A Riemann-Hilbert representation for Sobolev orthogonal polynomials

Abstract

In this article we consider polynomials orthogonal with respect to the inner product $$\langle P,Q \rangle_S = \int_{\mathbb{R}}P(x)Q(x) \mathrm{e}^{-V(x)} \, \mathrm{d}x + λ\int_{\mathbb{R}}P^\prime(x)Q^\prime(x) \mathrm{e}^{-V(x)} \, \mathrm{d}x$$ where $V$ is a polynomial of even degree (at least four) and positive leading coefficient, and $λ> 0$ a constant. The above inner product is a special case of the so-called Sobolev inner product. We show how one can represent the associated Sobolev orthogonal polynomials as a species of Type I multiple-orthogonal polynomial. This correspondence makes use of the WKB asymptotics of a certain second order ODE. From this we may write a Riemann--Hilbert problem for the Sobolev orthogonal polynomials, from which one can deduce a Christoffel--Darboux-type formula for the projection kernel.