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\(2 \times 2\) hypergeometric operators with diagonal eigenvalues. (English) Zbl 1426.42022

Summary: In this work we give all the order-two hypergeometric operators \(D\), symmetric with respect to some \(2 \times 2\) irreducible matrix-weight \(W\) on \((0, 1)\) such that \(D P_n = P_n ( \lambda_n 00 \mu_n )\) 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 a new family of such operators and weights, depending on three parameters, generalizing some older examples. We also give, in a very explicit way, the corresponding monic orthogonal polynomials, their three term recurrence relation and their squared matrix-norms.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
47S10 Operator theory over fields other than \(\mathbb{R}\), \(\mathbb{C}\) or the quaternions; non-Archimedean operator theory
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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