[Paper Review] On the Levy density function

Jung Hun Han|arXiv (Cornell University)|Feb 14, 2011

Fractional Differential Equations Solutions8 references3 citations

TL;DR

This paper derives the Lévy density function as the limit of a generalized Mittag-Leffler density function using fractional calculus and Mellin-Barnes integral representations. It identifies the 'Levy structure'—a specific gamma function ratio—as central to fractional differential equations and introduces a transformation linking ordinary and α-fractional spaces, enabling the mapping of exponential functions to α-exponential forms.

ABSTRACT

In this paper, we introduce the Levy density function as the limit of a generalized Mittag-Leffler density function. The fractional integral equation for the generalized Mittag-Leffler density function is also given. And the role of the Levy structure in the fractional calculus is described. Finally, a transformation is defined.

Motivation & Objective

• To establish the Lévy density function as the limiting case of a generalized Mittag-Leffler density function.
• To clarify the role of the Lévy structure in fractional calculus through H-function and Mellin-Barnes representations.
• To define a transformation that maps functions from ordinary space to α-fractional space, preserving structural analogies.
• To demonstrate how the limiting behavior of gamma and Pochhammer symbols leads to the explicit form of the Lévy density.

Proposed method

• Uses the generalized Mittag-Leffler function $ f(x) = x^{αγ-1} \gamma^{\gamma} E^{\gamma}_{(\alpha,\alpha\gamma)}(-\gamma x^{\alpha}) $ as a starting density function.
• Applies Laplace and Mellin transforms to derive integral representations, particularly the Mellin-Barnes form.
• Takes the limit as $ \gamma \to \infty $ in the Mellin-Barnes integral to extract the Lévy density structure.
• Identifies the key factor $ \frac{\Gamma(-\frac{s}{\alpha} + \frac{1}{\alpha})}{\u03b1 \Gamma(1-s)} $ as the 'Levy structure' in H-function representations.
• Introduces the 'Mathai transform' to map ordinary functions to α-fractional space, exemplified by transforming $ e^{-x} $ to $ x^{\alpha-1} E_{(\alpha,\alpha)}(-x^{\alpha}) $.
• Employs asymptotic analysis of the gamma function (Stirling's formula) and limits of Pochhammer symbols to justify convergence.

Experimental results

Research questions

• RQ1 How can the Lévy density function be derived as a limiting case of a generalized Mittag-Leffler density function?
• RQ2 What is the role of the H-function representation and Mellin-Barnes integral in characterizing the Lévy structure?
• RQ3 How does the pathway model and limiting behavior of parameters connect generalized Mittag-Leffler functions to the Lévy distribution?
• RQ4 What transformation maps functions in ordinary space to α-fractional space, and how does it preserve functional analogies?
• RQ5 In what way does the 'Levy structure' emerge naturally in fractional differential equations?

Key findings

• The Lévy density function is obtained as the limit of the generalized Mittag-Leffler density when $ \gamma \to \infty $, yielding the Mellin-Barnes integral $ \frac{1}{2\pi i} \oint \frac{\Gamma(-\frac{s}{\alpha} + \frac{1}{\alpha})}{\alpha \Gamma(1-s)} x^{-s} ds $.
• The factor $ \frac{\Gamma(-\u03b1^{-1}s + \alpha^{-1})}{\alpha \Gamma(1-s)} $ is defined as the 'Levy structure' and is essential in H-function representations of Lévy-type densities.
• The transformation defined in section 4 maps $ e^{-x} $ in ordinary space to $ x^{\alpha-1} E_{(\alpha,\alpha)}(-x^{\alpha}) $ in α-fractional space, preserving functional form under the limit.
• The limiting process $ \gamma \to \infty $, combined with asymptotic gamma function behavior, justifies the convergence of the generalized Mittag-Leffler function to the Lévy density.
• The paper shows that fractional differential equations naturally incorporate the Levy structure, suggesting its foundational role in fractional calculus.
• The Mathai transform is proposed as a formal tool to relate ordinary and α-fractional function spaces, with examples showing correspondence between exponential and α-exponential functions.