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