A Riemann–Hilbert problem for skew-orthogonal polynomials

Abstract

We find a local ( d + 1 ) × ( d + 1 ) Riemann–Hilbert problem characterizing the skew-orthogonal polynomials associated to the partition function of orthogonal ensembles of random matrices with a potential function of degree d. Our Riemann–Hilbert problem is similar to a local d × d Riemann–Hilbert problem found by Kuijlaars and McLaughlin characterizing the bi-orthogonal polynomials. This gives more motivation for finding methods to compute asymptotics of high order Riemann–Hilbert problems, and brings us closer to finding full asymptotic expansions of the skew-orthogonal polynomials.