Global Asymptotics For The Generalized Bessel Polynomials With Varying Large Negative Parameters

In this dissertation, we firstly study Riemann-Hilbert boundary value problem on the positive real axis with general growth condition at the infinity. To study this problem, we establish the definitions of the principal part and order at the infinity and the origin for the holomorphic function on C\[0,+?) and then discuss the behaviors of Cauchy type integral on the positive real axis at the infinity and the origin as well as the properties of the positive and negative boundary values. Based on those, the Riemann-Hilbert boundary value problem on the positive real axis is suitably presented and solved in detail. Secondly, we introduce Riemann-Hilbert boundary value problems for the matrix valued functions and discuss three special Riemann-Hilbert boundary value problems for the matrix valued functions, especially Riemann-Hilbert boundary value problems for lower triangular matrix valued functions on the positive real axis. Then we obtain the characterization for the orthogonal polynomials with respect to some weight functions on the positive real aixs. Thirdly, we prove the orthogonality of the generalized Bessel polynomials with varying large negative parameters on the positive real axis and get the characterization. Fourthly, by using some auxiliary functions and parabolic cylinder function, we construct the parametrix and then get the asymptotic expansion of the generalized Bessel polynomials with varying large negative parameters.This dissertation consists of six chapters. In Chapter1, we introduce some research background. In Chapter2, some preliminaries are presented, including the generalized Bessel polynomials, boundary value theory for analytic functions, several kind of spe-cial functions. In Chapter3, we study Riemann-Hilbert boundary value problem on the positive real axis with general growth condition at the infinity. In Chapter4, we intro-duce Riemann-Hilbert boundary value problems for matrix valued functions and discuss especially Riemann-Hilbert boundary value problems for lower triangular matrix valued functions on the positive real axis. In Chapter5, we prove the orthogonality of the main generalized Bessel polynomials with varying large negative parameters on the positive real axis and get the characterization. In Chapter6, we obtain the asymptotic expan-sion of the generalized Bessel polynomials with varying large negative parameters after introducing some auxiliary functions and constructing the parametrix.