# The $q$-multiple gamma functions of Barnes-Milnor type

## TL;DR

This paper introduces a new generalization called the $q$-BM multiple gamma function, unifying and extending properties of existing Barnes-Milnor and $q$-multiple gamma functions.

## Contribution

It defines the $q$-BM multiple gamma function and proves properties that connect it to existing gamma functions, advancing the theoretical understanding.

## Key findings

- Defined the $q$-BM multiple gamma function.
- Proved properties linking it to Barnes-Milnor functions.
- Unified existing gamma functions under a new framework.

## Abstract

The multiple gamma functions of BM (Barnes-Milnor) type and the $q$-multiple gamma functions have been studied independently. In this paper, we introduce a new generalization of the multiple gamma functions called the $q$-BM multiple gamma function including those functions and prove some properties the BM multiple gamma functions satisfy for them.