Asymptotic behavior of quadratic Hermite-Pade approximants to the exponential function and Riemann-Hilbert problems

We describe the asymptotic behavior of the polynomials p, q, r of degree n in type 1 Hermite-Pade approximation to the exponential function, i.e., p(z) e(-z) + q (z) + r (z) e(z) = O(z(3n+2)) as z --> 0. A steepest descent method for Riemann-Hilbert problems, due to Deift and Zhou, is used to obtain strong uniform asymptotics for the scaled polynomials p(3nz), q(3nz), r(3nz) in every domain of the complex plane. An important role is played by a three-sheeted Riemann surface and certain measures and functions defined on it. (C) 2003 Academie des sciences. Publie par Editions scientifiques et medicales Elsevier SAS. Tous droits reserves.