Asymptotic expansion of the Tricomi-Carlitz polynomials and their zeros

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:265

Article

Lee, K. F.¹  Wong, R.¹ 

[1] City Univ Hong Kong, Dept Math, Kowloon, Hong Kong, Peoples R China

关键词: Tricomi-Carlitz polynomials;    Uniform asymptotic expansions;    Airy function;   

DOI  :  10.1016/j.cam.2013.09.047

来源: Elsevier

【 摘 要 】

The Tricomi-Carlitz polynomials f(n)((alpha))(x) are non-classical discrete orthogonal polynomials on the real line with respect to the step function whose jumps are d psi((alpha)) (x) = (k+a)(k-1)e(-k)/k! at x = x(k) = +/-(k + alpha)(-1/2), k = 0, 1, 2, ... In this paper, we derive an asymptotic expansion for f(n)((alpha)) (t/root v) as n -> infinity, valid uniformly for bounded real t, where v = n + 2 alpha - 1/2 and alpha is a positive parameter. 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 given for one-term and two-term approximations. Asymptotic formulas are also presented for the zeros of f(n)((alpha)) (t/root v). (C) 2013 Elsevier B.V. All rights reserved.

【 授权许可】

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