Bounds for the logarithm of the Euler gamma function and its derivatives

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:433

Article

Diamond, Harold G.¹  Straub, Armin¹ 

[1] Univ Illinois, Dept Math, Urbana, IL 61801 USA

关键词: Gamma function;    Psi function;    Inequalities;    Asymptotic expansions;    Bernoulli polynomials;    Complete monotonicity;   

DOI  :  10.1016/j.jmaa.2015.08.034

来源: Elsevier

【 摘 要 】

We consider differences between log Gamma(x) and truncations of certain classical asymptotic expansions in inverse powers of x A whose coefficients are expressed in terms of Bernoulli polynomials Bn,(lambda), and we obtain conditions under which these differences are strictly completely monotonic. In the symmetric cases A = 0 and lambda = 1/2, we recover results of Sonin, Norlund and Alzer. Also we show how to derive these asymptotic expansions using the functional equation of the logarithmic derivative of the Euler gamma function, the representation of 1/x as a difference F(x +1) F(x), and a backward induction. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free