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

2015/08/13 by Diamond, Harold G., Straub, Armin · 1 citation
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1508.03267

Abstract

We consider differences between log Γ(x) and truncations of certain classical asymptotic expansions in inverse powers of x-λ whose coefficients are expressed in terms of Bernoulli polynomials B_n(λ), and we obtain conditions under which these differences are strictly completely monotonic. In the symmetric cases λ=0 and λ=1/2, we recover results of Sonin, Nörlund 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.

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• The Relationship Between Euler Numbers and Bernoulli Numbers with Ordered Partitions

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