Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 020, 18 pages      arXiv:2212.06526      https://doi.org/10.3842/SIGMA.2023.020
Contribution to the Special Issue on Evolution Equations, Exactly Solvable Models and Random Matrices in honor of Alexander Its' 70th birthday

Planar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials

Sergey Berezin ab, Arno B.J. Kuijlaars a and Iván Parra a
a) Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B box 2400, 3001 Leuven, Belgium
b) St. Petersburg Department of V.A. Steklov Mathematical Institute of RAS, Fontanka 27, 191023 St. Petersburg, Russia

Received December 14, 2022, in final form March 21, 2023; Published online April 12, 2023

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.

Key words: planar orthogonal polynomials; multiple orthogonal polynomials; Riemann-Hilbert problems; Hermite-Padé approximation; normal matrix model.

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