Partial solutions to several conjectures on completely monotonic degrees for remainders in asymptotic expansions of the digamma function

TL;DR

This paper investigates the completely monotonic degrees of remainders in asymptotic expansions of the gamma and digamma functions, addressing conjectures from prior research and providing partial solutions.

Contribution

It computes several completely monotonic degrees of remainders in asymptotic expansions, advancing understanding of these properties and partially resolving existing conjectures.

Findings

Computed multiple completely monotonic degrees for remainders

Provided partial solutions to several conjectures

Enhanced understanding of asymptotic expansion remainders

Abstract

Motivated by several conjectures posed in the paper "F. Qi and A.-Q. Liu, Completely monotonic degrees for a difference between the logarithmic and psi functions, J. Comput. Appl. Math., vol. 361, pp. 366--371 (2019); available online at https://doi.org/10.1016/j.cam.2019.05.001", the authors compute several completely monotonic degrees of the remainders in the asymptotic expansions of the logarithm of the gamma function and in the asymptotic expansions of the logarithm of the digamma function.