Elliptic hypergeometric functions associated with root systems

TL;DR

This paper surveys elliptic hypergeometric functions related to root systems, detailing key formulas and introducing Rains' elliptic Macdonald-Koornwinder theory, highlighting recent developments in the field.

Contribution

It provides a comprehensive overview of evaluation and transformation formulas for elliptic hypergeometric functions on root systems and introduces Rains' elliptic Macdonald-Koornwinder theory.

Findings

Annotated table of main formulas for elliptic hypergeometric functions

Introduction to Rains' elliptic Macdonald-Koornwinder theory

Summary of recent developments in elliptic hypergeometric functions

Abstract

We give a survey of elliptic hypergeometric functions associated with root systems, comprised of three main parts. The first two form in essence an annotated table of the main evaluation and transformation formulas for elliptic hypergeometric integeral and series on root systems. The third and final part gives an introduction to Rains' elliptic Macdonald-Koornwinder theory (in part also developed by Coskun and Gustafson).