# An elliptic hypergeometric beta integral transformation

## TL;DR

This paper proves a new elliptic hypergeometric integral identity, providing a different proof for a known conjecture and exploring its limits to derive identities related to the Selberg integral.

## Contribution

It introduces a novel elliptic hypergeometric integral identity and offers new proofs and classical limits, expanding the understanding of hypergeometric integrals.

## Key findings

- Proved a new elliptic hypergeometric integral identity
- Provided alternative proof for a known identity by Rains
- Derived new identities related to the Selberg integral in classical limits

## Abstract

In this article we prove a new elliptic hypergeometric integral identity. It previously appeared (as a conjecture) in articles by Rains, and Spiridonov and Vartanov. Moreover it gives a different proof of an identity in another article by Rains. We also give some basic hypergeometric and classical limits of this identity. The classical limit gives identities (some known, some new) between generalizations of the Selberg integral.