### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 019, 69 pages      arXiv:1408.0305      https://doi.org/10.3842/SIGMA.2018.019
Contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications

### Multivariate Quadratic Transformations and the Interpolation Kernel

Eric M. Rains
Department of Mathematics, California Institute of Technology, USA

Received September 12, 2017, in final form February 27, 2018; Published online March 08, 2018

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.

Key words: quadratic transformations; elliptic special functions.

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