# Nielsen's beta function and some infinitely divisible distributions

## TL;DR

This paper demonstrates that Nielsen's beta function and related special functions are generalized Stieltjes functions of order 2, revealing their logarithmic complete monotonicity and connections to infinitely divisible distributions.

## Contribution

It establishes new links between special functions, generalized Stieltjes functions, and infinitely divisible distributions, expanding the understanding of their properties and applications.

## Key findings

- Nielsen's beta function is a generalized Stieltjes function of order 2.
- Laplace transforms of certain functions are shown to be related to these special functions.
- Results apply to ratios of Gamma functions and asymptotic remainders of the double Gamma function.

## Abstract

We show that a large collection of special functions, in particular Nielsen's beta function, are generalized Stieltjes functions of order 2, and therefore logarithmically completely monotonic. This includes the Laplace transform of functions of the form $xf(x)$, where $f$ is itself the Laplace transform of a sum of dilations and translations of periodic functions. Our methods are also applied to ratios of Gamma functions, and to the remainders in asymptotic expansions of the double Gamma function of Barnes.