Some properties of the Mittag-Leffler functions and their relation with the Wright functions

Muhammet Kurulay; Mustafa Bayram

doi:10.1186/1687-1847-2012-181

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Some properties of the Mittag-Leffler functions and their relation with the Wright functions

Muhammet Kurulay^{1, 2}Email author and
Mustafa Bayram²

Advances in Difference Equations20122012:181

https://doi.org/10.1186/1687-1847-2012-181

Received: 6 August 2012

Accepted: 4 October 2012

Published: 17 October 2012

Abstract

This paper is a short description of our recent results on an important class of the so-called Mittag-Leffler functions, which became important as solutions of fractional order differential and integral equations, control systems and refined mathematical models of various physical, chemical, economical, management and bioengineering phenomena. We have studied the Mittag-Leffler functions as their typical representatives, including many interesting special cases that have already proven their usefulness in fractional calculus and its applications. We obtained a number of useful relationships between the Mittag-Leffler functions and the Wright functions. The Wright function plays an important role in the solution of a linear partial differential equation. The Wright function, which we denote by $W (z; α, β)$ , is so named in honor of Wright who introduced and investigated this function in a series of notes starting from 1933 in the framework of the asymptotic theory of partitions.

MSC:33E12.

Keywords

Mittag-Leffler functions the Wright functions

1 Introduction

1.1 The Mittag-Leffler function

The Mittag-Leffler function is an important function that finds widespread use in the world of fractional calculus. Just as the exponential naturally arises out of the solution to integer order differential equations, the Mittag-Leffler function plays an analogous role in the solution of non-integer order differential equations. In fact, the exponential function itself is a very special form, one of an infinite set of these seemingly ubiquitous functions. The standard definition of Mittag-Leffler [1] is given as follows:

E_{α} (z) = \sum_{k = 0}^{\infty} \frac{z^{k}}{Γ (α k + 1)}, α \in C, R (α) > 0, z \in C .

(1)

A two-parameter function of the M-L (Mittag-Leffler) type is defined by the series expansion [2]

E_{α, β} (z) = \sum_{k = 0}^{\infty} \frac{z^{k}}{Γ (α k + β)} (α, β \in C, R (α) > 0, R (β) > 0, z \in C) .

(2)

The M-L function provides a simple generalization of the exponential function because of the substitution of

n! = Γ (n + 1) n!

with

(n α)! = Γ (n α + 1)

. Particular cases of (2) recover elementary functions are recovered

E_{1, 2} = \frac{e^{z} - 1}{z}, E_{2, 2} (z^{2}) = \frac{sinh (z)}{z},

and

E_{1 / 2, 1} = e^{z^{2}} erfc (- z),

where erf (erfc) denotes the (complementary) error function defined as [3]

erfc (z) = \frac{2}{\sqrt{π}} \int_{z}^{\infty} e^{- t^{2}} d t .

By means of the series representation, a generalization of (1) and (2) is introduced by Prabhakar [4] as

E_{α, β}^{γ} (z) = \sum_{k = 0}^{\infty} \frac{{(γ)}_{n}}{Γ (α k + β)} . \frac{z^{k}}{k!} (α, β, γ \in C, R (α) > 0, R (β), R (γ) > 0, z \in C),

(3)

where

{(γ)}_{n}

is the Pochhammer symbol [5] given by

{(λ)}_{n} = \frac{Γ (λ + n)}{Γ (λ)} = {\begin{matrix} 1 & (n = 0; λ \neq 0), \\ {(γ)}_{n} = γ (γ + 1) (γ + 2) \dots (γ + n - 1) & (n \in N; λ \in C) . \end{matrix}

Note that

E_{1, 1}^{1} (z) = e^{z}, E_{α, 1}^{1} (z) = E_{α} (z), E_{α, β}^{1} (z) = E_{α, β} (z) .

Some new properties of the Mittag-Leffler function, including a definite integral and recurrence relation, were investigated in [6, 7].

1.2 The Wright function

The Wright function plays an important role in the solution of a linear partial differential equation. The Wright function, which we denote by $W (z; α, β)$ , is so named in honor of Wright, who introduced and investigated this function in a series of notes starting from 1933 in the framework of the asymptotic theory of partitions. This function was introduced that related Mittag-Leffler [8–11]. We obtained a number of useful relationships between the Mittag-Leffler functions and the Wright functions.

1.2.1 Definition

The Wright function is defined by the series representation, convergent in the whole z-complex plane [12]

W (z; α, β) = \sum_{k = 0}^{\infty} \frac{z^{k}}{k! Γ (α k + β)} .

1.2.2 The integral representation of the Wright function

W (z; α, β) = \frac{1}{2 π i} \int_{H a} exp {u + z u^{- α}} u^{- β} d u,

where Ha denotes the Hankel path. To prove the Hankel path, let us write the integrated function in the form of a power series in z and perform term-by-term integration using the integral representation formula for the reciprocal gamma function

\frac{1}{Γ (z)} = \int_{H a} e^{ζ} ζ^{z} d ζ .

In fact,

\begin{array}{rcl} W (z; α, β) & = & \frac{1}{2 π i} \int_{H a} exp {u + z u^{- α}} u^{- β} d u = \frac{1}{2 π i} \int_{H a} e^{u} [\sum_{n = 0}^{\infty} \frac{z^{n}}{n!} u^{- α n}] u^{- β} d u \\ = & \sum_{n = 0}^{\infty} \frac{z^{n}}{n!} [\frac{1}{2 π i} \int_{H a} e^{u} u^{- α n - β} d u] = \sum_{n = 0}^{\infty} \frac{z^{n}}{n! Γ (α n + β)} . \end{array}

1.2.3 The Laplace transform of the Wright function

We recall that the Mittag-Leffler function plays fundamental roles in applications of fractional calculus like fractional relaxation and fractional oscillation [13–16]. Kiryakova introduced and studied the multi-index Mittag-Leffler functions as their typical representatives, including many interesting special cases that have already proven their usefulness in FC and its applications [17]. Srivatava and Tomovski introduced and investigated the fractional calculus with an integral operator which contains the following family of generalized Mittag-Leffler functions [18]. Haubold, Mathaian and Saxena studied the Mittag-Leffler functions and their applications [19]. There is an interesting link between the Wright function and the Mittag-Leffler function. We now point out that the Wright function is related to the Mittag-Leffler function through the following Laplace transform pair:

2 Some properties of the Mittag-Leffler functions

Theorem 1 (Derivative of the Mittag-Leffler function)

If

α, β, γ \in C

,

R (α) > 0

,

R (β) > 0

,

R (γ) > 0

,

z \in C

and

r, n \in N

, then

\frac{d^{n}}{d x^{n}} [z^{β - 1} E_{α, β + r α}^{γ} (z)] = z^{β - n - 1} E_{α, β + r α - n}^{γ} (z) .

Proof Using definition (3), we have that

\frac{d^{n}}{d x^{n}} [z^{β - 1} E_{α, β + r α}^{γ} (z)] = \sum_{k = 0}^{\infty} \frac{{(γ)}_{k}}{Γ (α (k + r) + β - n)} \frac{z^{k + β - 1 - n}}{k!} = z^{β - n - 1} E_{α, β + r α - n}^{γ} (z) .

The Wright function is expressed with help of the Mittag-Leffler function:

\frac{d^{n}}{d x^{n}} [z^{β - 1} E_{α, β + r α}^{γ} (z)] = \sum_{k = 0}^{\infty} z^{β - 1 - n} {(γ)}_{k} W (z; α, β + r α - n) .

□

Theorem 2 (Integration of the Mittag-Leffler function)

If

α, β, γ \in C

,

R (α) > 0

,

R (β) > 0

,

R (γ) > 0

,

z \in C

and

r \in N

, then

\int_{0}^{z} z^{β + r α - 1} E_{α, β + r α}^{γ} (z) d z = z^{β + r α} E_{α, β + r α + 1}^{γ} (z) .

Proof According to the Mittag-Leffler function, we have

z^{β + r α - 1} E_{α, β + r α}^{γ} (z) = \sum_{k = 0}^{\infty} \frac{{(γ)}_{k}}{Γ (α (k + r) + β)} . \frac{z^{k + r α + β - 1}}{k!} .

Integrating both sides gives

\begin{array}{rcl} \int_{0}^{z} z^{β + r α - 1} E_{α, β + r α}^{γ} (z) d z & = & \int_{0}^{z} \sum_{k = 0}^{\infty} \frac{{(γ)}_{k}}{Γ (α (k + r) + β)} . \frac{z^{k + r α + β - 1}}{k!} d z \\ = & \sum_{k = 0}^{\infty} \frac{{(γ)}_{k}}{Γ (α (k + r) + β + 1)} . \frac{z^{k + r α + β}}{k!} \\ = & z^{β + r α} E_{α, β + r α + 1}^{γ} (z) . \end{array}

Relation with the Wright functions is as follows:

\begin{array}{rcl} \int_{0}^{z} z^{β + r α - 1} E_{α, β + r α}^{γ} (z) d z & = & \int_{0}^{z} \sum_{k = 0}^{\infty} \frac{{(γ)}_{k}}{Γ (α (k + r) + β)} . \frac{z^{k + r α + β - 1}}{k!} d z \\ = & \sum_{k = 0}^{\infty} z^{β + r α} {(γ)}_{k} W (z; α, β + r α + 1) . \end{array}

□

Theorem 3 Let

α, β, γ \in C

,

R (α) > 0

,

R (β) > 0

,

R (γ) > 0

,

z \in C

,

r \in N

, then

E_{α, β}^{γ} (z) = β E_{α, β + 1}^{γ} (z) + α z \frac{d}{d z} E_{α, β + 1}^{γ} (z) .

Proof By definition (3), we have that

\begin{array}{rcl} β E_{α, β + 1}^{γ} (z) + α z \frac{d}{d z} E_{α, β + 1}^{γ} (z) & = & β E_{α, β + 1}^{γ} (z) + α z \frac{d}{d z} \sum_{n = 0}^{\infty} \frac{{(γ)}_{n}}{Γ (α n + β + 1)} . \frac{z^{n}}{n!} \\ = & \sum_{n = 0}^{\infty} \frac{β {(γ)}_{n}}{Γ (α n + β + 1)} . \frac{z^{n}}{n!} + \sum_{n = 0}^{\infty} \frac{α n {(γ)}_{n}}{Γ (α n + β + 1)} . \frac{z^{n}}{n!} \\ = & E_{α, β}^{γ} (z) . \end{array}

Relation with the Wright functions is as follows:

\begin{array}{rcl} β E_{α, β + 1}^{γ} (z) + α z \frac{d}{d z} E_{α, β + 1}^{γ} (z) & = & β E_{α, β + 1}^{γ} (z) + α z \frac{d}{d z} \sum_{n = 0}^{\infty} \frac{{(γ)}_{n}}{Γ (α n + β + 1)} . \frac{z^{n}}{n!} \\ = & \sum_{n = 0}^{\infty} \frac{β {(γ)}_{n}}{Γ (α n + β + 1)} . \frac{z^{n}}{n!} + \sum_{n = 0}^{\infty} \frac{α n {(γ)}_{n}}{Γ (α n + β + 1)} . \frac{z^{n}}{n!} \\ = & \sum_{n = 0}^{\infty} {(γ)}_{n} W (z; α, β) . \end{array}

□

Theorem 4 For

α, β, γ \in C

,

R (α) > 0

,

R (β) > 0

,

R (γ) > 0

,

z \in C

,

r \in N

. ‘Note that’

(4)

Proof We have

(5)

(6)

Equation (6) can be written as follows:

\begin{array}{rcl} E_{α, β + r α + 2}^{γ} (z) & = & \sum_{k = 0}^{\infty} [\frac{1}{α (k + r) + β} - \frac{1}{α (k + r) + β + 1}] \frac{z^{k} {(γ)}_{k}}{Γ (α (k + r) + β) k!} \\ = & E_{α, β + r α + 1}^{γ} (z) - \sum_{k = 0}^{\infty} \frac{z^{k} {(γ)}_{k}}{(α (k + r) + β + 1) Γ (α (k + r) + β) k!} . \end{array}

(7)

We find from equation (7) that

\begin{array}{rcl} E_{α, β + r α + 1}^{γ} (z) - E_{α, β + r α + 2}^{γ} (z) & = & \sum_{k = 0}^{\infty} \frac{z^{k} {(γ)}_{k}}{(α (k + r) + β + 1) Γ (α (k + r) + β) k!} \\ = & \frac{z^{k} {(γ)}_{k}}{k!} (\frac{1}{(α (k + r) + β + 2) (α (k + r) + β + 1) Γ (α (k + r) + β)} \\ + \frac{1}{(α (k + r) + β + 2) Γ (α (k + r) + β)}) \end{array}

(8)

or

(9)

We now say each summation on the right-hand side of equation (9) is as follows:

\begin{array}{rcl} \frac{d^{2}}{d z^{2}} {z^{2} E_{α, β + r α + 3}^{γ} (z)} & = & \sum_{k = 0}^{\infty} \frac{(k + 2) (k + 1) {(γ)}_{k}}{Γ (α (k + r) + β + 3)} . \frac{z^{k}}{k!} \\ = & \sum_{k = 0}^{\infty} \frac{k^{2} {(γ)}_{k}}{Γ (α (k + r) + β + 3)} . \frac{z^{k}}{k!} \\ + 3 \sum_{k = 0}^{\infty} \frac{k {(γ)}_{k}}{Γ (α (k + r) + β + 3)} . \frac{z^{k}}{k!} + 2 E_{α, β + r α + 3}^{γ} (z) \end{array}

(10)

or

(11)

We find from equation (11) that

\sum_{k = 0}^{\infty} \frac{k^{2} {(γ)}_{k}}{Γ (α (k + r) + β + 3)} . \frac{z^{k}}{k!} = z \frac{d}{d z} E_{α, β + r α + 3}^{γ} (z) + z^{2} \frac{d^{2}}{d z^{2}} E_{α, β + r α + 3}^{γ} (z) .

(12)

Using equations (9), (11), and (12), we get

\begin{matrix} E_{α, β + r α + 1}^{γ} (z) - E_{α, β + r α + 2}^{γ} (z) \\ = (β + r α) (β + r α + 2) E_{α, β + r α + 3}^{γ} (z) + α^{2} z^{2} \frac{d^{2}}{d z^{2}} E_{α, β + r α + 3}^{γ} (z) \\ + α {α + 2 + 2 (β + r α)} z \frac{d}{d z} E_{α, β + r α + 3}^{γ} (z) . \end{matrix}

Relation with the Wright functions is as follows:

\begin{matrix} E_{α, β + r α + 1}^{γ} (z) - E_{α, β + r α + 2}^{γ} (z) \\ = \sum_{k = 0}^{\infty} (β + r α) (β + r α + 2) {(γ)}_{k} W (z; α, β + r α + 3) \\ + \sum_{k = 0}^{\infty} α^{2} z^{2} {(γ)}_{k} \frac{d^{2}}{d z^{2}} W (z; α, β + r α + 3) \\ + \sum_{k = 0}^{\infty} α {α + 2 + 2 (β + r α)} {(γ)}_{k} z \frac{d}{d z} W (z; α, β + r α + 3) . \end{matrix}

□

Theorem 5 If

α, β, γ \in C

,

R (α) > 0

,

R (β) > 0

,

R (γ) > 0

,

z \in C

,

r \in N

, then

z^{r} E_{α, β + r α}^{γ} (z) = E_{α, β}^{γ} (z) - \sum_{k = 0}^{r - 1} \frac{{(γ)}_{k}}{Γ (k α + β)} \frac{z^{k}}{k!} .

(13)

Proof We have from (13)

\sum_{k = r}^{\infty} \frac{{(γ)}_{k}}{Γ (k α + β)} \frac{z^{k}}{k!} = E_{α, β}^{γ} (z) - \sum_{k = o}^{r - 1} \frac{{(γ)}_{k}}{Γ (k α + β)} \frac{z^{k}}{k!} .

For

r = 2, 3, 4, \dots

, we obtain

\begin{matrix} z^{2} E_{α, β + 2 α}^{γ} (z) = E_{α, β}^{γ} (z) - \frac{1}{Γ (β)} - \frac{{(γ)}_{1} z}{Γ (α + β)}, \\ z^{3} E_{α, β + 3 α}^{γ} (z) = E_{α, β}^{γ} (z) - \frac{1}{Γ (β)} - \frac{{(γ)}_{1} z}{Γ (α + β)} - \frac{{(γ)}_{2} z^{2}}{Γ (2 α + β)}, \\ z^{4} E_{α, β + 4 α}^{γ} (z) = E_{α, β}^{γ} (z) - \frac{1}{Γ (β)} - \frac{{(γ)}_{1} z}{Γ (α + β)} - \frac{{(γ)}_{2} z^{2}}{Γ (2 α + β)} - \frac{{(γ)}_{3} z^{3}}{Γ (3 α + β)}, \\ ⋮ \\ \sum_{k = r}^{\infty} \frac{{(γ)}_{k}}{Γ (k α + β)} \frac{z^{k}}{k!} = E_{α, β}^{γ} (z) - \sum_{k = o}^{r - 1} \frac{{(γ)}_{k}}{Γ (k α + β)} \frac{z^{k}}{k!} . \end{matrix}

Relation with the Wright functions is as follows:

\sum_{k = r}^{\infty} \frac{{(γ)}_{k}}{Γ (k α + β)} \frac{z^{k}}{k!} = \sum_{k = 0}^{\infty} {(γ)}_{k} W (z; α, β) - \sum_{k = 0}^{r - 1} {(γ)}_{k} W (z; α, β) .

□

Declarations

Acknowledgements

This work was supported by the scientific and technological research council of Turkey (TUBITAK).

Authors’ Affiliations

(1)

Department of Mathematics, University of Connecticut

(2)

Department of Mathematics, Faculty of Art and Sciences, Yildiz Technical University

References

Mittag-Leffler G:Sur la nouvelle fontion $E_{α}$ . Comptes Rendus Hebdomadaires Des Seances Del Academie Des Sciences, Paris 2 1903, 137: 554–558.Google Scholar
Wiman A: Über den fundamental Satz in der Theorie der Funktionen. Acta Math. 1905, 29: 191–201. 10.1007/BF02403202MathSciNetView ArticleMATHGoogle Scholar
Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.MATHGoogle Scholar
Prabhakar TR: A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J. 1971, 19: 7–15.MathSciNetMATHGoogle Scholar
Rainville ED: Special Functions. Macmillan, New York; 1960.MATHGoogle Scholar
Gupta IS, Debnath L: Some properties of the Mittag-Leffer functions. Integral Transforms Spec. Funct. 2007, 18(5):329–336. 10.1080/10652460601090216MathSciNetView ArticleMATHGoogle Scholar
Peng J, Li K: A note on property of the Mittag-Leffler function. J. Math. Anal. Appl. 2010, 370(2):635–638. 10.1016/j.jmaa.2010.04.031MathSciNetView ArticleMATHGoogle Scholar
Wright EM: On the coefficients of power series having exponential singularities. J. Lond. Math. Soc. 1933, 8: 71–79. 10.1112/jlms/s1-8.1.71View ArticleMathSciNetMATHGoogle Scholar
Wright EM: The asymptotic expansion of the generalized Bessel function. Proc. Lond. Math. Soc. 1935, 38: 257–270. 10.1112/plms/s2-38.1.257View ArticleMathSciNetMATHGoogle Scholar
Wright EM: The asymptotic expansion of the generalized hypergeometric function. J. Lond. Math. Soc. 1935, 10: 287–293.Google Scholar
Wright EM: The generalized Bessel function of order greater than one. Q. J. Math., Oxford Ser. 1940, 11: 36–48. 10.1093/qmath/os-11.1.36View ArticleMathSciNetMATHGoogle Scholar
Erdelyi A, Magnus W, Oberhettinger F, Tricomi FG 3. In Higher Transcendental Functions. McGraw-Hill, New York; 1954. chapter 18Google Scholar
Mainardi F, Mura A, Pagnini G: The M -Wright function in time-fractional diffusion processes: a tutorial survey. Int. J. Differ. Equ. 2010. doi:10.1155/2010/104505Google Scholar
Mainardi F, Gorenflo R: Time-fractional derivatives in relaxation processes: a tutorial survey. Fract. Calc. Appl. Anal. 2007, 10: 269–308.MathSciNetMATHGoogle Scholar
Achar BNN, Hanneken JW, Clarke T: Damping characteristics of a fractional oscillator. Physica A 2004, 339: 311–319. 10.1016/j.physa.2004.03.030MathSciNetView ArticleGoogle Scholar
Mainardi F: Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London; 2010.View ArticleMATHGoogle Scholar
Kiryakova V: The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus. Comput. Math. Appl. 2010, 59: 1885–1895. 10.1016/j.camwa.2009.08.025MathSciNetView ArticleMATHGoogle Scholar
Srivastava HM, Tomovski Ž: Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel. Appl. Math. Comput. 2009, 211: 198–210. 10.1016/j.amc.2009.01.055MathSciNetView ArticleMATHGoogle Scholar
Haubold HJ, Mathaiand AM, Saxena RK: Mittag-Leffler functions and their applications. J. Appl. Math. 2011., 2011: Article ID 298628. doi:10.1155/2011/298628Google Scholar

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