Multivariate Quadratic Transformations and the Interpolation Kernel

TL;DR

This paper proves new quadratic transformations of elliptic Selberg integrals and explores the interpolation kernel, which is key for elliptic Fourier transforms and hypergeometric identities, advancing the understanding of elliptic hypergeometric functions.

Contribution

It introduces a novel approach to proving elliptic hypergeometric integral identities via reduction to a Zariski dense subset near the trigonometric limit.

Findings

Proved quadratic transformations of elliptic Selberg integrals.

Developed the theory of the interpolation kernel as an elliptic Fourier transform.

Provided new methods for elliptic hypergeometric integral proofs.

Abstract

We prove a number of quadratic transformations of elliptic Selberg integrals (conjectured in an earlier paper of the author), as well as studying in depth the "interpolation kernel", an analytic continuation of the author's elliptic interpolation functions which plays a major role in the proof as well as acting as the kernel for a Fourier transform on certain elliptic double affine Hecke algebras (discussed in a later paper). In the process, we give a number of examples of a new approach to proving elliptic hypergeometric integral identities, by reduction to a Zariski dense subset of a formal neighborhood of the trigonometric limit.