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

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 ${\mathrm{e}}^{-V(z)}$ 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.