$2\times2$ Hypergeometric operators with diagonal eigenvalues

TL;DR

This paper classifies all second-order hypergeometric operators with diagonal eigenvalues that are symmetric with respect to certain matrix weights, providing explicit families, orthogonal polynomials, recurrence relations, and norms.

Contribution

It introduces a complete classification of hypergeometric operators with diagonal eigenvalues and explicit construction of associated orthogonal polynomials and weights.

Findings

Explicit three-parameter family of operators and weights.

Derived orthogonal polynomials and their three-term recurrence relations.

Computed squared matrix-norms of the orthogonal polynomials.

Abstract

In this work we classify all the order-two Hypergeometric operators $D$, symmetric with respect to some $2\times 2$ irreducible matrix-weight $W$ such that $DP_n=P_n\left(\begin{smallmatrix} \lambda_n&0\\0&\mu_n \end{smallmatrix} \right)$ with no repetition among the eigenvalues $\{\lambda_n,\mu_n\}_{n\in\mathbb N_0}$, where $\{P_n\}_{n\in\mathbb N_0}$ is the (unique) sequence of monic orthogonal polynomials with respect to $W$. We obtain, in a very explicit way, a three parameter family of such operators and weights. We also give the corresponding monic orthongonal polynomials, their three term recurrence relation and their squared matrix-norms.