turkmath.org

Elliptic hypergeometric functions were introduced by Frenkel and Turaev two decades ago. They are generalizations of the basic (q-) and ordinary (plane) hypergeometric functions. During the last years, there has been a large increase of interest in elliptic hypergeometric functions both in the mathematics and mathematical physics community. In the talk, I will try to outline special functions of hypergeometric type, mainly in the context of mathematical physics. Then I will review the current status elliptic hypergeometric functions. In particular, I will discuss the elliptic gamma function and its properties, the elliptic beta integral (the top known generalization of the Euler beta integral), the elliptic analog of the Euler-Gauss hypergeometric function, and more complicated elliptic hypergeometric integrals. If time permits, I will discuss various applications of these functions in mathematical physics, mainly in integrable models, quantum field theory, and Painleve equations.