Asymptotics of orthogonal polynomials and Riemann-Hilbert problems

Abstract

In this thesis, we study the asymptotics of orthogonal polynomials as the degree grows to infinity. Our method is based on a recent and powerful method, the Riemann-Hilbert approach, introduced by Deift and Zhou. Both continuous and discrete orthogonal polynomials are discussed. To understand this new method well, we consider two specific examples, the Laguerre polynomials and the Krawtchouk polynomials. First, for the continuous case, we study the asymptotic behavior of the Laguerre polynomials Ln(αn)(nz) as n→∞. Here, αn is a sequence of negative numbers and-αn/n tends to a limit A > 1 as n→∞. An asymptotic expansion is obtained, which is uniformly valid in the upper half plane C+ = {z : Im z ≥ 0}. A corresponding expansion is also given for the lower half plane C- = {z : Im z ≤ 0}. For the discrete case, we study the asymptotics of the Krawtchouk polynomials KNn (z; p, q) as the degree n becomes large. Asymptotic expansions are obtained, when the ratio n/N tends to a limit c Є (0, 1) as n→∞. The results are valid in one or two regions in the complex plane depending on the values of c and p. Some modifications and improvements are made to this method. For instance, we do not require deformation of contours. Moreover, our results hold globally in the complex plane, particularly in regions containing the curve on which these polynomials are orthogonal. These are not available in the precious work using this method.