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A class of positive Fox H-functions

Filippo Giraldi

TL;DR

This work develops a framework for obtaining Fox $H$-functions that are strictly positive on $\mathbb{R}^+$ by leveraging Mellin-convolution products of nonnegative elementary functions built from stretched-exponential and power-law forms. Positivity is guaranteed under explicit existence-positivity (e.p.) conditions on the indexes and parameters, which align with the Mellin-transform structure of the $H$-function. By translating a Mellin-convolution construction into a Fox $H$-function with appropriate index choices, the paper also shows how positivity propagates through elementary transforms (Laplace, Euler) and through products of $H$-functions, enabling the generation of new positive instances. The results further connect these positive Fox $H$-functions to classical special functions, including Wright hypergeometric functions, MacRobert's $E$-functions, and Meijer $G$-functions, under explicit positivity criteria. Overall, the approach provides a constructive pathway to a broad class of positive densities and related objects in applications where Fox $H$-functions appear.

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 $\mathbb{R}^+$.

A class of positive Fox H-functions

TL;DR

This work develops a framework for obtaining Fox -functions that are strictly positive on by leveraging Mellin-convolution products of nonnegative elementary functions built from stretched-exponential and power-law forms. Positivity is guaranteed under explicit existence-positivity (e.p.) conditions on the indexes and parameters, which align with the Mellin-transform structure of the -function. By translating a Mellin-convolution construction into a Fox -function with appropriate index choices, the paper also shows how positivity propagates through elementary transforms (Laplace, Euler) and through products of -functions, enabling the generation of new positive instances. The results further connect these positive Fox -functions to classical special functions, including Wright hypergeometric functions, MacRobert's -functions, and Meijer -functions, under explicit positivity criteria. Overall, the approach provides a constructive pathway to a broad class of positive densities and related objects in applications where Fox -functions appear.

Abstract

The Fox -function is a special function which is defined via the Mellin-Barnes integrals and produces, as particular cases, Wright generalized hypergeometric functions, MacRobert's -functions and Meijer -functions, to name but few. Various cases of non-negative Fox -functions are obtained in literature by relying on the properties of integral transforms and the complete monotonicity. In the present scenario, Fox -functions, which are positive on , are determined via the Mellin convolution products of finite combinations, with possible repetitions, of elementary functions. The chosen elementary functions are non-negative on and are defined via stretched exponential and power laws. Further forms of positive Fox -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 -functions and Meijer -functions which are positive on .

Paper Structure

This paper contains 8 sections, 13 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. The Fox $H$-function
  3. The Mellin transform
  4. Positive Fox $H$-functions via Mellin convolution products
  5. Further forms of positive Fox $H$-functions
  6. Positive Fox $H$-functions via integral transforms
  7. Particular cases
  8. Summary and conclusions

Key Result

Theorem 4.1

The function $f: \mathbb{R}^+\to \mathbb{R}^+_0$, defined by Eq. (ftMellinConvTrunc) and constraints (cond_ab) - (cond_apppwv), is positive, $f(t)> 0$ for every $t>0$, for every $\left(n_1,n_2,n_3,n_4 \right)\in \mathbb{N}_0^4$, except for the following cases: $n_1,n_3,n_4=0$, and $n_2\geq 1$; or $n

Theorems & Definitions (25)

  • Theorem 4.1
  • proof
  • Proposition 4.2
  • proof
  • Theorem 4.3
  • proof
  • Corollary 5.1
  • proof
  • Corollary 5.2
  • proof
  • ...and 15 more