Classical elliptic hypergeometric functions and their applications

TL;DR

This paper reviews the theory of elliptic hypergeometric functions, focusing on classical analogues like the elliptic Gauss hypergeometric function, and discusses their properties and applications.

Contribution

It introduces an elliptic analogue of the Gauss hypergeometric function and explores its properties, expanding the classical special functions framework.

Findings

Elliptic hypergeometric series obey properties similar to classical functions

An elliptic analogue of the Gauss hypergeometric function is described

The paper discusses applications of elliptic hypergeometric functions

Abstract

General theory of elliptic hypergeometric series and integrals is outlined. Main attention is paid to the examples obeying properties of the "classical" special functions. In particular, an elliptic analogue of the Gauss hypergeometric function and some of its properties are described. Present review is based on author's habilitation thesis [Spi7] containing a more detailed account of the subject.