthogonal polynomials associated with the Sobolev-type inner product $$<p,q>_{s} = <p,q>_{\mathbf{u}_0} + <p’\mathbf{M}_1,q’\mathbf{M}_2>_{\mathbf{u}_1},$$ where $\mathbf{M}_1$ and $\mathbf{M}_2$ are non-singular matrices $p$ and $q$ are matrix polynomials.

Edinson Fuentes

Instructor