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

SIGMA 20 (2024), 076, 32 pages      arXiv:2306.14107      https://doi.org/10.3842/SIGMA.2024.076
Contribution to the Special Issue on Evolution Equations, Exactly Solvable Models and Random Matrices in honor of Alexander Its' 70th birthday

A Riemann-Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type

Alex Little
Unité de Mathématiques Pures et Appliquées, ENS de Lyon, France

Received December 27, 2023, in final form August 06, 2024; Published online August 16, 2024

Abstract
We present a representation of skew-orthogonal polynomials of symplectic type ($\beta=4$) in terms of a matrix Riemann-Hilbert problem, for weights of the form ${\rm e}^{-V(z)}$ where $V$ is a polynomial of even degree and positive leading coefficient. This is done by representing skew-orthogonality as a kind of multiple-orthogonality. From this, we derive a $\beta=4$ analogue of the Christoffel-Darboux formula. Finally, our Riemann-Hilbert representation allows us to derive a Lax pair whose compatibility condition may be viewed as a $\beta=4$ analogue of the Toda lattice.

Key words: Riemann-Hilbert problem; skew-orthogonal polynomials; random matrices.

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