Global asymptotics of orthogonal polynomials via Riemann-Hilbert approach

Abstract

The theory of orthogonal polynomials has long been an important subject in mathematical research. The study of the asymptotic behavior of orthogonal polynomials has attracted the most interest with many methods being developed for this purpose. One such method begins with the differential equations satisfied by orthogonal polynomials so that the existing asymptotic methods – long since established, such as WKB approximation, turning point theory and so on – can be applied. Similarly, there is also the method of difference equation. Since some orthogonal polynomials can be written in the integral form based on their generating functions, the integral approaches, such as the saddle point method and steepest descent method, can be used. There still remains, however, many instances wherein the above mentioned methods are inapplicable or have been ineffectual. The introduction of a new method known as Riemann-Hilbert approach by Deift and Zhou in 1993 marked a great change in asymptotic study of orthogonal polynomials. This method, which is based on a fundamental connection between orthogonal polynomials and Riemann-Hilbert problems (discovered by Fokas, Its and Kitaev in the 90’s) and which stems from the asymptotic study of nonlinear differential equations, is in fact a steepest descent method for the associated Riemann-Hilbert problems. Since its introduction, the method has been successful in solving many asymptotic problems as well as given way to various modifications and generalizations. This thesis begins with the observation that most of the analysis using Riemann- Hilbert approach are local. Hence, the asymptotic behavior of orthogonal polynomials derived is often represented by different (elementary or special) functions in many different regions. These various representations are sometimes unnecessary in the sense that the validity of the result in one region – usually in terms of special functions – actually covers other regions. By modifying Riemann-Hilbert approach, more global and elegant results for the asymptotics of certain orthogonal polynomials can be obtained. Two examples are considered to illustrate our modification in this thesis. One is polynomials orthogonal with respect to the weight function defined on the real line by

is a polynomial of degree 2m; the other is orthogonal polynomials associated with the varying quartic weight. Globally uniform asymptotic expansions are obtained for z in several regions, which together cover the whole complex z-plane. In particular, for the latter, the expansion involves the ª function affiliated with the Hastings¡McLeod solution of the second Painlev´e equation in the region containing the origin.