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We present a representation of skew-orthogonal polynomials of symplectic type (=4) in terms of a matrix Riemann-Hilbert problem, for weights of the form 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 =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 =4 analogue of the Toda lattice.
Alex Little (Fri,) studied this question.