Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 072, 20 pages      arXiv:1910.10686      https://doi.org/10.3842/SIGMA.2020.072
Contribution to the Special Issue on Elliptic Integrable Systems, Special Functions and Quantum Field Theory

Barnes-Ismagilov Integrals and Hypergeometric Functions of the Complex Field

Yury A. Neretin abcd
a)  Wolfgang Pauli Institut, c/o Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria
b)  Institute for Theoretical and Experimental Physics, Moscow, Russia
c)  Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Russia
d)  Institute for Information Transmission Problems, Moscow, Russia

Received April 09, 2020, in final form July 17, 2020; Published online August 02, 2020

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

Key words: Mellin-Barnes integrals; Mellin transform; hypergeometric functions; Lorentz group.

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