Asymptotic expansions for the gamma function

JOURNAL OF NUMBER THEORY

卷:169

Article

Xu, Aimin¹  Hu, Yongcai²  Tang, Peipei³ 

[1] Zhejiang Wanli Univ, Inst Math, Ningbo 315100, Zhejiang, Peoples R China

[2] Henan Polytech Inst, Dept Basic Educ, Nanyang 473000, Peoples R China

[3] Zhejing Univ City Coll, Sch Comp Sci, Hangzhou 310015, Zhejiang, Peoples R China

关键词: Gamma function;    Asymptotic expansion;    Cycle indicator polynomial;    Lagrange-Burmann formula;   

DOI  :  10.1016/j.jnt.2016.05.020

来源: Elsevier

【 摘 要 】

Mortici (2015) [31] proposed a new formula for approximating the gamma function and the convergence of the corresponding asymptotic series is very fast in comparison with other classical or recently discovered asymptotic series. In this paper, by the Lagrange-Burmann formula we give an explicit formula for determining the coefficients a(k) (k = 1,2,...) in Mortici's formula such that Gamma(x + 1)/root 2 pi x(x/e)(x) similar to exp{Sigma(infinity)(k=1) a(k) (x/12x(2) + 2/5)(k)}, x -> infinity. Moreover, by the cycle indicator polynomial of symmetric group, we give an explicit expression for the coefficients b(k) (k = 0,1,...) of the following expansion: Gamma(x + 1)/root 2 pi x(x/e)(x) similar to (Sigma(infinity)(k=0) b(k) (x/12x(2) + 2/5)(k))(1/r), x -> infinity. A recursive formula for calculating the coefficients b(k) (k = 0,1,...) is also given. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

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