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arXiv · 1810.08560

$2\times2$ Hypergeometric operators with diagonal eigenvalues

C. Calderón·Y. González·I. Pacharoni·S. Simondi·I. Zurrián
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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.

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C. Calderón, Y. González, I. Pacharoni, S. Simondi, I. Zurrián. 2018-10-11. $2\times2$ Hypergeometric operators with diagonal eigenvalues. https://doi.org/10.1016/j.jat.2019.105299

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