Planar Orthogonal Polynomials As Type II Multiple Orthogonal Polynomials

TL;DR

This paper demonstrates that planar orthogonal polynomials with logarithmic singularities are equivalent to Type II multiple orthogonal polynomials with multiple measures, and establishes a relation between their moment matrix determinants.

Contribution

It identifies a new correspondence between planar orthogonal polynomials with singularities and multiple orthogonal polynomials, providing explicit determinant relations.

Findings

Planar orthogonal polynomials with singularities are equivalent to Type II multiple orthogonal polynomials.

Derived the ratio between determinants of their moment matrices.

Established a connection between planar measures and multiple orthogonal polynomial systems.

Abstract

We show that the planar orthogonal polynomials with $l$ logarithmic singularities in the potential are the multiple orthogonal polynomials (Hermite-Pad\'e polynomials) of Type II with $l$ measures. We also find the ratio between the determinant of the moment matrix corresponding to the multiple orthogonal polynomials and the determinant of the moment matrix from the original planar measure.