A class of positive Fox H-functions

TL;DR

This paper characterizes a class of positive Fox H-functions using Mellin convolution products of elementary functions, providing new forms of well-known special functions that are positive on the positive real axis.

Contribution

It introduces a novel approach to determine positive Fox H-functions through Mellin convolutions of elementary functions, expanding the understanding of their structure and positivity.

Findings

Identifies conditions for positivity of Fox H-functions on +

Derives new positive forms of Wright hypergeometric, MacRobert's E, and Meijer G-functions

Provides explicit constructions of positive Fox H-functions from elementary functions

Abstract

The Fox $H$-function is a special function which is defined via the Mellin-Barnes integrals and produces, as particular cases, Wright generalized hypergeometric functions, MacRobert's $E$-functions and Meijer $G$-functions, to name but few. Various cases of non-negative Fox $H$-functions are obtained in literature by relying on the properties of integral transforms and the complete monotonicity. In the present scenario, Fox $H$-functions, which are positive on $\mathbb{R}^+$, are determined via the Mellin convolution products of finite combinations, with possible repetitions, of elementary functions. The chosen elementary functions are non-negative on $\mathbb{R}^+$ and are defined via stretched exponential and power laws. Further forms of positive Fox $H$-functions can be obtained from the former via elementary properties and integral transforms. As particular cases, we determine forms of Wright generalized hypergeometric functions, MacRobert's $E$-functions and Meijer $G$-functions which are positive on $\mathbb{R}^+$.