Pick Functions Related to the Multiple Gamma Functions of order $n$

TL;DR

This paper investigates the properties of functions related to Barnes multiple Gamma functions, proposing a conjecture for their Stieltjes representation and confirming it for the case n=3.

Contribution

It introduces a conjecture on the Stieltjes representation of functions derived from Barnes multiple Gamma functions and proves it for the specific case n=3.

Findings

Proposed a conjecture for the Stieltjes representation of related functions.

Confirmed the conjecture for the case n=3 by analyzing function properties.

Abstract

Let $G_n$ be the Barnes multiple Gamma function of order $n$ and the function $f_n(z)$ be defined as \begin{align*} f_n(z)=\dfrac{\log G_n(z+1)}{z^n\Log z},\quad z\in \mathbb{C}\setminus (-\infty,0]. \end{align*} In this work, a conjecture to find the Stieltjes representation is proposed such that $f_n(z)$ is a Pick function. The conjecture is established for the particular case $n=3$ by examining the properties of $f_3(z)$.