Several completely monotone functions related to DeTemple's sequence

TL;DR

This paper characterizes when certain functions involving the digamma function are completely monotone, generalizing previous results and confirming a conjecture by Chen, thus advancing understanding of special functions.

Contribution

It provides necessary and sufficient conditions for complete monotonicity of functions involving the digamma function with a parameter, extending prior work and verifying Chen's conjecture.

Findings

Identifies conditions for complete monotonicity of functions involving R(x)

Generalizes known results on monotonicity of special functions

Verifies a conjecture by Chen regarding these functions

Abstract

In this paper, we present the necessary and sufficient conditions such that several functions involving $R\left( x\right) =\psi \left( x+1/2\right) -\ln x$ with a parameter are completely monotone on $\left( 0,\infty \right) $, where $\psi $ is the digamma function. This generalizes some known results and verifies a conjecture posed by Chen.