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arXiv · 2306.14107

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

Alex Little
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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.

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Alex Little. 2023-06-25. A Riemann-Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type. https://doi.org/10.3842/sigma.2024.076

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