Using quasi-Darboux transformations to construct exceptional matrix polynomials

2025/10/15 by Ignacio Bono Parisi, Antonio J. Durán, Parisi, Ignacio Bono +3
Computer Science · Mathematics · #33C45 #47A56 #Advanced Optimization Algorithms Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Matrix Theory and Algorithms #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.2510.13487

openalex publication_date 2025/10/15 · openalex created_date 2025/10/17 · openalex updated_date 2026/07/28

Abstract

We introduce a couple of methods to construct exceptional matrix polynomials. One of them uses what we have called quasi-Darboux transformations. This seems to be a more powerful method to deal with the non-commutativity problems that appear when matrix-valued polynomials are considered. The other method does not use any transformation of Darboux type. Using both methods, we construct a collection of five illustrative examples that show how powerful our two methods are. The examples include exceptional matrix polynomials of Hermite, Laguerre, and Gegenbauer type, as well as an example with a weight matrix having a Dirac delta.

Citations

• Structure of operator algebras for matrix orthogonal polynomials
• Matrix exceptional Laguerre polynomials
• Matrix valued discrete-continuous functions with the prolate spheroidal property and bispectrality
• Exceptional Hahn and Jacobi polynomials with an arbitrary number of continuous parameters
• Durfee rectangles and pseudo-Wronskian equivalences for Hermite\n polynomials
• Higher order recurrence relation for exceptional Charlier, Meixner, Hermite and Laguerre orthogonal polynomials
• Wronskian type determinants of orthogonal polynomials, Selberg type formulas and constant term identities
• Zeroes of Wronskians of Hermite polynomials and Young diagrams
• Second order differential operators having several families of orthogonal matrix polynomials as eigenfunctions
• Matrix orthogonal polynomials whose derivatives are also orthogonal
• Matrix Valued Spherical Functions Associated to the Complex Projective Plane

Related

• Matrix exceptional Laguerre polynomials
• Darboux equivalence for matrix-valued orthogonal polynomials
• Exceptional Gegenbauer polynomials via isospectral deformation
• The single-indexed exceptional Krawtchouk polynomials
• A Bochner type classification theorem for exceptional orthogonal\n polynomials