2024/10/19 by Ignacio Bono Parisi, Parisi, Ignacio Bono, Inés Pacharoni +1 · 2 citations
Computer Science · Mathematics · #33C45 #34L05 #34L10 #42C05 #Advanced Topics in Algebra #Algebraic and Geometric Analysis #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.2411.00798
openalex publication_date 2024/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the theory of matrix-valued orthogonal polynomials, there exists a longstanding problem known as the Matrix Bochner Problem: the classification of all N × N weight matrices W(x) such that the associated orthogonal polynomials are eigenfunctions of a second-order differential operator. In [4], Casper and Yakimov made an important breakthrough in this area, proving that, under certain hypotheses, every solution to this problem can be obtained as a bispectral Darboux transformation of a direct sum of classical scalar weights. In the present paper, we construct three families of weight matrices W(x) of size N × N, associated with Hermite, Laguerre, and Jacobi weights, which can be considered 'singular' solutions to the Matrix Bochner Problem because they cannot be obtained as a Darboux transformation of classical scalar weights.