A Generalization of the Fox H-function
Abstract
In this paper we present a generalization of the Fox H-function called Fox-Barnes J-function. Like the Fox H-function, it is defined as a contour integral in the complex plane, but instead of an integrand given by a ratio of products of gamma functions involving several parameters, we use a ratio of products of double gamma functions. We study the conditions for its existence and how to choose a contour of integration based on the involved parameters. We discuss how the Fox H-function appears as a particular case and prove some properties of the Fox-Barnes J-function. As an application, we show how the Laplace transform of the Kilbas-Saigo function can be conveniently written in terms of the Fox-Barnes J-function, even in cases where the usual series representation is not convergent.
Cite
@article{arxiv.2510.15920,
title = {A Generalization of the Fox H-function},
author = {Jayme Vaz},
journal= {arXiv preprint arXiv:2510.15920},
year = {2026}
}
Comments
32 pages, 4 figures. Revised version based on reviewers' comments. The denomination Fox-Barnes I-function in the first version has been changed to Fox-Barnes J-function to avoid confusion with other function also called I-function