# Ladder relations for a class of matrix valued orthogonal polynomials

## TL;DR

This paper explores the algebraic and differential structures of matrix-valued orthogonal polynomials (MVOPs) using the framework of Casper and Yakimov, focusing on specific weight functions involving exponential matrices.

## Contribution

It extends the theory of MVOPs by deriving new algebraic and differential relations, especially for weights involving exponential matrices and polynomial potentials.

## Key findings

- Derived algebraic relations for MVOPs with exponential matrix weights
- Established differential relations for a class of MVOPs on the real line
- Analyzed the structure of differential and difference operator algebras acting on MVOPs

## Abstract

Using the theory introduced by Casper and Yakimov, we investigate the structure of algebras of differential and difference operators acting on matrix valued orthogonal polynomials (MVOPs) on $\mathbb{R}$, and we derive algebraic and differential relations for these MVOPs. A particular case of importance is that of MVOPs with respect to a matrix weight of the form $W(x)=e^{-v(x)}e^{xA} e^{xA^\ast}$ on the real line, where $v$ is a scalar polynomial of even degree with positive leading coefficient and $A$ is a constant matrix.