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

TL;DR

This paper investigates bounds for the logarithm of the Euler gamma function and its derivatives, focusing on differences from asymptotic expansions and conditions for complete monotonicity, extending classical results with new derivations.

Contribution

It provides new bounds and conditions for complete monotonicity of differences between log gamma and its asymptotic truncations, using novel derivations from functional equations.

Findings

Conditions for strict complete monotonicity are established.

Results recover classical cases for specific parameter values.

New derivations of asymptotic expansions are presented.

Abstract

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