Essays on the theory of elliptic hypergeometric functions

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.

62 pages

References in corpus (7)

• Properties of generalized univariate hypergeometric functions
• The importance of the Selberg integral
• Recurrences for elliptic hypergeometric integrals
• Summation, transformation, and expansion formulas for multibasic theta hypergeometric series
• Continuous biorthogonality of the elliptic hypergeometric function
• Sklyanin invariant integration
• Elliptic polynomials orthogonal on the unit circle with a dense point spectrum

Cited by in corpus (3)

• Superconformal indices for N=1 theories with multiple duals
• Elliptic Hypergeometric Laurent Biorthogonal Polynomials with a Dense Point Spectrum on the Unit Circle
• Elliptic Hypergeometric Solutions to Elliptic Difference Equations