Elliptic and Complex Hypergeometric Integrals in Quantum Field Theory

We consider relations between elliptic hypergeometric integrals and complex hypergeometric functions. Taking exactly computable type I elliptic beta integrals, describing superconformal indices of certain confining four-dimensional non-abelian gauge theories, we reduce them to the level of complex hypergeometric functions in the Mellin–Barnes representation. A connection of the complex hypergeometric functions to integrable many-body problems is indicated as well.