# Representations for the derivative at zero and finite parts of the Barnes zeta function

## TL;DR

This paper introduces new series, limit, and integral representations for the finite parts and derivatives of the Barnes zeta function and its related functions, enhancing analytical tools in special function theory.

## Contribution

It provides novel representations for the Barnes zeta function's finite parts and derivatives, including integral forms, applicable in any dimension and for related functions.

## Key findings

- Derived series and limit representations for Barnes zeta finite parts
- Established integral representations for poles of Barnes zeta function
- Connected results to the logarithm of Barnes Gamma and multiple Gamma functions

## Abstract

We provide new representations for the finite parts at the poles and the derivative at zero of the Barnes zeta function in any dimension in the general case. These representations are in the forms of series and limits. We also give an integral representation for the finite parts at the poles. Similar results are derived for an associated function, which we term homogeneous Barnes zeta function. Our expressions immediately yield analogous representations for the logarithm of the Barnes Gamma function, including the particular case also known as multiple Gamma function.