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A Generalization of the Fox H-function

Jayme Vaz

TL;DR

The paper introduces the Fox-Barnes $I$-function, a generalization of the Fox $H$-function built from the double gamma function within a Mellin–Barnes contour framework. It derives existence conditions, contour selection rules, and a suite of properties, while also showing how the Fox $H$-function arises as a special case under specific parameter mappings. A key contribution is the explicit Laplace-transform representation of the Kilbas–Saigo function in terms of the Fox-Barnes $I$-function, including cases where the standard series diverges, illustrating practical analytic power. This work provides a new analytic tool for representing complex transforms and distributions in fractional and anomalous dynamics, with potential broad applications in mathematical analysis and applied probability.

Abstract

In this paper we present a generalization of the Fox H-function called Fox-Barnes I-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 I-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 I-function, even in cases where the usual series representation is not convergent.

A Generalization of the Fox H-function

TL;DR

The paper introduces the Fox-Barnes -function, a generalization of the Fox -function built from the double gamma function within a Mellin–Barnes contour framework. It derives existence conditions, contour selection rules, and a suite of properties, while also showing how the Fox -function arises as a special case under specific parameter mappings. A key contribution is the explicit Laplace-transform representation of the Kilbas–Saigo function in terms of the Fox-Barnes -function, including cases where the standard series diverges, illustrating practical analytic power. This work provides a new analytic tool for representing complex transforms and distributions in fractional and anomalous dynamics, with potential broad applications in mathematical analysis and applied probability.

Abstract

In this paper we present a generalization of the Fox H-function called Fox-Barnes I-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 I-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 I-function, even in cases where the usual series representation is not convergent.
Paper Structure (10 sections, 9 theorems, 107 equations, 1 figure)

This paper contains 10 sections, 9 theorems, 107 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The double gamma function
  3. The Fox-Barnes $\boldsymbol{I}$-function
  4. Existence
  5. A particular case: the Fox $\boldsymbol{H}$-function
  6. Existence
  7. Another possible particular case
  8. Properties
  9. Application: Laplace Transform of the KS function
  10. Conclusions

Key Result

Proposition 1

Consider the Fox-Barnes $I$-function as in eq.fox-barnes and eq.fox-barnes.integrand with $\varepsilon = 0$. Suppose that and Suppose also that there exists $m_0$ and $n_0$ with $1 \leq m_0 \leq m$ and $1 \leq n_0 \leq n$ such that Then we have where

Figures (1)

  • Figure 1: Plots of the Laplace transform $\mathcal{L}_t[ E_{a,m,l}\left(-\lambda t^\nu\right);z]$ obtained from eq.\ref{['laplace.expression']} for $\lambda=1$ and $\alpha$ and $\gamma$ as above.

Theorems & Definitions (10)

  • Definition 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Proposition 8
  • Proposition 9