Two multivariate quadratic transformations of elliptic hypergeometric integrals

TL;DR

This paper proves two of Rains' conjectured quadratic transformations for multivariate elliptic hypergeometric functions in specific cases, and derives a third as a corollary, advancing understanding of elliptic integral identities.

Contribution

It establishes two new quadratic transformation formulas for elliptic hypergeometric integrals and derives a related third transformation, expanding the theoretical framework of elliptic hypergeometric functions.

Findings

Proved two quadratic transformations for elliptic hypergeometric functions.

Derived a third transformation as a corollary of the proofs.

Presented new equations for elliptic Selberg integrals with 10 parameters.

Abstract

Eric Rains conjectured several quadratic transformations between multivariate elliptic hypergeometric functions in "Elliptic Littlewood Identities", with the integrand multiplied by interpolation functions. In this article two of these conjectures are proven in the case where the interpolation functions are constant, and we obtain a third conjecture as a corollary. Two other equations for elliptic Selberg integrals with 10 parameters, two of which multiplying to pq/t, are given, as they are needed in the proof. The proofs consist essentially of a calculation which strings together many elliptic Dixon transformations. Some remarks are made about using Fubini in cases in which product contours do not exist.