Determinantal Formulas for Rational Perturbations of Multiple Orthogonality Measures

TL;DR

This paper derives determinantal formulas for multiple orthogonal polynomials under rational perturbations of their measures, extending classical results and allowing arbitrary polynomial modifications and point mass additions.

Contribution

It provides new determinantal formulas for multiple orthogonal polynomials after rational measure perturbations, generalizing previous results and including arbitrary polynomial modifications.

Findings

Derived Uvarov-type determinantal formulas for perturbed multiple orthogonal polynomials.

Established necessary and sufficient conditions for normality of indices.

Allowed arbitrary polynomial perturbations and addition of point masses to measures.

Abstract

Given multiple orthogonal polynomials on the real line with respect to a system $\bm{\mu} = (\mu_1,\ldots,\mu_r)$, we investigate multiple orthogonal polynomials associated with any rational perturbation of the form $$ \widetilde{\bm\mu}=\Big(\frac{\Phi_1}{\Psi_{1}} \mu_1,\dots,\frac{\Phi_r}{\Psi_r}\mu_r\Big), $$ for any polynomials $\Phi_1,\dots,\Phi_r$ and $\Psi_1,\dots,\Psi_r$. We derive the analogues of Uvarov's determinantal formula for the multiple orthogonal polynomials of type I and type II for $\widetilde{\bm\mu}$ and establish necessary and sufficient condition for normality of the indices. The result allows the polynomials $\{\Phi_j,\Psi_j\}_{j=1}^r$ to be arbitrary and permits the addition of finitely many point masses to each of the measures $\mu_j$. Moreover, the measures $\mu_j$ may be taken as quasi-definite linear functionals, which is of interest even in the case $r=1$.