Planar 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.

References in corpus (5)

• Characteristic Polynomials of Complex Random Matrix Models
• Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane
• Orthogonal and multiple orthogonal polynomials, random matrices, and Painlevé equations
• Strong Asymptotics of Planar Orthogonal Polynomials: Gaussian Weight Perturbed by Finite Number of Point Charges
• Progress on the study of the Ginibre ensembles I: GinUE

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