The Fox H-function is a special function which is defined via the Mellin-Barnes integrals and produces, as particular cases, Mittag-Leffler functions, MacRobert’s E-functions, Wright generalized hypergeometric functions and Meijer G-functions. 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 the set of positive real numbers, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {R}^+$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>+</mo> </msup> </math></EquationSource> </InlineEquation>, or the interval <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\left. \right] 0,1\left[ \right. $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfenced close="]" /> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mfenced open="[" /> </mrow> </math></EquationSource> </InlineEquation>, or <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\left. \right] 1,+\infty \left[ \right. $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfenced close="]" /> <mn>1</mn> <mo>,</mo> <mo>+</mo> <mi>∞</mi> <mfenced open="[" /> </mrow> </math></EquationSource> </InlineEquation>, and vanish otherwise, are determined via the Mellin convolution products of finite combinations, with possible repetitions, of elementary functions. The chosen elementary functions are non-negative on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {R}^+$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>+</mo> </msup> </math></EquationSource> </InlineEquation>, and are defined via stretched exponential and power laws. Further forms of positive Fox H-functions can be obtained from the former via elementary properties and integral transforms. As particular cases, we determine forms of Wright generalized hypergeometric functions, MacRobert’s E-functions and Meijer G-functions which are positive on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {R}^+.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>+</mo> </msup> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation>

A class of positive Fox H-functions

期刊论文
发表日期：

2026年2月13日

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Filippo Giraldi

摘要

The Fox H-function is a special function which is defined via the Mellin-Barnes integrals and produces, as particular cases, Mittag-Leffler functions, MacRobert’s E-functions, Wright generalized hypergeometric functions and Meijer G-functions. 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 the set of positive real numbers, $\mathbb {R}^+$ $R^{+}$ , or the interval $\left. \right] 0,1\left[ \right. $ $(] 0, 1 [)$ , or $\left. \right] 1,+\infty \left[ \right. $ $(] 1, + \infty [)$ , and vanish otherwise, are determined via the Mellin convolution products of finite combinations, with possible repetitions, of elementary functions. The chosen elementary functions are non-negative on $\mathbb {R}^+$ $R^{+}$ , 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 $\mathbb {R}^+.$ $R^{+} .$