Quadratic Hermite–Padé Approximation to the Exponential Function: A Riemann–Hilbert Approach

Abstract

We investigate the asymptotic behavior of the polynomials p, q, r of degrees n in type I Hermite–Pade approximation to the exponential function, defined by p(z)e-z + q(z) + r(z) ez = O(z3n+2) as z → 0. These polynomials are characterized by a Riemann–Hilbert problem for a 3 × 3 matrix valued function. We use the Deift–Zhou steepest descent method for Riemann–Hilbert problems to obtain strong uniform asymptotics for the scaled polynomials p(3nz), q(3nz), and r(3nz) in every domain in the complex plane. An important role is played by a three-sheeted Riemann surface and certain measures and functions derived from it. Our work complements the recent results of Herbert Stahl.