Essays on the theory of elliptic hypergeometric functions
Abstract
This is a brief survey of the main results of the theory of elliptic hypergeometric functions  a new class of special functions of mathematical physics. A proof is given of the most general known univariate exact integration formula generalizing Euler's beta integral. It 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. The known elliptic beta integrals on root systems are listed, and symmetry transformations are considered for the corresponding higherorder elliptic hypergeometric functions.
 Publication:

Russian Mathematical Surveys
 Pub Date:
 June 2008
 DOI:
 10.1070/RM2008v063n03ABEH004533
 arXiv:
 arXiv:0805.3135
 Bibcode:
 2008RuMaS..63..405S
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematical Physics
 EPrint:
 62 pages