General Christoffel Perturbations for Mixed Multiple Orthogonal Polynomials

TL;DR

This paper develops Christoffel formulas for mixed multiple orthogonal polynomials under general matrix polynomial perturbations, extending classical results to non-monic and rank-deficient cases.

Contribution

It introduces a framework for perturbing mixed multiple orthogonal polynomials with general matrix polynomials and establishes criteria for their orthogonality preservation.

Findings

Derived Christoffel formulas for general matrix polynomial perturbations

Established a divisibility criterion for the existence of perturbed orthogonality

Extended classical orthogonal polynomial theory to non-monic and rank-deficient cases

Abstract

Performing both right and left multiplication operations using general regular matrix polynomials, which need not be monic and may possess leading coefficients of arbitrary rank, on a rectangular matrix of measures associated with mixed multiple orthogonal polynomials, reveals corresponding Christoffel formulas. These formulas express the perturbed mixed multiple orthogonal polynomials in relation to the original ones. Utilizing the divisibility theorem for matrix polynomials, we establish a criterion for the existence of perturbed orthogonality, expressed through the non-cancellation of certain $\tau$ determinants.