Aspects of elliptic hypergeometric functions

TL;DR

This paper surveys the development of elliptic hypergeometric functions, highlighting their definitions, key properties, and applications, including solutions to the Yang-Baxter equation via integral operators.

Contribution

It provides a comprehensive overview of elliptic hypergeometric functions, including their integrals, extensions, and the generalization of Bailey chains to elliptic integrals.

Findings

Elliptic hypergeometric functions include elliptic beta integrals and analogues of classical hypergeometric functions.

The Bailey chain technique has been extended to elliptic integrals.

Elliptic hypergeometric kernels can solve the Yang-Baxter equation.

Abstract

General elliptic hypergeometric functions are defined by elliptic hypergeometric integrals. They comprise the elliptic beta integral, elliptic analogues of the Euler-Gauss hypergeometric function and Selberg integral, as well as elliptic extensions of many other plain hypergeometric and $q$-hypergeometric constructions. In particular, the Bailey chain technique, used for proving Rogers-Ramanujan type identities, has been generalized to integrals. At the elliptic level it yields a solution of the Yang-Baxter equation as an integral operator with an elliptic hypergeometric kernel. We give a brief survey of the developments in this field.