Sesión Análisis Real, Armónico y Teoría de AproximaciónUsing quasi-Darboux transformations to construct exceptional matrix polynomials
Ignacio Bono Parisi
FAMAF - UNC - CIEM, Argentina - Esta dirección de correo electrónico está siendo protegida contra los robots de spam. Necesita tener JavaScript habilitado para poder verlo.
Exceptional polynomials are a generalization of the classical Hermite, Jacobi, and Laguerre polynomials. They are sequences \(\{p_n(x)\}\) of polynomials that are orthogonal with respect to a positive weight \(w(x)\) and are also eigenfunctions of a second-order differential operator \[p_{n}(x)\cdot D = p_{n}''(x)\,f_{2}(x) + p_{n}'(x)\,f_{1}(x) + p_{n}(x)\,f_{0}(x) = \lambda_{n} \, p_{n}(x),\] while allowing finitely many degrees (the exceptional degrees) to be missing from the sequence. In this talk, based on [1], we consider a matrix-valued extension of these exceptional families and introduce a new method, which we call the quasi-Darboux transformation, for constructing exceptional matrix-valued orthogonal polynomials. In contrast with the scalar setting, the noncommutativity of matrix coefficients introduces additional difficulties in the construction process. We will illustrate the quasi-Darboux approach through some explicit examples.
Trabajo en conjunto con: Ignacio Zurrián (Universidad de Sevilla, España) y Antonio Durán (Universidad de Sevilla, España).
Referencias
[1] Ignacio Bono Parisi, Antonio Durán y Ignacio Zurrián, ``Using quasi-Darboux transformations to construct exceptional matrix polynomials'', Preprint, arXiv:2510.13487 (2025)