Determinantal Formulas for Rational Perturbations of Multiple Orthogonality Measures

Abstract

In a previous paper, we studied the Christoffel transforms of multiple orthogonal polynomials by means of adding a finitely supported measure to the multiple orthogonality system. This approach was able to handle the Christoffel transforms of the form $(\Phi\mu_1,\dots,\Phi\mu_r)$ for a polynomial $\Phi$, where $\Phi\mu_j$ is the linear functional defined by $$f(x)\mapsto \int f(x)\Phi(x)d\mu_j(x).$$ For these systems we derived determinantal formulas generalizing Christoffel's classical theorem. In the current paper, we generalize these formulas to consider the case of rational perturbations $$\Big(\frac{\Phi_1}{\Psi_{1}} \mu_1,\dots,\frac{\Phi_r}{\Psi_r}\mu_r\Big),$$ for any polynomials $\Phi_1,\dots,\Phi_r$ and $\Psi_1,\dots,\Psi_r$. This includes the general Christoffel transforms $(\Phi_1\mu_1,\dots,\Phi_r\mu_r)$ with $r$ arbitrary polynomials {$\Phi_1,\dots,\Phi_r$,} as well as the analogous Geronimus transforms. This generalizes a theorem of Uvarov to the multiple orthogonality setting. We allow zeros of the numerators and denominators to overlap which permits addition of pure point measure. The formulas are derived for multiple orthogonal polynomials of type I and type II for any multi-index.