(2\times2\) Hypergeometric operators with diagonal eigenvalues

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 \(\{...

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Bibliographic Details
Published in:arXiv.org 2018-10
Main Authors: Calderón, C, González, Y, Pacharoni, I, Simondi, S, Zurrián, I
Format: Article
Language:English
Subjects:
Eigenvalues
Mathematical analysis
Matrix methods
Norms
Operators (mathematics)
Polynomials
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.1810.08560