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.

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