Bernoulli polynomials and asymptotic expansions of the quotient of gamma functions

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:235

Article

Buric, Tomislav¹  Elezovic, Neven¹ 

[1] Univ Zagreb, Fac Elect Engn & Comp, Zagreb 10000, Croatia

关键词: Gamma function;    Wallis quotient;    Wallis power function;    Bernoulli polynomials;    Asymptotic expansion;    Stirling formula;   

DOI  :  10.1016/j.cam.2011.01.045

来源: Elsevier

【 摘 要 】

The main subject of this paper is the analysis of asymptotic expansions of Wallis quotient function Gamma(x+t)/Gamma(x+s) and Wallis power function [Gamma(x+t)/Gamma(x+s)](1/(t-s)) when x tends to infinity. Coefficients of these expansions are polynomials derived from Bernoulli polynomials. The key to our approach is the introduction of two intrinsic variables alpha = 1/2 (t + s - 1) and beta = 1/2 (1 + t - s)(1 - t + s) which are naturally connected with Bernoulli polynomials and Wallis functions. Asymptotic expansion of Wallis functions in terms of variables t and s and also alpha and beta is given. Application of the new method leads to the improvement of many known approximation formulas of the Stirling's type. (C) 2011 Elsevier B.V. All rights reserved.

【 授权许可】

Free