Global asymptotics of Hermite polynomials via Riemann-Hilbert approach

Abstract

In this thesis, we study the asymptotic behavior of the Hermite polynomials Hn((2n+ 1)1/2z) as n → ∞. A globally uniform asymptotic expansion is obtained for z in an unbounded region containing the right half-plane Re z ≥ 0. A corresponding expan- sion can also be given for z in the left half-plane by using the symmetry property of the Hermite polynomials. Our approach is based on the steepest-descent method for Riemann-Hilbert problems introduced by Deift and Zhou. At the end of the thesis, we compare our result with that of Olver obtained by using differential equation theory.