Determinantal Formulas for Rational Perturbations of Multiple Orthogonality Measures

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 $r=1$.