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Quadratic HermitePadé approximation to the exponential function: a Riemann–Hilbert approach
 Constr. Approx
"... We investigate the asymptotic behavior of the polynomials p, q, r of degrees n in type I HermitePadé approximation to the exponential function, defined by p(z)e −z + q(z) + r(z)e z = O(z 3n+2) as z → 0. These polynomials are characterized by a Riemann– Hilbert problem for a 3 × 3 matrix valued func ..."
Abstract

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We investigate the asymptotic behavior of the polynomials p, q, r of degrees n in type I HermitePadé approximation to the exponential function, defined by p(z)e −z + q(z) + r(z)e z = O(z 3n+2) as z → 0. These polynomials are characterized by a Riemann– Hilbert problem for a 3 × 3 matrix valued function. We use the DeiftZhou 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 threesheeted Riemann surface and certain measures and functions derived from it. Our work complements recent results of Herbert Stahl. Contents