$2\times 2$ Hypergeometric operators with diagonal eigenvalues

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 $D{P}_n=P_n\left(\begin{array}{cc}{\lambda }_n& 0\\ 0& {\mu }_n\end{array}\right)$ 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.