Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function

TL;DR

This paper introduces completely monotonic functions of positive order and demonstrates their connection to the remainders in asymptotic expansions of the logarithms of Barnes double gamma and Euler's gamma functions.

Contribution

It establishes a new class of completely monotonic functions of any positive order and links them to asymptotic expansion remainders of important special functions.

Findings

Remainders in asymptotic expansions are completely monotonic of any positive integer order.

New class of completely monotonic functions of order r>0 introduced.

Provides insights into the structure of special functions' asymptotic behaviors.

Abstract

We introduce completely monotonic functions of order $r>0$ and show that the remainders in asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function give rise to completely monotonic functions of any positive integer order.