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arXiv · 2502.10483

A class of positive Fox H-functions

Filippo Giraldi
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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 $\mathbb{R}^+$.

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Filippo Giraldi. 2025-02-14. A class of positive Fox H-functions. https://arxiv.org/abs/2502.10483

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