General Geronimus Perturbations for Mixed Multiple Orthogonal Polynomials

TL;DR

This paper introduces generalized Geronimus transformations for mixed multiple orthogonal polynomials, establishing new formulas and conditions that relate perturbed and original polynomials, with applications to Jacobi-Pineiro polynomials.

Contribution

It develops a broad framework for Geronimus transformations using matrix polynomials, extending previous results to non-monic and rank-agnostic cases, and analyzes their impact on orthogonality and matrix functions.

Findings

Derived Christoffel-type formulas for perturbed polynomials.

Proved equivalence between Geronimus-perturbed orthogonality and non-cancellation of $ au$-determinants.

Analyzed effects on Markov-Stieltjes matrix functions and applied to Jacobi-Pineiro polynomials.

Abstract

General Geronimus transformations, defined by regular matrix polynomials that are neither required to be monic nor restricted by the rank of their leading coefficients, are applied through both right and left multiplication to a rectangular matrix of measures associated with mixed multiple orthogonal polynomials. These transformations produce Christoffel-type formulas that establish relationships between the perturbed and original polynomials. Moreover, it is proven that the existence of Geronimus-perturbed orthogonality is equivalent to the non-cancellation of certain $\tau$-determinants. The effect of these transformations on the Markov-Stieltjes matrix functions is also determined. As a case study, we examine the Jacobi-Pi\~neiro orthogonal polynomials with three weights.