# Elliptic extension of Gustafson's $q$-integral of type $G_2$

## TL;DR

This paper proves an elliptic extension of Gustafson's $q$-integral of type $G_2$, expressing it via elliptic gamma functions and connecting it to known $q$-beta integrals as a special case.

## Contribution

It introduces a new elliptic beta integral of type $G_2$ and provides a proof using elliptic gamma functions and existing $BC_1$ integrals.

## Key findings

- Evaluation formula expressed with elliptic gamma functions
- Includes Gustafson's $q$-beta integral as a limit case
- Utilizes elliptic beta integral of type $BC_1$ in proof

## Abstract

The evaluation formula for an elliptic beta integral of type $G_2$ is proved. The integral is expressed by a product of Ruijsenaars' elliptic gamma functions, and the formula includes that of Gustafson's $q$-beta integral of type $G_2$ as a special limiting case as $p\to 0$. The elliptic beta integral of type $BC_1$ by van Diejen and Spiridonov is effectively used in the proof of the evaluation formula.