Uvarov Perturbations for Multiple Orthogonal Polynomials of the Mixed Type on the Step-Line

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-Pi\~neiro 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.