[Paper Review] On Mittag-Leffler function and associated polynomials

D. Babusci, G. Dattoli+1 more|arXiv (Cornell University)|Jun 15, 2012

Mathematical functions and polynomials2 references3 citations

TL;DR

This paper investigates the Mittag-Leffler function (MLF) and its connection to fractional calculus, introducing a new family of fractional heat polynomials and fractional Laguerre polynomials. Using operational calculus and integral representations, the authors derive generating functions, differential equations, and integral identities, establishing links between MLF, Wright functions, and classical orthogonal polynomials, with key results including closed-form expressions for integrals of $ E_\alpha(-x^2) $ and $ W_{\alpha,\beta}(-x^2) $.

ABSTRACT

The Mittag-Leffler function plays a role of central importance in the theory of fractional derivatives. In this brief note we discuss the properties of this function and its connection with the Wright-Bessel functions and with a new family of associated heat polynomials.

Motivation & Objective

• Address the growing need for deeper analytical and computational tools for fractional derivatives by studying the Mittag-Leffler function (MLF) and its associated polynomials.
• Establish connections between the MLF and Wright-Bessel functions, and define new families of fractional polynomials with applications in anomalous diffusion.
• Develop operational techniques to derive integral representations and generating functions for MLF-related polynomials.
• Generalize classical polynomial families (Hermite, Laguerre) to fractional orders using fractional derivatives and operational calculus.
• Provide closed-form expressions for integrals involving $ E_\alpha(-x^2) $ and $ W_{\alpha,\beta}(-x^2) $, extending Gaussian-type function analysis.

Proposed method

• Define the fractional heat polynomials $ {}_{\alpha}H_n(x,y) $ as solutions to fractional Fokker-Planck equations via the evolution operator $ E_\alpha(k_\alpha t^\alpha \partial_x^2) $.
• Derive the generating function $ \sum_{n=0}^\infty \frac{\xi^n}{n!} {}_{\alpha}H_n(x,y) = E_\alpha(\xi^2 y) e^{\xi x} $, placing them in the Appell family.
• Introduce an operational calculus formalism using a shift-like operator $ \hat{c} $ satisfying $ \hat{c}^\alpha \varphi(0) = \varphi(\alpha+1) $ for $ \varphi(\mu) = 1/\Gamma(\mu) $.
• Use the identity $ E_\alpha(-x^2) = \frac{1}{1 + \hat{c}^\alpha x^2} \varphi(0) $ to derive integral representations via the gamma function identity $ \frac{1}{A^\nu} = \frac{1}{\Gamma(\nu)} \int_0^\infty e^{-sA} s^{\nu-1} ds $.
• Define the Wright function $ W_{\alpha,\beta}(x) $ via $ \hat{c}^{\alpha-1} e^{\hat{c}^\beta x} \varphi(0) $, and derive its series expansion and differential properties.
• Construct fractional Laguerre polynomials $ {}_{\alpha}L_n(x,y) $ using a fractional Laguerre derivative $ {}_{\alpha}\hat{D}_{L,x} = -(1/\alpha) \partial_x^\alpha x \partial_x $, and derive their generating functions.

Experimental results

Research questions

• RQ1 How can the Mittag-Leffler function be systematically linked to fractional heat polynomials and their generating functions?
• RQ2 What are the integral properties of $ E_\alpha(-x^2) $, and how can they be derived using operational calculus?
• RQ3 How do fractional derivatives modify classical polynomial families such as Hermite and Laguerre, and what are the resulting generalized polynomials?
• RQ4 What is the operational calculus framework that enables the derivation of closed-form expressions for integrals and derivatives of MLF-related functions?
• RQ5 How do the new fractional polynomials relate to classical orthogonal polynomials and Wright functions?

Key findings

• The integral $ \int_{-\infty}^\infty E_\alpha(-x^2) dx = \frac{\pi}{\Gamma(1 - \alpha/2)} $ is derived using operational calculus and gamma function identities.
• The fractional heat polynomials $ {}_{\alpha}H_n(x,y) $ satisfy the recurrence $ \partial_x {}_{\alpha}H_n = n \, {}_{\alpha}H_{n-1} $ and a fractional differential equation involving $ \partial_y^\alpha $.
• The generating function $ \sum_{n=0}^\infty \frac{\xi^n}{n!} {}_{\alpha}H_n(x,y) = E_\alpha(\xi^2 y) e^{\xi x} $ confirms their membership in the Appell family.
• The fractional Laguerre polynomials $ {}_{\alpha}L_n(x,y) $ are defined via the exponential of the fractional derivative operator $ e^{y \, {}_{\alpha}\hat{D}_{L,x}} $, with generating function $ \sum_{n=0}^\infty t^n {}_{\alpha}L_n(x,y) = \frac{1}{1 - yt} E_\alpha\left( -\frac{x^\alpha t}{1 - yt} \right) $.
• The integral $ \int_{-\infty}^\infty W_{\alpha,\beta}(-x^2) dx = \frac{\sqrt{\pi}}{\Gamma(\alpha - \beta/2)} $ is established for the Wright function.
• The fractional derivative of the Wright function satisfies $ \partial_x^m W_{\alpha,\beta}(x) = W_{\alpha + m\beta, \beta}(x) $, generalizing the behavior of classical special functions.