Type II Hermite–Padé approximation to the exponential function

摘要

We obtain strong and uniform asymptotics in every domain of the complex plane for the scaled polynomials a(3nz), b(3nz), and c(3nz) where a, b, and c are the type II Hermite–Padé approximants to the exponential function of respective degrees 2n+2, 2n and 2n, defined by a(z)e-z-b(z)=O(z3n+2) and a(z)ez-c(z)=O(z3n+2) as z→0. Our analysis relies on a characterization of these polynomials in terms of a 3×3 matrix Riemann–Hilbert problem which, as a consequence of the famous Mahler relations, corresponds by a simple transformation to a similar Riemann–Hilbert problem for type I Hermite–Padé approximants. Due to this relation, the study that was performed in previous work, based on the Deift–Zhou steepest descent method for Riemann–Hilbert problems, can be reused to establish our present results.