anar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials

Abstract

A recent result of S.-Y. Lee and M. Yang states that the planar orthogonal polynomials orthogonal with respect to a modified Gaussian measure are multiple orthogonal polynomials of type II on a contour in the complex plane. We show that the same polynomials are also type I orthogonal polynomials on a contour, provided the exponents in the weight are integer. From this orthogonality, we derive several equivalent Riemann-Hilbert problems. The proof is based on the fundamental identity of Lee and Yang, which we establish using a new technique.

Spherical Induced Ensembles with Symplectic Symmetry

“…Additionally, the insertion of a point charge, also known as the Christoffel perturbation has a physical application for instance, in the context of the massive quantum field theory [3,6]. Let us also mention that the insertion of a point charge has been extensively studied in the context of planar (skew-)orthogonal polynomials, see, e.g., [6,14,53] and references therein. On the one hand, the local statistics of the ensemble also depends on the local geometry of the droplet.…”