Elliptic Hypergeometric Functions, III

Elliptic hypergeometric functions are the most complicated special
 functions of hypergeometric type. In these introductory lectures I'll
try to present the ideas behind their construction and outline some
of the applications. The topical content is given below.

Multiple zeta and gamma functions of Barnes and infinite basic
products. Finite difference equations of the first order with
elliptic coefficients and the elliptic gamma functions.
The elliptic beta integral as the top know generalization of
the Euler beta integral. An elliptic analogue of the Euler-Gauss
 hypergeometric function and its W(E_7) symmetry. An elliptic
analogue of the Selberg integral. Relation to the representation theory
of Lie groups (and supergroups) via the interpretation of elliptic
hypergeometric integrals as superconformal indices of four
dimensional supersymmetric gauge field theories.