Barnes-Ismagilov integrals and hypergeometric functions of the complex field

TL;DR

This paper introduces a new family of Mellin-Barnes type integrals related to hypergeometric functions of the complex field, with applications in Lorentz group representation theory.

Contribution

It expresses Barnes-Ismagilov integrals as quadratic forms of generalized hypergeometric functions and explores their properties.

Findings

Representation of integrals as hypergeometric quadratic forms

Connections to Lorentz group representations

Analysis of properties of complex hypergeometric functions

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