Essays on the theory of elliptic hypergeometric functions

TL;DR

This paper reviews the theory of elliptic hypergeometric functions, introduces a unifying elliptic beta integral, constructs an elliptic analogue of the Gauss hypergeometric function, and explores their properties and symmetries.

Contribution

It presents the most general univariate elliptic beta integral and develops the elliptic hypergeometric function theory with new integral formulas and symmetry analyses.

Findings

Proved the elliptic beta integral generalizing Euler's beta integral.

Constructed the elliptic hypergeometric function and derived its differential equation.

Described biorthogonality relations and symmetry transformations.

Abstract

We give a brief review of the main results of the theory of elliptic hypergeometric functions -- a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler's beta integral, which is called the elliptic beta integral. An elliptic analogue of the Gauss hypergeometric function is constructed together with the elliptic hypergeometric equation for it. Biorthogonality relations for this function and its particular subcases are described. We list known elliptic beta integrals on root systems and consider symmetry transformations for the corresponding elliptic hypergeometric functions of the higher order.