Barnes-Ismagilov integrals and hypergeometric functions of the complex field

Abstract: We examine a family ${}pG{q}^{\mathbb C}\big[\genfrac{}{}{0pt}{}{(a)}{(b)};z\big]$ of integrals of Mellin-Barnes type over the space ${\mathbb Z}\times {\mathbb R}$, such functions $G$ naturally arise in representation theory of the Lorentz group. We express ${}pG{q}^{\mathbb C}(z)$ as quadratic expressions in the generalized hypergeometric functions ${}{p}F{q-1}$ and discuss further properties of the functions ${}pG{q}^{\mathbb C}(z)$.