Barnes–Ismagilov Integrals and Hypergeometric Functions of the Complex Field

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 ${}_pG_{q}^{\mathbb C}(z)$.