Barnes-Ismagilov integrals and hypergeometric functions of the complex field

We examine a family ${}_pG_{q}\big[\genfrac{}{}{0pt}{}{(a)}{(b)};z\big]$ 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 ${}_pG_q(z)$ as quadratic expressions in generalized hypergeometric functions ${}_pF_{q-1}$ and discuss further properties of functions ${}_pG_q(z)$. The talk is based on https://arxiv.org/abs/1910.10686