Singular examples of the Matrix Bochner Problem
Journal of Approximation Theory(2024)
Abstract
The Matrix Bochner Problem aims to classify which weight matrices have their sequence of orthogonal polynomials as eigenfunctions of a second-order differential operator. Casper and Yakimov, in Casper and Yakimov (2022), demonstrated that, under certain hypotheses, all solutions to the Matrix Bochner Problem are noncommutative bispectral Darboux transformations of a direct sum of classical scalar weights. This paper aims to provide the first proof that there are solutions to the Matrix Bochner Problem that do not arise through a noncommutative bispectral Darboux transformation of any direct sum of classical scalar weights. This initial example could contribute to a more comprehensive understanding of the general solution to the Matrix Bochner Problem.
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Key words
Matrix-valued orthogonal polynomials,Matrix Bochner Problem,Discrete-continuous bispectrality,Matrix-valued bispectral functions
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