Determinants of elliptic hypergeometric integrals

TL;DR

This paper introduces a new approach to understanding elliptic hypergeometric integrals by interpreting them as solutions to difference equations, leading to novel proofs of key identities.

Contribution

It extends the interpretation of elliptic hypergeometric integrals to higher dimensions and orders, providing new proofs for the elliptic beta integral and transformation formulas.

Findings

Proved the elliptic beta integral using difference equations.

Extended the interpretation to higher-dimensional integrals.

Established Galois-theoretical conditions for uniqueness of solutions.

Abstract

We start from an interpretation of the $BC_2$-symmetric "Type I" (elliptic Dixon) elliptic hypergeometric integral evaluation as a formula for a Casoratian of the elliptic hypergeometric equation, and give an extension to higher-dimensional integrals and higher-order hypergeometric functions. This allows us to prove the corresponding elliptic beta integral and transformation formula in a new way, by proving both sides satisfy the same difference equations, and that the difference equations satisfy a Galois-theoretical condition that ensures uniqueness of simultaneous solution.