Scalar and matrix Riemann–Hilbert approach to the strong asymptotics of Padé approximants and complex orthogonal polynomials with varying weight

Abstract

We describe methods for the derivation of strong asymptotics for the denominator polynomials and the remainder of Padé approximants for a Markov function with a complex and varying weight. Two approaches, both based on a Riemann–Hilbert problem, are presented. The first method uses a scalar Riemann–Hilbert boundary value problem on a two-sheeted Riemann surface, the second approach uses a matrix Riemann–Hilbert problem. The result for a varying weight is not with the most general conditions possible, but the loss of generality is compensated by an easier and transparent proof.