Uvarov Perturbations for Multiple Orthogonal Polynomials of the Mixed Type on the Step-Line
Manuel Mañas, Miguel Rojas
TL;DR
The paper develops a matrix-analytic framework for Uvarov-type perturbations of mixed-type MOPs on the step line, unifying Christoffel, Geronimus, and Uvarov transformations through left/right matrix polynomials $L(x)$ and $R(x)$ and discrete masses. It derives explicit connection formulas between perturbed and unperturbed polynomials, analyzes the induced finite-band changes to moment matrices and block-banded recurrence operators, and describes the rational spectral transformations of the Markov–Stieltjes function via matrix polynomials. The main contributions include the construction of the perturbation framework, determinant criteria (the $ au$-determinants) for the existence of perturbed orthogonality, and a detailed case study of Jacobi–Piñeiro polynomials illustrating the algebraic and spectral effects. The work provides a linear-algebraic perspective that clarifies how rational and additive spectral modifications deform structured moment matrices and recurrence operators while preserving much of the underlying block structure. It also highlights open questions about sufficiency of the nonvanishing $ au$-determinants beyond the LU-factorization assumption and points to future directions in multivariate and higher-dimensional analogues of mixed-type perturbations.
Abstract
Uvarov-type perturbations for mixed-type multiple orthogonal polynomials on the step line are investigated within a matrix-analytic framework. The transformations considered involve both rational and additive modifications of a rectangular matrix of measures, implemented through left and right multiplication by regular matrix polynomials together with the addition of finitely many discrete matrix masses. These operations induce structured finite-band modifications of the corresponding moment matrix and rational transformations of the associated Markov-Stieltjes matrix function. Explicit Uvarov-type connection formulas are obtained, relating the perturbed and unperturbed families of mixed-type multiple orthogonal polynomials, and determinant conditions ensuring the existence of the new orthogonality are established. The algebraic consequences of these transformations are analyzed, showing their effect on the structure of the block-banded recurrence operators and on the spectral data encoded in the Markov-Stieltjes matrix functions. As an illustrative example, nontrivial Uvarov-type perturbations of the Jacobi-Piñeiro multiple orthogonal polynomials are presented. The results unify Christoffel, Geronimus, and Uvarov transformations within a single matrix framework, providing a linear-algebraic perspective on rational and additive spectral transformations of structured moment matrices and their corresponding recurrence operators.