Uniform asymptotic expansions of the Tricomi-Carlitz polynomials-猫眼课题宝

DOI：

10.1090/s0002-9939-10-10301-3

发表时间：

2010-07

期刊：

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中科院分区：

其他

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作者：

K. Lee;R. Wong

本文给出了Tricomi-Carlitz多项式f（α）n（x）和修正的洛梅尔多项式hn，ν（x）在n → ∞时的一致渐近展开式，对x在（0，∞）中成立.由于这两个多项式不满足二阶微分方程，因此为微分方程开发的强大工具不适用。我们的讨论分为三个部分。在第一部分中，我们直接从三项递归关系（n+1）f（α）n+1（x）−（n+α）xf（α）n（x）+f（α）n−1（x）= 0导出了f（α）n（x）的一个渐近展开式，它在包含临界值x=±2/v的区域内一致成立，其中v =n+2α−1/2。这种方法是基于Wang和Wong [Numer. 91（2002）和94（2003）]。在第二部分中，利用三次变换导出了积分函数exp[ν τ（s; t）] ds的展开式，其中J（s;t）和exp（s;t）是s的解析函数，t是有界的真实的参数，且exp（s; t）有两个鞍点s±（t），当t趋于某个真实的数t0时，这两个鞍点s±（t）合并.然后，我们应用集成的部分技术建议的Bleistein。作为应用，导出了Tricomi-Carlitz多项式零点的渐近展开式。有界t的有效性可以通过使用由Olde Daalhuis和Temme引入的有理函数序列来扩展到无界t。展开式涉及Airy函数及其导数。误差界也给出了一项和两项近似。最后研究了修正的洛梅尔多项式hn，ν（t/N）的渐近展开式，它在包含临界值x=±1/N的区域内一致成立，其中N=n+ν.这种方法再次基于三项递推的转折点理论;它们的零点也被导出。城大索书号：QA404.5 .L44 2012%iv，148叶：ill。30厘米。%

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.%%%%CityU Call Number: QA404.5 .L44 2012%%%%iv, 148 leaves : ill. 30 cm.%%%%%%%%

Uniform asymptotic expansions of the Tricomi-Carlitz polynomials

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