Uniform asymptotic expansions of the Tricomi-Carlitz polynomials and the Modified Lommel polynomials

Abstract

In this thesis, we derive uniform asymptotic expansions of the Tricomi-Carlitz poly- nomials f(α)n(x) and the modified Lommel polynomials hn,ν(x), as n → ∞, valid for x in (0,∞). Since these two polynomials do not satisfy a second-order differential equation, the powerful tools developed for differential equations are not applicable. Our discussion is divided into three parts. In the first part, we derive directly from the three-term recurrence relation (n+1)f(α)n+1(x)−(n+α)xf(α)n(x)+f(α)n−1(x) = 0, an asymptotic expansion for f(α)n(x) which holds uniformly in regions containing the critical values x=±2/√ν, where ν=n+2α−1/2. This method is based on the turning-point theory for three-term recurrence introduced by Wang and Wong [Numer. Math. 91 (2002) and 94 (2003)]. In the second part, the expansion is derived by using the cubic transformation for the integral ∫cJ(s;t) exp[νϕ(s;t)] ds, where J(s;t) and ϕ(s;t) are analytic functions of s, t is a bounded real parameter and ϕ(s; t) have two saddle points s±(t) which coalesce as t tends to some real number t0. Then we apply the integration-by-part technique suggested by Bleistein. As an application, an asymptotic expansion for the zeros of the Tricomi-Carlitz polynomials is derived. The validity for bounded t can be extended to unbounded t by using a sequence of rational functions introduced by Olde Daalhuis and Temme. The expansion involves the Airy functions and their derivatives. Error bounds are also given for one-term and two-term approximations. We finally study a asymptotic expansion for the modified Lommel polynomials hn,ν(t/N) which holds uniformly in regions containing the critical values x=±1/N, where N=n+ν. This method is again based on the turning-point theory for three-term recurrence; their zeros are also derived.

Date of Award	16 Jul 2012
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Sue Cheun Roderick WONG (Supervisor)