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MITTAG-LEFFLER FUNCTIONS AND THEIR APPLICATIONS

H.J. HAUBOLDOffice for Outer Space Affairs, United Nations

Vienna International Centre, P.O. Box 500, A-1400 Vienna, Austria.and Centre for Mathematical Sciences, Pala Campus,

Arunapuram P.O., Pala, Kerala-686574, India.hans.haubold@unvienna.org

A.M. MATHAIDepartment of Mathematics and Statistics, McGill University

Montreal, Canada H3A 2K6and Centre for Mathematical Sciences, Pala Campus,

Arunapuram P.O., Pala, Kerala-686574, India.mathai@math.mcgill.ca

R.K. SAXENADepartment of Mathematics and Statistics, Jai Narain Vyas University

Jodhpur-342005, India.ram.saxena@yahoo.com

Abstract

Motivated essentially by the success of the applications of the Mittag-Leffler functions in many areas ofscience and engineering, the authors present in a unified manner, a detailed account or rather a brief surveyof the Mittag-Leffler function, generalized Mittag-Leffler functions, Mittag-Leffler type functions, and theirinteresting and useful properties. Applications of Mittag-Leffler functions in certain areas of physical andapplied sciences are also demonstrated. During the last two decades this function has come into prominenceafter about nine decades of its discovery by a Swedish Mathematician G.M. Mittag-Leffler, due its vastpotential of its applications in solving the problems of physical, biological, engineering and earth sciences etc.In this survey paper, nearly all types of Mittag-Leffler type functions existing in the literature are presented.An attempt is made to present nearly an exhaustive list of references concerning the Mittag-Leffler functionsto make the reader familiar with the present trend of research in Mittag-Leffler type functions and theirapplications.

1. Introduction

The special function

Eα(z) =

∞∑

k=0

zk

Γ(1 + αk), α ∈ C,ℜ(α) > 0, z ∈ C (1.1)

and its general form

Eα,β(z) =

∞∑

k=0

zk

Γ(β + αk), α, β ∈ C,ℜ(α) > 0,ℜ(β) > 0, z ∈ C (1.2)

with C being the set of complex numbers are called Mittag-Leffler functions (Erdelyi, et al., 1955, Section18.1). The former was introduced by Mittag-Leffler (Mittag-Leffler,1903), in connection with his methodof summation of some divergent series. In his papers (Mittag-Leffler, 1903, 1905), he investigated certainproperties of this function. The function defined by (1.2) first appeared in the work of Wiman (Wiman,1905). The function (1.2) is studied, among others, by Wiman (1905), Agarwal (1953), Humbert (1953)and Humbert and Agarwal (1953) and others. The main properties of these functions are given in the bookby Erdelyi et al. (1955, Section 18.1) and a more comprehensive and a detailed account of Mittag-Leffler

1

functions are presented in Dzherbashyan (1966, Chapter 2). In particular, the functions (1.1) and (1.2) areentire functions of order ρ = 1/α and type σ = 1; see, for example, (Erdelyi, et al., 1955, p.118).

The Mittag-Leffler function arises naturally in the solution of fractional order integral equations orfractional order differential equations, and especially in the investigations of the fractional generalizationof the kinetic equation, random walks, Levy flights, super-diffusive transport and in the study of complexsystems. The ordinary and generalized Mittag-Leffler functions interpolate between a purely exponentiallaw and power-law like behavior of phenomena governed by ordinary kinetic equations and their fractionalcounterparts, see Lang (1999a, 1999b), Hilfer (2000), Saxena et al. (2002).

The Mittag-Leffler function is not given in the tables of Laplace transforms, where it naturally occursin the derivation of the inverse Laplace transform of the functions of the type pα(a + bpβ), where p is theLaplace transform parameter and a and b are constants. This function also occurs in the solution of certainboundary value problems involving fractional integro-differential equations of Volterra type (Samko et al.,1993). During the various developments of fractional calculus in the last four decades this function hasgained importance and popularity on account of its vast applications in the fields of science and engineering.Hille and Tamarkin (1930) have presented a solution of the Abel-Volterra type equation in terms of Mittag-Leffler function. During the last 15 years the interest in Mittag-Leffler function and Mittag-Leffler typefunctions is considerably increased among engineers and scientists due to their vast potential of applicationsin several applied problems, such as fluid flow, rheology, diffusive transport akin to diffusion, electric networks,probability, statistical distribution theory etc. For a detailed account of various properties, generalizations,and application of this function, the reader may refer to earlier important works of Blair (1974), Bagley andTorvik (1984), Caputo and Mainardi (1971), Dzherbashyan (1966), Gorenflo and Vessella (1991), Gorenfloand Rutman (1994), Kilbas and Saigo (1995), Gorenflo et al. (1997), Gorenflo and Mainardi (1994, 1996,1997), Gorenflo, Luchko and Rogosin (1997), Gorenflo, Kilbas and Rogosin (1998), Luchko (1999), Luchkoand Srivastava (1995), Kilbas, Saigo and Saxena (2002, 2004), Saxena and Saigo (2005), Kiryakova (2008a,2008b), Saxena, Kalla and Kiryakova (2003), Saxena, Mathai and Haubold (2002, 2004, 2004a, 2004b, 2006),Saxena and Kalla (2008), Mathai, Saxena and Haubold (2006), Haubold and Mathai (2000), Haubold, Mathaiand Saxena (2007), Srivastava and Saxena (2001), and others.

This paper is organized as follows: Section 2 deals with special cases of Eα(z). Functional relations ofMittag-Leffler functions are presented in Section 3. Section 4 gives the basic properties. Section 5 is devotedto the derivation of recurrence relations for Mittag-Leffler functions. In Section 6, asymptotic expansionsof the Mittag-Leffler functions are given. Integral representations of Mittag-Leffler functions are given inSection 7. Section 8 deals with the H-function and its special cases. The Mellin-Barnes integrals for theMittag-Leffler functions are established in Section 9. Relations of Mittag-Leffler functions with Riemann-Liouville fractional calculus operators are derived in Section 10. Generalized Mittag-Leffler functions andsome of their properties are given in Section 11. Laplace transform, Fourier transform, and fractionalintegrals and derivatives are discussed in Section 12. Section 13 is devoted to the application of Mittag-Leffler function in fractional kinetic equations. In Section 14, time-fractional diffusion equation is solved.Solution of space-fractional diffusion equation is discussed in Section 15. In Section 16, solution of a fractionalreaction-diffusion equation is investigated in terms of the H-function. Section 17 is devoted to the applicationof generalized Mittag-Leffler functions in nonlinear waves. Recent generalizations of Mittag-Leffler functionsare discussed in Section 18.

2. Some special cases

We begin our study by giving the special cases of the Mittag-Leffler function Eα(z).

(i) E0(z) =1

1 − z, |z| < 1

(ii) E1(z) = ez

(iii) E2(z) = cosh(√z), z ∈ C

2

(iv) E2(−z2) = cos z, z ∈ C

(v) E3(z) =1

2

[

ez13 + 2e−

12 z

13 cos

(√3

2z

13

)]

, z ∈ C

(vi) E4(z) =1

2

[

cos(z14 ) + cosh(z

14 )]

, z ∈ C

(vii) E 12(±z 1

2 ) = ez[

1 + erf(±z 12 )]

= ezerfc(∓z 12 ), z ∈ C

where erfc denotes the complimentary error function and the error function is defines as

erf(z) =2√π

∫ z

0

exp(−t2)dt, erfc(z) = 1 − erf(z), z ∈ C.

For half-integer n/2 the function can be written explicitly as

(viii) En2(z) = 0Fn−1

(

:1

n,2

n, ...,

n− 1

n;z2

nn

)

+2(n+1)/2z

n!√π

1F2n−1

(

1;n+ 2

2n,n+ 3

2n, ...,

3n

2n;z2

nn

)

(ix) E1,2(z) =ez − 1

z, E2,2(z) =

sinh(√z)√

z.

3. Functional relations for the Mittag-Leffler functions

In this section, we discuss the Mittag-Leffler functions of rational order α = m/n, with m,n ∈ Nrelatively prime. The differential and other properties of these functions are described in Erdelyi, et al.(1955)and Dzherbashyan (1966).

Theorem 3.1. The following results hold:

dm

dzmEm(zm) = Em(zm) (3.1)

dm

dzmEm

n(z

mn ) = Em

n(z

mn ) +

n−1∑

r=1

z−rmn

Γ(1 − rm/n), n = 2, 3, ... (3.2)

Emn

(z) =1

m

m−1∑

r=1

E 1n(z

1m exp(i2πr/m)) (3.3)

E 1n(z

1n ) = ez

[

1 +

n−1∑

r=1

γ(1 − rn , z)

Γ(1 − r/n)

]

, n = 2, 3, ... (3.4)

where γ(a, z) denotes the incomplete gamma function, defined by,

γ(a, z) =

∫ z

0

e−tta−1dt. (3.5)

3

In order to establish the above formulas, we observe that (3.1) and (3.2) readily follow from the definition(1.2). For proving the formula (3.3), we recall the identity

m−1∑

r=0

exp[i2πkr/m] =

m if k = 0 (mod m)0 if k 6= 0 (mod m).

(3.6)

By virtue of the results (1.1) and (3.6), we find that

m−1∑

r=0

Eα(zei2πr/m) = mEαm(zm),m ∈ N (3.7)

which can be written as

Eα(z) =1

m

m−1∑

r=0

E αm

(z1m ei2πr/m), m ∈ N (3.8)

and the result (3.3) now follows by taking α = m/n. To prove the relation (3.4), we set m = 1 in (3.1) andmultiply it by exp(−z) to obtain

d

dz

[

e−zE 1n(z

1n )]

= e−zm−1∑

r=1

z−r/m

Γ(1 − r/m). (3.9)

On integrating both sides of the above equation with respect to z and using the definition of incompletegamma function (3.5), we obtain the desired result (3.4). An interesting case of (3.8) is given by

E2α(z2) =1

2[Eα(z) + Eα(−z)]. (3.10)

4. Basic properties

This section is based on the paper of Berberan-Santos (2005). From (1.1) and (1.2) it is not difficult toprove that

Eα(−x) = E2α(x2) − xE2α,1+α(x2) (4.1)

and

Eα(−ix) = E2x(−x2) − ixE2α,1+α(−x2), i =√−1. (4.2)

It is shown in Berberan-Santos (2005, p.631) that the following three equations can be used for the directinversion of a function I(x) to obtain its inverse H(k):

H(k) =eck

π

∫ ∞

0

[ℜ[I(c+ iω)] cos(kω) −ℑ[I(c+ iω)] sin(kω)dω (4.3)

=2eck

π

∫ ∞

0

ℜ[I(c+ iω) cos(kω)dω, k > 0 (4.4)

= −2eck

π

∫ ∞

0

ℑ[I(c+ iω)] sin(kω)dω, k > 0. (4.5)

With the help of the results (4.2) and (4.4), it yields the following formula for the inverse Laplace transformH(k) of the function Eα(−x).

Hα(k) =2

π

∫ ∞

0

E2α(−t2) cos(kt)dt, k > 0, 0 ≤ α ≤ 1. (4.6)

4

In particular, the following interesting results can be derived from the above result.

H1(k) =2

π

∫ ∞

0

cosh(it) cos(kt)dt =2

π

∫ ∞

0

cos(t) cos(kt)dt = δ(k − 1), i =√−1 (4.7)

H 12(k) =

2

π

∫ ∞

0

e−t2 cos(kt)dt =1√π

e−k2

4 (4.8)

H 14(k) =

2

π

∫ ∞

0

et2erfc(t2) cos(kt)dt. (4.9)

Another integral representation of Hα(k) in terms of the Levy one-sided stable distribution Lα(k) was givenby Pllard (1948) in the form

Hα(k) =1

αk−(1+ 1

α)Lα(k−

1α ). (4.10)

The inverse Laplace transform of Eα(−xβ), denoted by Hβα(k) with 0 < α ≤ 1, is obtained as

Hβα(k) =

∫ ∞

0

tαβ Lα(t)Lβ(kt

αβ )dt, (4.11)

where Lα(t) is the one-sided Levy probability density function. From Berberan-Santos (2005, p.432) we have

Hα(k) =1

π

∫ ∞

0

[E2α(−ω2) cos(kω) + ωE2α,1+α(−ω2) sin(kω)]dω, 0 < α ≤ 1. (4.12)

Expanding the above equation in a power series, it gives

Hα(k) =1

π

∞∑

n=0

bn(α)kn, 0 ≤ α < 1 (4.13)

with

b0(α) =

∫ ∞

0

E2α(−t2)dt. (4.14)

The Laplace transform of the equation (4.13) is the asymptotic expansion of Eα(−x) as

Eα(−x) =1

π

∞∑

n=0

bn(α)

xn+1, 0 ≤ α < 1. (4.15)

5. Recurrence relations

By virtue of the definition (1.2), the following relations are obtained in the form of

Theorem 5.1. We have

Eα,β(z) = zEα,α+β(z) +1

Γ(β)(5.1)

Eα,β(z) = βEα,β+1(z) + αzd

dzEα,β+1(z) (5.2)

dm

dzm

[

zβ−1Eα,β(zα)]

= zβ−m−1Eα,β−m(zα),ℜ(β −m) > 0,m ∈ N. (5.3)

d

dzEα,β(z) =

Eα,β−1(z) − (β − 1)Eα,β(z)

αz. (5.4)

5

The above formulae are useful in computing the derivative of the Mittag-Leffler function Eα,β(z). Thefollowing theorem has been established by Saxena (2002):

Theorem 5.2. If ℜ(α) > 0,ℜ(β) > 0 and r ∈ N then there holds the formula

zrEα,β+rα(z) = Eα,β(z) −r−1∑

n=0

zn

Γ(β + nα). (5.5)

Proof. We have from the right side of (5.5),

Eα,β(z) −r−1∑

n=0

zn

Γ(β + nα)=

∞∑

n=r

zn

Γ(β + nα).

Put n− r = k or n = k + r. Then

∞∑

n=r

zn

Γ(β + nα)=

∞∑

k=0

zk+r

Γ(β + rα + kα)

= zrEα,β+rα(z).

For r = 2, 3, 4 we obtain the following corollaries:

Corollary 5.1. If ℜ(α) > 0,ℜ(β) > 0 then there holds the formula

z2Eα,β+2α(z) = Eα,β(z) − 1

Γ(β)− z

Γ(α+ β). (5.6)

Corollary 5.2. If ℜ(α) > 0,ℜ(β) > 0 then there holds the formula

z3Eα,β+3α(z) = Eα,β(z) − 1

Γ(β)− z

Γ(α+ β)− z2

Γ(2α+ β). (5.7)

Corollary 5.3. If ℜ(α) > 0,ℜ(β) > 0 then there holds the formula

z4Eα,β+4α(z) = Eα,β(z) − 1

Γ(β)− z

Γ(α+ β)− z2

Γ(2α+ β)− z3

Γ(3α+ β). (5.8)

Remark 5.1. For a generalization of the result (5.5), see Saxena, Kalla and Kiryakova (2003).

6. Asymptotic expansions

The asymptotic behavior of Mittag-Leffler functions plays a very important role in the interpretation ofthe solution of various problems of physics connected with fractional reaction, fractional relaxation, fractionaldiffusion and fractional reaction-diffusion etc in complex systems. The asymptotic expansion of Eα(z) isbased on the integral representation of the Mittag-Leffler function in the form

Eα(z) =1

2πi

∫

Ω

tα−1 exp(t)

tα − zdt,ℜ(α) > 0, α, z ∈ C, (6.1)

where the path of integration Ω is a loop starting and ending at −∞ and encircling the circular disk |t| ≤ |z| 1α

in the positive sense, | arg t| < π on Ω. The integrand has a branch point at t = 0. The complex t-plane iscut along the negative real axis and in the cut plane the integrand is single-valued, the principal branch oftα is taken in the cut plane. (6.1) can be proved by expanding the integrand in powers of t and integrating

6

term by term by making use of the well-known Hankel’s integral for the reciprocal of the gamma function,namely

1

Γ(β)=

1

2πi

∫

Ha

eζ

ζβdζ. (6.2)

The integral representation (6.1) can be used to obtain the asymptotic expansion of the Mittag-Lefflerfunction at infinity (Erdelyi, et at., 1955). Accordingly, the following cases are mentioned below: (i): If0 < α < 2 and µ is a real number such that

πα

2< µ < min[π, πα], (6.3)

then for N∗ ∈ N,N∗ 6= 1 there holds the following asymptotic expansion:

Eα(z) =1

αz(1−β)/α exp(z

1α ) −

N∗

∑

r=1

1

Γ(1 − αr)

1

zr+O

[

1

zN∗+1

]

, (6.4)

as |z| → ∞, | arg z| ≤ µ; and

Eα(z) = −N∗

∑

r=1

1

Γ(1 − αr)

1

zr+O

[

1

zN∗+1

]

, (6.5)

as |z| → ∞, µ ≤ | arg z| ≤ π. (ii) : When α ≥ 2 then there holds the following asymptotic expansion:

Eα(z) =1

α

∑

n

z1n exp[exp(

2nπi

α)z

1α ] −

N∗

∑

r=1

1

Γ(1 − αr)

1

zr+O

[

1

zN∗+1

]

(6.6)

as |z| → ∞, | arg z| ≤ απ2 , and where the first sum is taken over all integers n such that

| arg(z) + 2πn| ≤ απ

2. (6.7)

The asymptotic expansion of Eα,β(z) is based on the integral representation of the Mittag-Leffler functionEα,β(z) in the form

Eα,β(z) =1

2πi

∫

Ω

tα−β exp(t)

tα − zdt,ℜ(α) > 0,ℜ(β) > 0, z, α, β ∈ C, (6.8)

which is an extension of (6.1) with the same path. As in the previous case, the Mittag-Leffler function hasthe following asymptotic estimates: (iii): If 0 < α < 2 and µ is a real number such that

πα

2< µ < min[π, πα], (6.9)

then there holds the following asymptotic expansion:

Eα,β(z) =1

αz(1−β)/α exp(z

1α ) −

N∗

∑

r=1

1

Γ(β − αr)

1

zr+O

[

1

zN∗+1

]

(6.10)

as |z| → ∞, | arg z| ≤ µ; and

Eα,β(z) = −N∗

∑

r=1

1

Γ(β − αr)

1

zr+O

[

1

zN∗+1

]

, (6.11)

as |z| → ∞, µ ≤ | arg z| ≤ π. (iv): When α ≥ 2 then there holds the following asymptotic expansion:

Eα,β(z) =1

α

∑

n

z1n exp[exp(

2nπi

α)z

1α ]1−β −

N∗

∑

r=1

1

Γ(β − αr)

1

zr+O

[

1

zN∗+1

]

, (6.12)

7

as |z| → ∞, | arg z| ≤ απ2 and where the first sum is taken over all integers n such that

| arg(z) + 2πn| ≤ απ

2. (6.13)

7. Integral representations

In this section several integrals associated with Mittag-Leffler functions are presented, which can beeasily established by the application by means of beta and gamma function formulas and other techniques,see Erdelyi, et al. (1955), Gorenflo et al. (1997, 2002).

∫ ∞

0

e−ζEα(ζαz)dζ =1

1 − z, |z| < 1, α ∈ C,ℜ(α) > 0 (7.1)

∫ ∞

0

e−xxβ−1Eα,β(xαz)dx =1

1 − z, |z| < 1, α, β ∈ C,ℜ(α) > 0,ℜ(β) > 0 (7.2)

∫ x

0

(x− ζ)β−1Eα(ζα)dζ = Γ(β)xβEα,β+1(xα),ℜ(β) > 0 (7.3)

∫ ∞

0

e−sζEα(−ζα)dζ =sα−1

1 + sα,ℜ(s) > 0 (7.4)

∫ ∞

0

e−sζζmα+β−1E(m)α,β (±aζα)dζ =

m!sα−β

(sα ∓ a)m+1,ℜ(s) > 0,ℜ(α) > 0,ℜ(β) > 0, (7.5)

where α, β ∈ C and

E(m)α,β (z) =

dm

dzmEα,β(z)

Eα(−xα) =2

πsin(απ/2)

∫ ∞

0

ζα−1 cos(xζ)

1 + 2ζα cos(απ/2) + ζ2αdζ, α ∈ C,ℜ(α) > 0 (7.6)

Eα(−x) =1

πsin(απ)

∫ ∞

0

ζα−1

1 + 2ζα cos(απ) + ζ2αe−ζx

1α ζ dζ, α ∈ C,ℜ(α) > 0 (7.7)

Eα(−x) = 1 − 1

2α+x

1α

π

∫ ∞

0

arctan

[

ζα + cos(απ)

sin(απ)

]

e−ζx1α ζ dζ, α ∈ C,ℜ(α) > 0. (7.8)

Note 7.1. Equation (7.7) can be employed to compute the numerical coefficients of the leading term of theasymptotic expansion of Eα(−x). Equation (7.8) yields

b0(α) =α

πΓ(α) sin(2απ)

∫ ∞

0

ζα−1

1 + 2ζ2α cos(2απ) + ζ4αdζ, α <

1

2. (7.9)

From Berberan-Santos (2005) and Gorenflo et al.(1997) the following results hold:

Eα(−x) =2x

π

∫ ∞

0

E2α(−t2)x2 + t2

dt, 0 ≤ α ≤ 1, α ∈ C. (7.10)

In particular, the following cases are of importance

E1(−x) =2x

π

∫ ∞

0

cosh(it)

x2 + t2dt = exp(−x) (7.11)

E 12(−x) =

2x

π

∫ ∞

0

exp(−t2)x2 + t2

dt = ex2

erfc(x). (7.12)

E 14(−x) =

2x

π

∫ ∞

0

et2erfc(−t2)x2 + t2

dt. (7.13)

8

Note 7.2. Some new properties of the Mittag-Leffler functions are recently obtained by Gupta and Debnath(2007).

8. The H-function and its special cases

The H-function is defined by means of a Mellin-Barnes type integral in the following manner (Mathaiand Saxena, 1978):

Hm,np,q (z) = Hm,n

p,q

[

z

∣

∣

∣

∣

(ap,Ap)

(bq,Bq)

]

= Hm,np,q

[

z

∣

∣

∣

∣

(a1,A1),...,(ap,Ap)

(b1,B1),...,(bq,Bq)

]

=1

2πi

∫

Ω

Θ(ζ)z−ζdζ (8.1)

where i =√

(−1) and

Θ(s) =

∏mj=1 Γ(bj + sBj)

∏nj=1 Γ(1 − aj − sAj)

∏qj=m+1 Γ(1 − bj − sBj)

∏pj=n+1 Γ(aj + sAj)

, (8.2)

and an empty product is interpreted as unity; m,n, p, q ∈ N0 with 0 ≤ n ≤ p, 1 ≤ m ≤ q, Ai, Bj ∈ R+, ai, bj ∈C, i = 1, ..., p; j = 1, ..., q such that

Ai(bj + k) 6= Bj(ai − λ− 1), k, λ ∈ N0; i = 1, ..., n; j = 1, ...,m, (8.3)

where we employ the usual notations: N0 = 0, 1, 2, ...;R = (−∞,∞), R+ = (0,∞), and C being the complexnumber field. The contour Ω is the infinite contour which separates all the poles of Γ(bj + sBj), j = 1, ...,mfrom all the poles of Γ(1−ai +sAi), i = 1, ..., n. The contour Ω could be Ω = L−∞ or Ω = L+∞ or Ω = Liγ∞

where L−∞ is a loop starting at −∞ encircling all the poles of Γ(bj + sBj), j = 1, ...,m and ending at −∞.L+∞ is a loop starting at +∞, encircling all the poles of Γ(1−ai−sAi), i = 1, ..., n and ending at +∞. Liγ∞

is the infinite semicircle starting at γ − i∞ and going to γ + i∞. A detailed and comprehensive account ofthe H-function is available from the monographs of Mathai and Saxena (1978), Prudnikov et al. (1990) andKilbas and Saigo (2004). The relation connecting the Wright’s function pψq(z) and the H-function is givenfor the first time in the monograph of Mathai and Saxena (1978, p.11, eq(1.7.8)) as

pψq

[

(a1, A1), ..., (ap, Ap)(b1, B1), ..., (bq, Bq)

|z]

= H1,pp,q+1

[

−z∣

∣

∣

∣

(1−a1.A1),...,(1−ap,Ap)

(0,1),(1−b1,B1),...,(1−bq,Bq)

]

, (8.4)

where pψq(z) is the Wright’s generalized hypergeometric function (Wright, 1935, 1940); also see Erdelyi, etal. (1953, Section 4.1), defined by means of the series representation in the form

pψq(z) =

∞∑

r=0

∏pj=1 Γ(aj +Ajr)

∏qj=1 Γ(bj + Bjr)

zr

r!(8.5)

where z ∈ C, ai, bj ∈ C,Ai, Bj ∈ R+, Ai 6= 0, Bj 6= 0; i = 1, ..., p; j = 1, ..., q,

q∑

j=1

Bj −p∑

i=1

Ai = ∆ > −1.

The Mellin-Barnes contour integral for the generalized Wright function is given by

pψq

[

(ap, Ap)(bq, Bq)

|z]

=1

2πi

∫

Ω

Γ(s)∏p

j=1 Γ(aj −Ajs)∏q

j=1 Γ(bj − sBj)(−z)−sds (8.6)

where the path of integration separates all the poles of Γ(s) at the points s = −ν, ν ∈ N0 lying to the leftand all the poles of

∏pj=1 Γ(aj − sAj), j = 1, ..., p at the points s = (Aj + νj)/Aj , νj ∈ N0, j = 1, ..., p lying

9

to the right. If Ω = (γ − i∞, γ + i∞) then the above representation is valid if either of the conditions aresatisfied:

(i) ∆ < 1, | arg(−z)| < (1 − ∆)π

2, z 6= 0,

(ii) ∆ = 1, (1 + ∆)γ +1

2< ℜ(δ), arg(−z) = 0, z 6= 0, δ =

q∑

j=1

bj −p∑

j=1

aj +p− q)

2.

This result was proved by Kilbas, Saigo and Trujillo (2002).

The generalized Wright function includes many special functions besides the Mittag-Leffler functionsdefined by the equations (1.1) and (1.2). It is interesting to observe that for Ai = Bj = 1, i = 1, ..., p; j =1, ..., q, (8.5) reduces to a generalized hypergeometric function pFq(z). Thus

pψq

[

(ap, 1)(bq, 1)

|z]

=

∏pj=1 Γ(aj)

∏qj=1 Γ(bj)

pFq(a1, ..., ap; b1, ..., bq; z) (8.7)

where aj 6= −v, j = 1, ..., p, v = 0, 1, ...; bj 6= −λ, j = 1, ..., q, λ = 0, 1, ...; p ≤ q or p = q + 1, |z| < 1. Wright(1933) introduced a special case of (8.5) in the form

φ(a, b; z) = 0ψ1

[

(b, a)|z]

=

∞∑

r=0

1

Γ(ar + b)

zr

r!, (8.8)

which widely occurs in problems of fractional diffusion. It has been shown by Saxena, Mathai and Haubold(2004a), also see Kiryakova (1994), that

Eα,β(z) = 1ψ1

[

(1, 1)(β, α)

|z]

= H1,11,2

[

−z∣

∣

∣

∣

(0,1)

(0,1),(1−β,α)

]

. (8.9)

If we further take β = 1 in (8.9) we find that

Eα,1(z) = Eα(z) = 1ψ1

[

(1, 1)(1, α)

|z]

= H1,11,2

[

−z∣

∣

∣

∣

(0,1)

(0,1),(0,α)

]

(8.10)

where α ∈ C,ℜ(α) > 0.

Remark 8.1. A series of papers are devoted to the application of the Wright function in partial differentialequation of fractional order extending the classical diffusion and wave equations. Mainardi (1997) hasobtained the result for a fractional diffusion wave equation in terms of the fractional Green function involvingthe Wright function. The scale-variant solutions of some partial differential equations of fractional order wereobtained in terms of special cases of the generalized Wright function by Buckwar and Luchko (1998) andLuchko and Gorenflo (1998).

9. Mellin-Barnes integrals for Mittag-Leffler functions

These integrals can be obtained from the identities (8.9) and (8.10).

Lemma 9.1. If ℜ(α) > 0,ℜ(β) > 0 and z ∈ C the following representations are obtained:

Eα(z) =1

2πi

∫ γ+i∞

γ−i∞

Γ(s)Γ(1 − s)

Γ(1 − αs)(−z)−sds (9.1)

Eα,β(z) =1

2πi

∫ γ+i∞

γ−i∞

Γ(s)Γ(1 − s)

Γ(β − αs)(−z)−sds (9.2)

10

where the path of integration separates all the poles of Γ(s) at the points s = −ν, ν = 0, 1, ... from those ofΓ(1 − s) at the points s = 1 + v, v = 0, 1, ....

On evaluating the residues at the poles of the gamma function Γ(1− s) we obtain the following analyticcontinuation formulas for the Mittag-Leffler functions:

Eα(z) =1

2πi

∫ γ+i∞

γ−i∞

Γ(s)Γ(1 − s)

Γ(1 − αs)(−z)−sds = −

∞∑

k=1

z−k

Γ(1 − αk)(9.3)

and

Eα,β(z) =1

2πi

∫ γ+i∞

γ−i∞

Γ(s)Γ(1 − s)

Γ(β − αs)(−z)−sds = −

∞∑

k=1

z−k

Γ(β − αk). (9.4)

10. Relation with Riemann-Liouville fractional calculus operators

In this section, we present the relations of Mittag-Leffler functions with the left and right-sided operatorsof Riemann-Liouville fractional calculus, which are defined below.

(Iα0+φ)(x) =

1

Γ(α)

∫ x

0

(x− t)α−1φ(t)dt,ℜ(α) > 0 (10.1)

(Iα−)(x) =

1

Γ(α)

∫ ∞

x

(t− x)α−1φ(t)dt,ℜ(α) > 0 (10.2)

(Dα0+φ)(x) =

(

d

dx

)[α]+1

[I1−α0+

φ](x)

=1

Γ(1 − α)

(

d

dx

)[α]+1 ∫ x

0

(x− t)α−1φ(t)dt,ℜ(α) > 0 (10.3)

(Dα−φ)(x) =

(

− d

dx

)[α]+1

[I1−α− φ](x)

=1

Γ(1 − α)

(

− d

dx

)[α]+1 ∫ ∞

x

(t− x)−αφ(t)dt,ℜ(α) > 0 (10.4)

where [α] means the maximal integer not exceeding α and α is the fractional part of α.

Note 10.1. The fractional integrals (10.1) and (10.2) are connected by the relation (Kilbas, 2005, p.118)

[Iα−φ(1/t)](x) = xα−1

(

Iα0+

[t−α−1φ(t)])

(

1

x

)

.

Theorem 10.1. Let ℜ(α) > 0 and ℜ(β) > 0 then there holds the formulas

(Iα0+

[tα−1Eα,β(atα)])(x) =xβ−1

a

[

Eα,β(axα) − 1

Γ(β)

]

, a 6= 0 (10.5)

(Iα0+

[Eα(atα)])(x) =1

a[Eα(axα) − 1] , a 6= 0 (10.6)

which by virtue of the definitions (1.1) and (1.2) can be written as

(Iα0+

[tβ−1Eα,β(atα)])(x) = xα+β−1Eα,α+β(axα) (10.7)

(Iα0+

[Eα(atα)])(x) = xαEα,α+1(axα). (10.8)

11

Theorem 10.2. Let ℜ(α) > 0 and ℜ(β) > 0 then there holds the formulas

(Iα−[t−α−βEα,β(at−α)])(x) =

xα−β

a

[

Eα,β(ax−α) − 1

Γ(β)

]

, a 6= 0 (10.9)

(Iα−[t−α−1Eα(at−α)])(x) =

1

axα−1[Eα(ax−α) − 1], a 6= 0. (10.10)

Theorem 10.3. Let 0 < ℜ(α) < 1 and ℜ(β) > ℜ(α) then there holds the formulas

(Dα0+

[tβ−1Eα,β(atα)])(x) =xβ−α−1

Γ(β − α)+ axβ−1Eα,β(axα) (10.11)

(Dα0+

[Eα(atα)])(x) =x−α

Γ(1 − α)+ aEα(axα). (10.12)

Theorem 10.4. Let ℜ(α) > 0 and ℜ(β) > [ℜ(α)] + 1 then there holds the formula

(Dα−[tα−βEα,β(at−α)])(x) =

x−β

Γ(β − α)+ ax−α−βEα,β(ax−α). (10.13)

11. Generalized Mittag-Leffler type functions

By means of the series representation a generalization of (1.1) and (1.2) is introduced by Prabhakar(1971) as

Eγα,β(z) =

∞∑

n=0

(γ)n

n!Γ(β + αn), α, β, γ ∈ C,ℜ(α) > 0,ℜ(β) > 0, (11.1)

where

(γ)n = γ(γ + 1)...(γ + n− 1) =Γ(γ + n)

Γ(γ)

whenever Γ(γ) is defined, (γ)0 = 1, γ 6= 0. It is an entire function of order ρ = [ℜ(α)]−1 and type σ =1ρ

[

ℜ(α)ℜ(α)]−ρ

. It is a special case of Wright’s generalized hypergeometric function, Wright (1934,1935)

as well as the H-function (Mathai and Saxena, 1978). For various properties of this function with applications,see Prabhakar (1971). Some special cases of this function are enumerated below.

Eα(z) = E1α,1(z). (11.2)

Eα,β(z) = E1α,β(z). (11.3)

αγEγ+1α,β (z) = (1 + αγ − β)Eγ

α,β(z) + Eγα,β−1(z). (11.4)

φ(α, β; z) = Γ(β)Eα1,β(z) (11.5)

where φ(α, β; z) is the Kummer’s confluent hypergeometric function. Eγα,β(z) has the following representa-

tions in terms of the Wright’s function and H-function.

Eγα,β(z) =

1

Γ(γ)1ψ1

[

(γ, 1)(β, α)

; z

]

(11.6)

=1

Γ(γ)H1,1

1,2

[

−z∣

∣

∣

∣

(1−γ,1)

(0,1),(1−β,α)

]

(11.7)

=1

2πωΓ(γ)

∫ c+i∞

c−i∞

Γ(s)Γ(γ − s)

Γ(β − αs)(−z)−sds,ℜ(γ) > 0 (11.8)

12

where 1ψ1(·) and H1,11,2 (·) are respectively Wright generalized hypergeometric function and the H-function. In

the Mellin-Barnes integral representation, ω =√−1 and the c in the contour is such that 0 < c < ℜ(γ) and

it is assumed that the poles of Γ(s) and Γ(γ − s) are separated by the contour. The following two theoremsare given by Kilbas, Saigo and Saxena (2004).

Theorem 11.1. If α, β, γ, a ∈ C,ℜ(α) > 0,ℜ(β) > n,ℜ(γ) > 0 then for n ∈ N the following results hold:

dn

dzn[zβ−1Eγ

α,β(azα)] = zβ−n−1Eγα,β−n(azα). (11.9)

In particular,

dn

dzn[zβ−1Eα,β(azα)] = zβ−n−1Eα,β−n(azα) (11.10)

anddn

dzn[zβ−1φ(γ, β; az)] =

Γ(β)

Γ(β − n)zβ−n−1φ(γ;β − n; az). (11.11)

Theorem 11.2. If α, β, γ, a, ν, σ ∈ C,ℜ(α) > 0,ℜ(β) > 0,ℜ(γ) > 0,ℜ(ν) > 0,ℜ(σ) > 0 then

∫ x

0

(x − t)β−1Eγα,β [a(x− t)α]tν−1Eσ

α,ν(atα)dt = xβ+ν−1Eγ+σα,β+ν(axα). (11.12)

The proof of (11.12) can be developed with the help of the Laplace transform formula

L[

xβ−1Eγα,β(axα)

]

(s) = s−β(1 − as−α)−γ (11.13)

where α, β, γ, a ∈ C, ℜ(α) > 0,ℜ(β) > 0,ℜ(γ) > 0, s ∈ C,ℜ(s) > 0, |as−α| < 1. For γ = 1, (11.13) reducesto

L[

xβ−1Eα,β(axα)]

(s) = s−β(1 − as−α)−1. (11.14)

Generalization of the above two results is given by Saxena (2002).

L[

tρ−1Eδβ,γ(atα)

]

(s) =s−ρ

Γ(δ)2ψ1

[

(δ, 1), (ρ, α)(γ, β)

;a

sα

]

, (11.15)

where ℜ(β) > 0,ℜ(γ) > 0,ℜ(s) > 0,ℜ(ρ) > 0, s > |a|1

ℜ(α) ,ℜ(δ) > 0.

Relations connecting the function defined by (11.1) and the Riemann-Liouville fractional integrals andderivatives are given by Saxena and Saigo (2005) in the form of nine theorems. Some of the interestingtheorems are given below.

Theorem 11.3. Let α > 0, β > 0, γ > 0 and a ∈ R. Let Iα0+

be the left-sided operator of Riemann-Liouvillefractional integral. Then there holds the formula

(Iα0+

[tγ−1Eδβ,γ(atβ)])(x) = xα+γ−1Eδ

β,α+γ(axβ). (11.16)

Theorem 11.4. Let α > 0, β > 0, γ > 0 and a ∈ R. Let Iα− be the right-sided operator of Riemann-Liouville

fractional integral. Then there holds the formula

(Iα−[t−α−γEδ

β,γ(at−β)])(x) = x−γEδβ,α+γ(ax−β). (11.17)

13

Theorem 11.5. Let α > 0, β > 0, γ > 0 and a ∈ R. Let Dα0+

be the left-sided operator of Riemann-Liouvillefractional derivative. Then there holds the formula

(Dα0+

[tγ−1Eδβ,γ(atβ)])(x) = xγ−α−1Eδ

β,γ−α(axβ). (11.18)

Theorem 11.6. Let α > 0, β > 0, γ − α + α > 1 and a ∈ R. Let Dα− be the right-sided operator of

Riemann-Liouville fractional derivative. Then there holds the formula

(Dα−[tα−γEδ

β,γ(at−β)])(x) = x−γEδβ,γ−α(ax−β). (11.19)

In a series of papers by Luchko and Yakubovich (1990, 1994), Luckho and Srivastava (1995), Al-Bassam andLuchko (1995), Hadid and Luchko (1996), Gorenflo and Luchko (1997), Gorenflo et al. (2000), Luchko andGorenflo (1999), the operational method was developed to solve in closed forms certain classes of differentialequations of fractional order and also integral equations. Solutions of the equations and problems consideredare obtained in terms of generalized Mittag-Leffler functions. The exact solution of certain differentialequation of fractional order is given by Luchko and Srivastava (1995) in terms of the function (11.1) by usingoperational method. In other papers, the solutions are established in terms of the following functions ofMittag-Leffler type: If z, ρ, βj ∈ C,ℜ(αj) > 0, j = 1, ...,m and m ∈ N then

Eρ((αj , βj)1,m; (z)) =

∞∑

k=0

(ρ)k∏m

j=1 Γ(ajk + βj)

zk

k!. (11.20)

For m = 1, (11.20) reduces to (11.1). The Mellin-Barnes integral for this function is given by

Eρ((αj , βj)1,m; (z)) =1

2πiΓ(ρ)

∫ γ+i∞

γ−i∞

Γ(s)Γ(ρ− s)∏m

j=1 Γ(βj − αjs)(−z)−sds, i =

√−1 (11.21)

=1

Γ(ρ)H1,1

1,m+1

[

−z∣

∣

∣

∣

(1−ρ,1)

(0,1),(1−βj,αj),j=1,...,m

]

(11.22)

where 0, γ < ℜ(ρ),ℜ(ρ) > 0 and the contour separates the poles of Γ(s) from those of Γ(ρ − s). ℜ(αj) >0, j = 1, ...,m, arg(−z) < π. The Laplace transform of the function defined by (11.20) is given by

L[Eρ((αj , βj)1,m;−t)](s) =1

sΓ(ρ)2ψm

[

(ρ, 1), (1, 1)(αj , βj)1,m

∣

∣

∣

∣

1

s

]

, (11.23)

where ℜ(s) > 0.

Remark 11.1. In a recent paper, Kilbas, Saigo and Saxena (2002) obtained a closed form solution of afractional generalization of a free electron equation of the form:

Dατ a(τ) = λ

∫ τ

0

tδa(τ − t)Ebρ,δ+1(iνt

ρ)dt+ βτσEγρ,σ+1(iνt

ρ), 0 ≤ τ ≤ 1, i =√−1 (11.24)

where b, λ ∈ C, ν, β ∈ R+, α > 0, ρ > 0;α > −1, ρ > −1, δ > −1 and Ebρ,δ+1(·) is the generalized Mittag-

Leffler function given by (11.1), and α(τ) is the unknown function to be determined.

Remark 11.2. The solution of fractional differential equations by the operational methods are also obtainedin terms of certain multivariate Mittag-Leffler functions defined below: The multivariate Mittag-Lefflerfunction of n complex variables z1, ..., zn with complex parameters a1, ..., an, b ∈ C is defined as

E(a1,...,an),b(z1, ..., zn) =

∞∑

k=0

L1+...+Ln=k∑

L1,...,Ln≥0

(

k

L1, ..., Ln

)

∏nj=1 z

Lj

j

Γ(b±∑nj=1 ajLj)

(11.25)

14

in terms of the multinomial coefficients

(

k

L1, ..., Ln

)

=k!

L1!...Ln!, k, Lj,∈ N0, j = 1, ...,m. (11.26)

Another generalization of the Mittag-Leffler function (1.2) was introduced by Kilbas and Saigo (1995)in terms of a special function of the form

Eα,m,β(z) =

∞∑

k=0

ckzk, c0 = 1, ck =

k−1∏

i=0

Γ(α(im+ β) + 1)

Γ(α(im+ β + 1) + 1), k ∈ N0 = 0, 1, 2, ... (11.27)

where an empty product is to be interpreted as unity; α, β ∈ C are complex numbers and m ∈ R,ℜ(α) >0,m > 0, α(im+ β) /∈ Z− = 0,−1,−2, ..., i = 0, 1, 2, ... and for m = 1 the above defined function reducesto a constant multiple of the Mittag-Leffler function, namely

Eα,1,β(z) = Γ(αβ + 1)Eα,αβ+1(z), (11.28)

where ℜ(α) > 0 and α(i + β) /∈ Z−. It is an entire function of z of order [ℜ(α)]−1 and type σ = 1/m,see Gorenflo et al. (1998). Certain properties of this function associated with Riemann-Liouville fractionalintegrals and derivatives are obtained and exact solutions of certain integral equations of Abel-Volterra typeare derived by their applications (Kilbas and Saigo, 1995, 1996). Its recurrence relations, connection withhypergeometric functions and differential formulas are obtained by Gorenflo, Kilbas and Rogosin (1998), alsosee, Gorenflo and Mainardi (1996). In order to present the applications of Mittag-Leffler functions we givedefinitions of Laplace transform, Fourier transform, Riemann-Liouville fractional calculus operators, Caputooperator and Weyl fractional operators in the next section.

12. Laplace and Fourier transforms, fractional calculus operators

We will need the definitions of the well-known Laplace and Fourier transforms of a function N(x, t) andtheir inverses, which are useful in deriving the solution of fractional differential equations governing certainphysical problems. The Laplace transform of a function N(x, t) with respect to t is defined by

LN(x, t) = N(x, s) =

∫ ∞

0

e−stN(x, t)dt, t > 0, x ∈ R (12.1)

where ℜ(s) > 0 and its inverse transform with respect to s is given by

L−1N(x, s) = L−1N(x, s); t =1

2πi

∫ c+i∞

c−i∞

estN(x, s)ds. (12.2)

The Fourier transform of a function N(x, t) with respect to x is defined by

FN(x, t) = F ∗(k, t) =

∫ ∞

−∞

eikxN(x, t)dx. (12.3)

The inverse Fourier transform with respect to k is given by the formula

F−1F ∗(k, t) =1

2π

∫ ∞

−∞

e−ikxF ∗(k, t)dk. (12.4)

From Mathai and Saxena (1978) and Prudnikov et al.(1990, p.355, eq(2.25.3)) it follows that the Laplacetransform of the H-function is given by

Ltρ−1Hm,np,q

[

ztσ∣

∣

∣

∣

(ap,Ap)

(bq,Bq)

]

= s−ρHm,n+1p+1,q

[

zs−σ

∣

∣

∣

∣

(1−ρ,σ),(ap,Ap)

(bq ,Bq)

]

, (12.5)

15

where ℜ(s) > 0,ℜ(ρ+ σmin1≤j≤m(bj

Bj) > 0, σ > 0,

| arg z| < 1

2πΩ, Ω > 0,Ω =

n∑

i=1

Ai −p∑

i=n+1

Ai +

m∑

j=1

Bj −q∑

j=m+1

Bj . (12.6)

By virtue of the cancelation law for the H-function (Mathai and Saxena, 1978) it can be readily seen that

L−1s−ρHm,np,q

[

zsσ

∣

∣

∣

∣

(ap,Ap)

(bq ,Bq)

]

= tρ−1Hm,np+1,q

[

zt−σ

∣

∣

∣

∣

(ap,Ap),(ρ,σ)

(bq ,Bq)

]

, (12.7)

where σ > 0,ℜ(s) > 0,ℜ[ρ + σmax1≤i≤n(1−ai

Ai)] > 0, | arg z| < 1

2πΩ1, Ω1 > 0,Ω1 = Ω − ρ. In view of theresults

J− 12(x) =

√

2

πxcosx (12.8)

the cosine transform of the H-function (Mathai and Saxena, 1978, p.49) is given by

∫ ∞

0

tρ−1 cos(kt)Hm,np,q [atµ

∣

∣

(ap,Ap)

(bq,Bq)]dt =

π

kρHm+1,n

q+1,p+2

[

kµ

a

∣

∣

∣

∣

(1−bq,Bq),( 1+ρ2 , µ

2 )

(ρ,µ),(a−ap,Ap),( 1+ρ2 , µ

2 )

]

(12.9)

where ℜ[ρ + µmin1≤j≤m(bj

Bj)] > 0, ℜ[ρ + µmax1≤j≤n(

aj−1Aj

) < 0, | arg a| < 12πΩ,Ω > 0; Ω =

∑mj=1 Bj −

∑qj=m+1Bj +

∑nj=1 Aj −

∑pj=n+1 Aj , and k > 0.

The definitions of fractional integrals used in the analysis are defined below. The Riemann-Liouvillefractional integral of order ν is defined by (Miller and Ross, 1993, p.45)

0D−νt f(x, t) =

1

Γ(ν)

∫ t

0

(t− u)ν−1f(x, u)du, (12.10)

where ℜ(ν) > 0. Following Samko, Kilbas and Marichev (1993, p.37) we define the Riemann-Liouvillefractional derivative for α > 0 in the form

0Dαt f(x, t) =

1

Γ(n− α)

dn

dtn

∫ t

0

(t− u)n−α−1f(x, u)du, n = [α] + 1 (12.11)

where [α] means the integral part of the number α. From Erdelyi, et al. (1954b, p.182) we have

L0D−νt f(x, t) = s−νF (x, s), (12.12)

where F (x, s) is the Laplace transform of f(x, t) with respect to t, ℜ(s) > 0,ℜ(ν) > 0. The Laplace transformof the fractional derivative defined by (12.11) is given by Oldham and Spanier (1974, p.134,eq (8.1.3))

L0Dαt f(x, t) = sαF (x, s) −

n∑

k=1

sk−10D

α−kt f(x, t)|t=0, n− 1 < α ≤ n, (12.13)

In certain boundary-value problems arising in the theory of visco-elasticity and in the hereditary solidmechanics the following fractional derivative of order α > 0 is introduced by Caputo (1969) in the form

Dαt f(x, t) =

1

Γ(m− α)

∫ t

0

(t− τ)m−α−1f (m)(x, t)dt,m− 1 < α ≤ m,ℜ(α) > 0,m ∈ N (12.14)

=∂m

∂tmf(x, t), if α = m, (12.15)

where ∂m

∂tm f is the m-th partial derivative of the function f(x, t) with respect to t. The Laplace transformof this derivative is given by Podlubny (1999) in the form

16

LDαt f(t); s = sαF (s) −

m−1∑

k=0

sα−k−1f (k)(0+), m− 1 < α ≤ m. (12.16)

The above formula is very useful in deriving the solution of differintegral equations of fractional ordergoverning certain physical problems of reaction and diffusion. Making use of the definitions (12.10) and(12.11) it readily follows that for f(t) = tρ we obtain

0D−νt tρ =

Γ(ρ+ 1)

Γ(ρ+ ν + 1)tρ+ν ,ℜ(ν) > 0,ℜ(ρ) > −1; t > 0 (12.17)

and

0Dνt t

ρ =Γ(ρ+ 1)

Γ(ρ− ν + 1)tρ−ν ,ℜ(ν) < 0,ℜ(ρ) > −1; t > 0. (12.18)

On taking ρ = 0 in (12.18) we find that

0Dνt [1] =

1

Γ(1 − ν)t−ν , t > 0,ℜ(ν) < 1. (12.19)

From the above result, we infer that the Riemann-Liouville derivative of unity is not zero. We also need theWeyl fractional operator defined by

−∞Dµx =

1

Γ(n− µ)

dn

dtn

∫ t

−∞

(t− u)n−µ−1f(u)du, (12.20)

where n = [µ] + 1 is the integer part of µ > 0. Its Fourier transform is (Metzler and Klafter, 2000, p.59,A.11)

F−∞Dµxf(x) = (ik)µf∗(k), (12.21)

where we define the Fourier transform as

h∗(q) =

∫ ∞

−∞

h(x) exp(iqx)dx. (12.22)

Following the convention initiated by Compte (1996) we suppress the imaginary unit in Fourier space byadopting a slightly modified form of the above result in our investigations (Metzler and Klafter, 2000,p.59,A.12).

F−∞Dµxf(x) = −|k|µf∗(k), (12.23)

instead of (12.21).

We now proceed to discuss the various applications of Mittag-Leffler functions in applied sciences. Inorder to discuss the application of Mittag-Leffler function in kinetic equations, we derive the solution of twokinetic equations in the next section.

13. Application in kinetic equations

Theorem 13.1. If ℜ(ν) > 0 then the solution of the integral equation

N(t) −N0 = −cν0D−νt N(t) (13.1)

is given by

N(t) = N0Eν(−cνtν) (13.2)

where Eν(t) is the Mittag-Leffler function defined in (1.1).

17

Proof. Applying Laplace transform to both sides of (13.1) and using (12.12) it gives

N(s) = LN(t); s = N0s−1[1 + (s/c)−ν ]−1. (13.3)

By virtue of the relation

L−1s−ρ =tρ−1

Γ(ρ),ℜ(ρ) > 0,ℜ(s) > 0, s ∈ C, (13.4)

it is seen that

L−1[N0s−1[1 + (s/c)−ν ]−1] = N0

∞∑

k=0

(−1)kcνkL−1s−νk−1 (13.5)

= N0

∞∑

k=0

(−1)kcνk tνk

Γ(1 + νk)= N0Eν(−cνtν). (13.6)

This completes the proof of Theorem 13.1.

Remark 13.1. If we apply the operator 0Dνt from the left to (13.1) and make use of the formula

0Dνt [1] =

1

Γ(1 − ν)t−ν , t > 0,ℜ(ν) < 1 (13.7)

we obtain the fractional diffusion equation

0DνtN(t) −N0

t−ν

Γ(1 − ν)= −cνN(t), t > 0,ℜ(ν) < 1 (13.8)

whose solution is also given by (13.6).

Remark 13.2. We note that Haubold and Mathai (2000) have given the solution of (13.1) in terms of theseries given by (13.5). The solution in terms of the Mittag-Leffler function is given in Saxena, Mathai andHaubold (2002).

Alternate procedure. We now present an alternate method similar to that followed by Al-Saqabi andTuan (2006) for solving some differintegral equations, also, see Saxena and Kalla (2008) for details.

Applying the operator (−cν)m0D

−mνt to both sides of (13.1) we find that

(−cν)m0D

−mνt N(t) − (−cν)m+1

0D−ν(m+1)t N(t) = N0 0D

−mνt [1],m = 0, 1, 2, ... (13.9)

Summing up the above expression with respect to m from 0 to ∞, it gives

∞∑

m=0

(−cν)m0D

−mνt N(t) −

∞∑

m=0

(−cν)m+10D

−ν(m+1)t N(t) = N0

∞∑

m=0

(−cν)m0D

−mνt [1]

which can be written as

∞∑

m=0

(−cν)m0D

−mνt N(t) −

∞∑

m=1

(−cν)m0D

−νmt N(t) = N0

∞∑

m=0

(−cν)m0D

−mνt [1].

Simplifying the above equation by using the result

0D−νt tµ−1 =

Γ(µ)

Γ(µ+ ν)tµ+ν−1 (13.10)

where minℜ(ν),ℜ(µ) > 0, we obtain

18

N(t) = N0

∞∑

m=0

(−cν)m tmν

Γ(1 +mν)(13.11)

for m = 0, 1, 2, ... Rewriting the series on the right in terms of the Mittag-Leffler function, it yields thedesired result (13.6). The next theorem can be proved in a similar manner.

Theorem 13.2. If minℜ(ν),ℜ(µ) > 0 then the solution of the integral equation

N(t) −N0tµ−1 = −cν0D

−νt N(t) (13.12)

is given by

N(t) = N0Γ(µ)tµ−1Eν,µ(−cνtν), (13.13)

where Eν,µ(t) is the generalized Mittag-Leffler function defined in (1.2).

Proof. Applying Laplace transform to both sides of (13.12) and using (1.11), it gives

N(s) = N(t); s = N0Γ(µ)s−µ[1 + (s/c)−ν ]−1. (13.14)

Using the relation (13.4), it is seen that

L−1N0Γ(µ)s−µ[1 + (s/c)−ν ]−1 = N0

∞∑

k=0

(−1)kcνkL−1s−µ−νk

= N0Γ(µ)tµ−1∞∑

k=0

(−1)kcνk tνk

Γ(µ+ νk)

= N0Γ(µ)tµ−1Eν,µ(−cνtν). (13.15)

This completes the proof of Theorem 13.2.

Alternate procedure. We now give an alternate method similar to that followed by Al-Saqabi and Tuan(2006) for solving the differintegral equations. Applying the operator (−cν)m

0D−mνt to both sides of (13.12),

we find that

(−cν)m0D

−mνt N(t) − (−cν)m+1

0D−ν(m+1)t N(t) = N0(−cν)m

0D−mνt tµ−1 (13.16)

for m = 0, 1, 2, .... Summing up the above expression with respect to m from 0 to ∞, it gives

∞∑

m=0

(−cν)m0D

−mνt N(t) −

∞∑

m=0

(−cν)m+10D

−ν(m+1)t N(t)

= N0

∞∑

m=0

(−cν)m0D

−mνt tµ−1

which can be written as

∞∑

m=0

(−cν)m0D

−mνt N(t) −

∞∑

m=1

(−cν)m0D

−mνt N(t) = N0

∞∑

m=0

(−cν)m0D

−mνt tµ−1.

Simplifying by using the result (13.10) we obtain

N(t) = N0Γ(µ)

∞∑

m=0

(−cν)m tµ+mν−1

Γ(mν + µ);m = 0, 1, 2, ... (13.17)

19

Rewriting the series on the right of (13.17) in terms of the generalized Mittag-Leffler function, it yields thedesired result (13.5). Next we present a general theorem given by Saxena et al.(2004).

Theorem 13.3. If c > 0,ℜ(ν) > 0 then for the solution of the integral equation

N(t) −N0f(t) = −cν0D−νt N(t) (13.18)

where f(t) is any integrable function on the finite interval [0, b], there exists the formula

N(t) = c N0

∫ t

0

H1,11,2

[

cν(t− τ)ν

∣

∣

∣

∣

(−1/ν,1)

(−1/ν,1),(−1,ν)

]

f(τ)dτ, (13.19)

where H1,11,2 (·) is the H-function defined by (8.1).

The proof can be developed by identifying the Laplace transform of N()+cν0D−νt N(t) as an H-function

and then using the convolution property for the Laplace transform. In what follows, Eδβ,γ(·) will be employed

to denote the generalized Mittag-Leffler function, defined by (11.1).

Note 13.1. For an alternate derivation of this theorem see Saxena and Kalla (2008).

Next we will discuss time-fractional diffusion.

14. Application to time-fractional diffusion

Theorem 14.1. Consider the following time-fractional diffusion equation

∂α

∂tαN(x, t) = D

∂2

∂x2N(x, t), 0 < α < 1, (14.1)

where D is the diffusion constant and N(x, t = 0) = δ(x) is the Dirac delta function and limx→±∞N(x, t) =0. Then its fundamental solution is given by

N(x, t) =1

|x|H1,01,1

[

|x|2Dtα

∣

∣

∣

∣

(1,α)

(1,2)

]

. (14.2)

Proof. In order to find a closed form representation of the solution in terms of the H-function, we use themethod of joint Laplace-Fourier transform, defined by

N∗(k, s) =

∫ ∞

0

∫ ∞

−∞

e−st+ikxN(x, t)dx dt (14.3)

where, according to the convention followed, ∼ will denote the Laplace transform and * the Fourier transform.Applying the Laplace transform with respect to time variable t, Fourier transform with respect to spacevariable x and using the given condition, we find that

sαN∗(k, s) − sα−1 = −Dk2N∗(k, s)

and then

N∗(k, s) =sα−1

sα +Dk2.

Inverting the Laplace transform, it yields

N∗(k, t) = L−1 sα−1

sα +Dk2 = Eα(−Dk2tα), (14.4)

where Eα(·) is the Mittag-Leffler function defined by (1.1). In order to invert the Fourier transform, we willmake use of the integral

20

∫ ∞

0

cos(kt)Eα,β(−at2)dt =π

kH1,0

1,1

[

k2

a

∣

∣

∣

∣

(β,α)

(1,2)

]

, (14.5)

where ℜ(α) > 0,ℜ(β) > 0, k > 0, a > 0; and the formula

1

2π

∫ ∞

−∞

e−ikxf(k)dk =1

π

∫ ∞

0

f(k) cos(kx)dk. (14.6)

Then it yields the required solution as

N(x, t) =1

|x|H2,13,3

[ |x|2Dtα

∣

∣

(1,1),(1,α),(1,1)

(1,2),(1,1),(1,1)

]

=1

|x|H1,01,1

[ |x|2Dtα

∣

∣

(1,α)

(1,2)

]

. (14.7)

Note 14.1. For α = 1, (14.7) reduces to the Gaussian density

N(x, t) =1

2(πDt)12

exp(− |x|24Dt

). (14.8)

Fractional space-diffusion will be discussed in the next section.

15. Application to fractional-space diffusion

Theorem 15.1. Consider the following fractional space-diffusion equation

∂

∂tN(x, t) = D

∂α

∂xαN(x, t), 0 < α < 1, (15.1)

where D is the diffusion constant, ∂α

∂xα is the operator defined by (12.20) and N(x, t = 0) = δ(x) is the Diracdelta function and limx→±∞N(x, t) = 0. Then its fundamental solution is given by

N(x, t) =1

a|x|H1,12,2

[

|x|(Dt)

1α

∣

∣

∣

∣

(1, 1α

),(1, 12 )

(1,1),(1, 12 )

]

. (15.2)

The proof can be developed on similar lines to that of the theorem of the preceding section.

16. Application to fractional reaction-diffusion model

In the same way, we can establish the following theorem, which gives the fundamental solution of thereaction-diffusion model given below.

Theorem 16.1. Consider the following reaction-diffusion model

∂β

∂tβN(x, t) = η −∞D

αxN(x, t), 0 < β ≤ 1 (16.1)

with the initial condition N(x, t = 0) = δ(x), limx→±∞N(x, t) = 0, where η is a diffusion constant and δ(x)is the Dirac delta function. Then for the solution of (16.1) there holds the formula

N(x, t) =1

a|x|H2,13,3

[

|x|(ηtβ)

1α

∣

∣

∣

∣

(1, 1α

),(1, βα

),(1, 12 )

(1,1),(1, 1α

),(1, 12 )

]

. (16.2)

For details of the proof, the reader is referred to the original paper by Saxena, Mathai and Haubold (2006).

Corollary 16.1. For the solution of the fraction reaction-diffusion equation

∂

∂tN(x, t) = η −∞D

αxN(x, t), (16.3)

21

with initial condition N(x, t = 0) = δ(x) there holds the formula

N(x, t) =1

α|x|H1,12,2

[

|x|(ηt)

1α

∣

∣

∣

∣

(1, 1α

),(1, 12 )

(1,1),(1, 12 )

]

(16.4)

where α > 0.

Note 16.1. It may be noted that (16.4) is a closed form representation of a Levy stable law. It is interestingto note that as α→ 2 the classical Gaussian solution is recovered since

N(x, t) =1

2|x|H1,12,2

[

|x|(ηt)

1α

∣

∣

∣

∣

(1, 12 ),(1, 12 )

(1,1),(1, 12 )

]

=1

2π12 |x|

∞∑

k=0

(−1)k

k!

[ |x|2(ηt)

1α

]2k+1

(16.5)

= (4π(ηt)2α )−

12 exp

[

− |x|24(ηt)

2α

]

. (16.6)

17. Application to nonlinear waves

It will be shown in this section that by the application of the inverse Laplace transforms of certainalgebraic functions derived in Saxena, Mathai and Haubold (2006), we can establish the following theoremfor nonlinear waves.

Theorem 17.1. Consider the fractional reaction-diffusion equation

0Dαt N(x, t) + a0D

βt N(x, t)

= ν2−∞D

γxN(x, t) + ζ2N(x, t) + φ(x, t) (17.1)

for x ∈ R, t > 0, 0 ≤ α ≤ 1, 0 ≤ β ≤ 1 with initial conditions N(x, 0) = f(x), limx→±∞N(x, t) = 0 forx ∈ R, where ν2 is a diffusion constant, ζ is a constant which describes the nonlinearity in the system, andφ(x, t) is nonlinear function which belongs to the area of reaction-diffusion, then there holds the followingformula for the solution of (17.1).

N(x, t) =

∞∑

r=0

(−a)r

2π

∫ ∞

−∞

t(α−β)rf∗(k) exp(−kx)

× [Er+1α,(α−β)r+1(−bt

α) + tα−βEr+1α,(α−β)(r+1)+1(−bt

α)]dk

+

∞∑

r=0

(−a)r

2π

∫ t

0

ζα+(α−β)r−1

∫ ∞

−∞

φ(k, t− ζ) exp(−ikx)

× Er+1α,α+(α−β)r(−bζα)dk dζ (17.2)

where α > β and Eδβ,γ(·) is the generalized Mittag-Leffler function, defined by (11.1), and b = ν2|k|γ − ζ2.

Proof. Applying the Laplace transform with respect to the time variable t and using the boundary condi-tions, we find that

sαN(x, s) − sα−1f(x) + asβN(x, s) − asβ−1f(x)

= ν2−∞D

γxN(x, s) + ζ2N(x, s) + f(x, s). (17.3)

22

If we apply the Fourier transform with respect to the space variable x to (17.3) it yields

sαN∗(k, s) − sα−1f∗(k) + asβN∗(k, s) − asβ−1f∗(k)

= −ν2|k|γN∗(k, s) + ζ2N∗(k, s) + f∗(k, s). (17.4)

Solving for N∗ it gives

N∗(k, s) =(sα−1 + asβ−1)f∗(k) + f∗(k, s)

sα + asβ + b(17.5)

where b = ν2|k|γ − ζ2. For inverting (17.5) it is convenient to first invert the Laplace transform and then theFourier transform. Inverting the Laplace transform with the help of the result Saxena et al.(2004a, eq(28))

L−1

sρ−1

sα + asβ + b; t

= tα−ρ∞∑

r=0

(−a)rt(α−β)rEr+1α,α+(α−β)r−ρ+1(−btα), (17.6)

where ℜ(α) > 0,ℜ(β) > 0,ℜ(ρ) > 0, | asβ

sα+b | < 1 and provided that the series in (17.6) is convergent, it yields

N∗(k, t) =

∞∑

r=0

(−a)rt(α−β)rf∗(k)

× [Er+1α,(α−β)r+1(−btα) + tα−βEr+1

α,(α−β)(r+1)+1(−btα)]

+

∞∑

r=0

(−a)r

∫ t

0

φ∗(k, t− ζ)ζα+(α−β)r−1Er+1α,(α−β)r+α(−bζα)dζ. (17.7)

Finally, the inverse Fourier transform gives the desired solution (17.3).

18. Generalized Mittag-Leffler type functions

The multiindex (m-tuple) Mittag-Leffler function is defined in Kiryakova (2000) by means of the powerseries

E( 1ρi

),(µi)(z) =

∞∑

k=0

φkzk =

∞∑

k=0

zk

∏mj=1 Γ(µj + k

ρj). (18.1)

Here m > 1 is an integer, ρ1, ..., ρm and µ1, ..., µm are arbitrary real parameters.

The following theorem is proved by Kiryakova (2000, p.244) which shows that the multiindex Mittag-Leffler function is an entire function and also gives its asymptotic estimate, order and type.

Theorem 18.1. For arbitrary sets of indices ρi > 0,−∞ < µi < ∞, i = 1, ...,m the multiindex Mittag-Leffler function defined by (18.1) is an entire function of order

ρ = [

m∑

i=1

1

ρi]−1, that is,

1

ρ=

1

ρ1+ ...+

1

ρm(18.2)

and type

σ = (ρ1

ρ)

ρρ1 ...(

ρm

ρ)

ρρm . (18.3)

Furthermore, for every positive ǫ, the asymptotic estimate

|E(1/ρi),(µi)(z)| < exp((σ + ǫ)|z|ρ), (18.4)

23

holds for |z| ≥ r0(ǫ), r0(ǫ) sufficiently large.

It is interesting to note that for m = 2, (18.2) reduces to the generalized Mittag-Leffler function con-sidered by Dzherbashyan (1960) denoted by φρ1,ρ2(z;µ1, µ2) and defined in the following form (Kiryakova,1994, Appendix)

E(1/ρ1,1/ρ2);(µ1,µ2)(z) = φρ1,ρ2(z;µ1, µ2)

=

∞∑

k=0

zk

Γ(µ1 + kρ1

)Γ(µ2 + kρ2

), (18.5)

and shown to be an entire function of order

ρ =ρ1ρ2

ρ1 + ρ2and type σ = (

ρ1

ρ2)

ρ2ρ1 (

ρ2

ρ1)

ρ1ρ2 . (18.6)

Relations between multiindex Mittag-Leffler function with H-function, generalized Wright function and otherspecial functions are given by Kiryakova; for details, see the original papers Kiryakova (1999, 2000). Saxena,Kalla and Kiryakova (2003) investigated the relations between the multiindex Mittag-Leffler function andthe Riemann-Liouville fractional integrals and derivatives. The results derived are of general nature and giverise to a number of known as well as unknown results in the theory of generalized Mittag-Leffler functions,which serve as a backbone for the fractional calculus. Two interesting theorems established by Saxena et al.(2003) are described below.

Theorem 18.2. Let α > 0, ρi > 0, µi > 0, i = 1, ...,m and further, let Iα0+

be the left-sided Riemann-Liouville fractional integral. Then there holds the relation

(Iα0+

[tρi−1E(1/ρi),(µi)(at1/ρi)])(x)

= xα+ρi−1E(1/ρi),(µ1+α,µ2,...,µm)(at1

ρi ). (18.7)

Theorem 18.3. Let α > 0, ρi > 0, µi > 0, i = 1, ...,m and further, let Iα− be the right-sided Riemann-

Liouville fractional integral. Then there holds the relation

(Iα−[t−ρi−αE(1/ρi),(µi)(at

− 1ρi )])(x)

= x−µiE(1/ρi),(µ1+α,µ2,...,µm)(at− 1

ρi ). (18.8)

Another generalization of the Mittag-Leffler function is recently given by Sharma (2008) in terms of theM-series defined by

pMαq = pM

αq (a1, ..., ap; b1, ..., bq;α; z)

=

∞∑

r=0

(a1)r...(ap)r

(b1)r...(bq)rΓ(αr + 1)zr, p ≤ q + 1. (18.9)

Remark 18.1. According to Saxena (2009), the M-series discussed by Sharma (2008) is not a new specialfunction. It is, in disguise, a special case of the generalized Wright function pψq(z), which was introducedby Wright (1935), as shown below.

24

κ p+1ψq+1

[

(a1, 1), ..., (ap, 1), (1, 1)(b1, 1), ..., (bq, 1), (1, α)

; z

]

=

∞∑

r=0

(a1)r...(ap)r(1)r

(b1)r...(bq)rΓ(αr + 1)

zr

r!

=

∞∑

r=0

(a1)r...(ap)r

(b1)r...(bq)rΓ(αr + 1)zr

= pMαq (a1, ..., ap; b1, ..., bq;α; z) (18.10)

where

κ =

∏qj=1 Γ(bj)

∏pj=1 Γ(aj)

.

Fractional integration and fractional differentiation of the M-series are discussed by Sharma (2008). The tworesults proved in Sharma (2008) for the function defined by (18.9) are reproduced below. For ℜ(ν) > 0

(Iν0+

[pMαq (a1, ..., ap;b1, ..., bq;α; z)])(x)

=zν

Γ(1 + ν)p+1M

αq+1(a1, ..., ap, 1; b1, ..., bq, 1 + ν;α; z)

and for ℜ(ν) < 0

(Dν0+

[pMαq (a1, ..., ap;b1, ..., bq;α; z)])(x)

=zν

Γ(1 − ν)p+1M

αq+1(a1, ..., ap, 1; b1, ..., bq, 1 − ν;α; z).

19. Mittag-Leffler Distributions and Processes

19.1. Mittag-Leffler Statistical Distribution and Its Properties

A statistical distribution in terms of the Mittag-Leffler function Eα(y) was defined by Pillai (1990) interms of the distribution function or cumulative density function as follows:

Gy(y) = 1 − Eα(−yα) =

∞∑

k=1

(−1)k+1yαk

Γ(1 + αk), 0 < α ≤ 1, y > 0 (19.1.1)

and Gy(y) = 0 for y ≤ 0. Differentiating on both sides with respect to y we obtain the density function f(y)as follows:

f(y) =d

dyGy(y)

=d

dy

[

∞∑

k=1

(−1)k+1yαk

Γ(1 + αk)

]

=

∞∑

k=1

(−1)k+1αkyαk−1

Γ(1 + αk)

=

∞∑

k=1

(−1)k+1yαk

Γ(1 + αk)=

∞∑

k=0

(−1)kyα+αk−1

Γ(α+ αk)

by replacing k by k + 1

= yα−1Eα,α(−yα), 0 < α ≤ 1, y > 0 (19.1.2)

where Eα,β(x) is the generalized Mittag-Leffler function.

25

It is straightforward to observe that for the density in (19.1.2) the distribution function is that in(19.1.1). The Laplace transform of the density in (19.1.2) is the following:

Lf (t) =

∫ ∞

0

e−txf(x)dx =

∫ ∞

0

e−txxα−1Eα,α(−xα)dx

= (1 + tα)−1, |tα| < 1. (19.1.3)

Note that (19.1.3) is a special case of the general class of Laplace transforms discussed in Section 2.3.7[Haubold and Mathai (2008)]. From (19.1.3) one can also note that there is a structural representation interms of positive Levy distribution. A positive Levy random variable u > 0, with parameter α is such thatthe Laplace transform of the density of u > 0 is given by e−tα

. That is,

E[e−tu] = e−tα

(19.1.4)

where E(·) denotes the expected value of (·) or the statistical expectation of (·). When α = 1 the randomvariable is degenerate with the density function

f1(x) =

1, for x = 10, elsewhere.

Consider an exponential random variable with density function

f1(x) =

e−x, for 0 ≤ x <∞0, elsewhere;

Lf1(t) = (1 + t)−1 (19.1.5)

and with the Laplace transform Lf1(t).

Theorem 19.1.1 Let y > 0 be a Levy random variable with Laplace transform as in (19.1.4) and let x

and y be independently distributed. Then u = yx1α is distributed as a Mittag-Leffler random variable with

Laplace transform as in (19.1.3).

Proof. For establishing this result we will make use of a basic result on conditional expectations, whichwill be stated as a lemma.

Lemma 19.1.1 For two random variables x and y having a joint distribution,

E(x) = E[E(x|y)] (19.1.6)

whenever all the expected values exist, where the inside expectation is taken in the conditional space of xgiven y and the outside expectation is taken in the marginal space of y.

Now by applying (19.1.6) we have the following: Let the density of u be denoted by g(u). Then theLaplace transform of g is given by

E[e−(tx1α )y|x] = e−tαx. (19.1.8)

But the right side of (19.1.8) is in the form of a Laplace transform of the density of x with parameter tα.Hence the expected value of the right side is

Lg(t) = (1 + tα)−1 (19.1.9)

which establishes the result. From (19.1.8) one property is obvious. Suppose that we consider an arbitraryrandom variable y with the Laplace transform of the form

Lg(t) = e−[φ(t)] (19.1.10)

whenever the expected value exists, where φ(t) be such that

φ(tx1α ) = xφ(t), lim

t→0φ(t) = 0.

26

Then from (19.1.8) we have

E[e−(tx1α )y|x] = e−x[φ(t)]. (19.1.11)

Now, let x be an arbitrary positive random variable having Laplace transform, denoted by Lx(t) whereLx(t) = ψ(t). Then from (19.1.9) we have

Lg(t) = ψ[φ(t)]. (19.1.12)

For example, if y is a random variable whose density has the Laplace transform, denoted by Ly(t) = φ(t),

with φ(tx1α ) = xφ(t), and if x is a real random variable having the gamma density,

fx(x) =xβ−1e−

xδ

δβΓ(β), x ≥ 0, β > 0, δ > 0 (19.1.13)

and fx(x) = 0 elsewhere, and if x and y are statistically independently distributed and if u = yx1α then the

Laplace transform of the density of u, denoted by Lu(t) is given by

Lu(t) = [1 + δφ(t)]−β. (19.1.14)

Note 19.1.1. Since we did not put any restriction on the nature of the random variables, except that theexpected values exist, the result in (19.1.12) holds whether the variables are continuous, discrete or mixed.

Note 19.1.2. Observe that for the result in (19.1.12) to hold we need only the conditional Laplace transformof y given x be of the form in (19.1.11) and the marginal Laplace transform of x be ψ(t). Then the resultin (19.1.12) will hold. Thus statistical independence of x and y is not a basic requirement for the result in(19.1.12) to hold

Thus from (19.1.13) we may write a particular case as

z = yx1α (19.1.15)

where x is distributed as in (19.1.5) and y as in (19.1.4) then z will be distributed as in (19.1.3) or (19.1.9)when x and y are assumed to be independently distributed.

Note 19.1.3 The representation of the Mittag-Leffler variable as well as the properties described on page1432 of Jayakumar (2003) and on page 53 of Jayakumar and Suresh (2003) are to be rewritten and correctedbecause the exponential variable and Levy variable seem to be interchanged there.

By taking the natural logarithms on both sides of (19.1.15) we have

1

αlnx+ ln y = ln z. (19.1.16)

Then the first moment of ln z is available from (19.1.16) by computing E[lnx] and E[ln y]. But E[lnx] isavailable from the following procedure:

E[e−t ln x] = E[eln x−t

] = E[x−t] =

∫ ∞

0

x−te−xdx

= Γ(1 − t) for ℜ(1 − t) > 0 (19.1.17)

which will be Γ(β − t)/Γ(β) for the density in (19.1.13). Hence

E[lnx] = − d

dtE[e−t ln x]|t=0 = − d

dtΓ(1 − t)|t=0.

But

27

d

dtΓ(1 − t) = −Γ(1 − t)

d

dtln Γ(1 − t) = −Γ(1 − t)ψ(1 − t) (19.1.18)

where ψ(·) is the psi function of (·), see Mathai (1993) for details. Hence by taking the limits t→ 0

E[ln x] = −Γ(1 − t)ψ(1 − t)|t=0 = −ψ(1) = γ (19.1.19)

where γ is Euler’s constant, see Mathai (1993) for details.

19.2. Mellin-Barnes representation of the Mittag-Leffler density

Consider the density function in (19.1.2). After writing in series form and then looking at the corre-sponding Mellin-Barnes representation we have the following:

g(x) = xα−1Eα,α(−xα) =

∞∑

k=0

Γ(1 + k)(−1)k

k!

xα−1+αk

Γ(α+ αk)(19.2.1)

=1

2πi

∫ c+i∞

c−i∞

1

α

Γ( 1α − s

α )Γ(1 − 1α + s

α )

Γ(1 − s)x−sds, 1 − α < c < 1 (19.2.2)

[by expanding as the sum of residues at the poles of Γ(1 − 1α + s

α )]

=1

2πi

∫ c1+i∞

c1−i∞

Γ(s)Γ(1 − s)

Γ(αs)xαs−1ds = g(x), 0 < c1 < 1, 0 < α ≤ 1 (19.2.3)

by putting 1α − s

α = s1. Here the point s = 0 is removable. By taking the Laplace transform of g(x) from(19.2.1) we have

Lg(t) =

∞∑

k=0

(−1)k

Γ(α+ αk)

∫ ∞

0

xα+αk−1e−txdx

=

∞∑

k=0

(−1)kt−α−αk = t−α(1 + t−α)−1 = (1 + tα)−1, |tα| < 1. (19.2.4)

19.2.1. Generalized Mittag-Leffler density

Consider the generalized Mittag-Leffler function

g1(x) =1

Γ(γ)

∞∑

k=0

(−1)kΓ(γ + k)

k!Γ(αk + αγ)xαγ−1+αk

= xαγ−1Eγαγ,α(−xα), α > 0, γ > 0. (19.2.6)

Laplace transform of g1(x) is the following:

Lg1(t) =

∞∑

k=0

(−1)k

k!

(γ)k

Γ(αγ + αk)

∫ ∞

0

xαγ+αk−1e−txdx

=

∞∑

k=0

(−1)k (γ)k

k!t−αγ−αk = (1 + tα)−γ , |tα| < 1. (19.2.7)

In fact, this is a special case of the general class of Laplace transforms connected with Mittag-Leffler functionconsidered in Mathai, et al. (2006).

28

19.3. Mittag-Leffler Density as an H-function

g1(x) of (19.2.6) can be written as a Mellin-Barnes integral and then as an H-function.

g1(x) =1

Γ(η)

1

2πi

∫ c+i∞

c−i∞

Γ(s)Γ(η − s)

Γ(αη − αs)xαη−1(xα)−sds,ℜ(η) > 0, 0 < c < ℜ(η)

=xαη−1

Γ(η)H1,1

1,2

[

xα

∣

∣

∣

∣

(1−η,1)

(0,1),(1−αη,α)

]

(19.3.1)

=1

αΓ(η)

1

2πi

∫ c1+i∞

c1−i∞

Γ(η − 1α + s

α )Γ( 1α − s

α )

Γ(1 − s)x−sds (19.3.2)

[by taking αη − 1 − αs = −s1]

=1

αΓ(η)H1,1

1,2

[

x

∣

∣

∣

∣

(1− 1α

, 1α

)

(η− 1α

, 1α

),(0,1)

]

. (19.3.3)

Since g and g1 are represented as inverse Mellin transforms, in the Mellin-Barnes representation, one canobtain the (s− 1)-th moments of g and g1 from (19.3.2). That is,

Mg1(s) = E(xs−1) in g1

=1

Γ(γ)

Γ(η − 1α + s

α )Γ( 1α − s

α )

αΓ(1 − s), (19.3.4)

for 1 − α < ℜ(s) < 1, 0 < α ≤ 1, η > 0.

Mg(t) = E(xs−1) in g

=Γ(1 − 1

α + sα )Γ( 1

α − sα )

αΓ(1 − s)(19.3.5)

for 1 − α < ℜ(s) < 1, 0 < α ≤ 1, obtained by putting η = 1 in (19.3.4) also. Since

limα→1

Γ( 1α − s

α )

Γ(1 − s)= 1

for α→ 1, (19.3.4) reduces to

Mg1(t) =1

Γ(η)Γ(η − 1 + s) for α→ 1. (19.3.6)

Its inverse Mellin transform is then

g1 =1

Γ(η)

1

2πi

∫ c+i∞

c−i∞

Γ(η − 1 + s)x−sds =1

Γ(η)xη−1e−x, x ≥ 0, η > 0 (19.3.7)

which is the one-parameter gamma density and for η = 1 it reduces to the exponential density. Hencethe generalized Mittag-Leffler density g1 can be taken as an extension of a gamma density such as theone in (19.3.7) and the Mittag-Leffler density g as an extension of the exponential density for η = 1. Isthere a structural representation for the random variable giving rise to the Laplace transform in (19.2.4)corresponding to (19.1.10)? The answer is in the affirmative and it is illustrated in (19.1.14).

Note 19.3.1. Pillai (1990, Theorem 2.2), Lin (1998, Lemma 3) and others list the ρ-th moment of theMittag-Leffler density g in (19.1.2) as follows:

E(xρ) =Γ(1 − ρ

α )Γ(1 + ρα )

Γ(1 − ρ),−α < ℜ(ρ) < α < 1.

Therefore

29

E(xs−1) =Γ(1 + 1

α − sα )Γ(1 − 1

α + sα )

Γ(2 − s)

=( 1

α − sα )Γ( 1

α − sα )Γ(1 − 1

α + sα )

(1 − s)Γ(1 − s)=

1

α

Γ( 1α − s

α )Γ(1 − 1α + s

α )

Γ(1 − s)

which is the expression in (19.3.5). Hence the two expressions are one and the same.

Note 19.3.2. If y = ax, a > 0 and if x has a Mittag-Leffler distribution then the density of y can also berepresented as a Mittag-Leffler function with the Laplace transform

Lx(t) = (1 + tα)−1 ⇒ Ly(t) = (1 + (at)α)−1, a > 0, |(at)α| < 1.

Note 19.3.3. From the representation that

E(xh) =Γ(1 − h

α )Γ(1 + hα )

Γ(1 − h),−α < ℜ(h) < α < 1

we have

E(x0) =Γ(1)Γ(1)

Γ(1)= 1.

Further, g(x) in (19.2.1) is a non-negative function for all x, with

E(xh) =

∫ ∞

0

xhg(x)dx = 1 for h = 0.

Hence g(x) is a density function for a positive random variable x. Note that from the series form for theMittag-Leffler function it is not possible to show that

∫∞

0g(x)dx = 1 directly.

19.4. Structural Representation of the Generalized Mittag-Leffler Variable

Let u be the random variable corresponding to the Laplace transform (19.2.7) with tα replaced by δtα

and γ by η. Let u be a positive Levy variable with the Laplace transform e−tα

, 0 < α ≤ 1 and let v be agamma random variable with parameters η and δ or with he Laplace transform (1 + δt)−η, η > 0, δ > 0. Letu and v be statistically independently distributed.

Lemma 19.4.1. Let u, v as defined above. Then

w ∼ uv1α (19.4.1)

where w is a generalized Mittag-Leffler variable with Laplace transform (1+δtα)−β , |δtα| < 1, where ∼ means‘distributed as’ or both sides have the same distribution.

Proof. Denoting the Laplace transform of the density of w by Lw(t) and treating it as an expected value

Lw(t) = E[e−tv1α u] = E[E[e−(tv

1α u)|v]]

= E[e−(tv1α )α

] = E[e−tαv] = (1 + δtα)−η (19.4.2)

from the Laplace transform of a gamma variable. This establishes the result.

From the structural representation in (19.4.1), taking the Mellin transforms and writing as expectedvalues, we have

E(ws−1) = E(us−1)E(v1α )s−1 (19.4.2)

due to statistical independence of u and v. The left side is available from (19.3.4) as

30

E(ws−1) =Γ(η − 1

α + sα )Γ( 1

α − sα )δ

s−1α

αΓ(η)Γ(1 − s). (19.4.3)

Let us compute E[v1α ]s−1 from the gamma density. That is,

E[v1α ]s−1 =

1

δηΓ(η)

∫ ∞

0

(v1α )s−1vη−1e−

vδ dv =

Γ(η − 1α + s

α )δs−1

α

Γ(η)(19.4.4)

for ℜ(s) > 1 − αη, 0 < α ≤ 1, η > 0. Comparing (19.4.3) and (19.4.4) we have the (s − 1)-th moment of aLevy variable

E[us−1] =Γ( 1

α − sα )

αΓ(1 − s)=

Γ(1 + 1α − s

α )

Γ(2 − s),ℜ(s) < 1, 0 < α ≤ 1. (19.4.5)

Note 19.4.1. Lin (1998) gives the ρ-th moment of a Levy variable with parameter α as

E[uρ] =Γ(1 − ρ

α )

Γ(1 − ρ). (19.4.6)

Hence for ρ = s− 1 we have

E[us−1] =Γ(1 + 1

α − sα )

Γ(2 − s)=

( 1α − s

α )Γ( 1α − s

α )

(1 − s)Γ(1 − s)

=Γ( 1

α − sα )

αΓ(1 − s)for ℜ(s) < 1.

This is (19.4.5) and hence both the representations are one and the same.

Hence the Levy density, denoted by g2(u), can be written as

g2(u) =1

2πi

∫ c+i∞

c−i∞

Γ( 1α − s

α )

αΓ(1 − s)u−sds. (19.4.7)

Its Laplace transform is then

Lg2(t) =1

2πi

∫ c+i∞

c−i∞

Γ( 1α − s

α )

αΓ(1 − s)[

∫ ∞

0

u1−s−1e−tudu]ds

=1

2πi

∫ c+i∞

c−i∞

1

αΓ(

1

α− s

α)t−1+sds

=1

2πi

∫ c1+i∞

c1−i∞

1

αΓ(s

α)t−sds (19.4.7)

by making the substitution −1+s = −s1. Then evaluating as the sum of the residues at sα = −ν, ν = 0, 1, 2, ...

we have

Lg2(t) =∞∑

ν=0

(−1)ν

ν!tαν = e−tα

. (19.4.8)

This verifies the result about the Laplace transform of the positive Levy variable with parameter α. Notethat when α = 1, (19.4.8) gives the Laplace transform of a degenerate random variable taking the value 1with probability 1.

Mellin convolution of certain Levy variables can be seen to be again a Levy variable.

31

Lemma 19.4.2. Let xj be a positive Levy variable with parameter αj , 0 < αj ≤ 1 and let x1, ..., xp bestatistically independently distributed. Then

u = x1x1

α12 ...x

1α1α2...αp−1p

is distributed as a Levy variable with parameter α1α2...αp.

Proof. From (19.4.5)

E[e−txj ] = e−tαj

, 0 < αj ≤ 1, j = 1, ..., p

E[e−tu] = E[e−tx1x1

α12 ...x

1α1...αp−1p ] = E[E[e−tx1...x

1α1...αp−1p |x2,...,xp

]]

= E[e−tα1x2x1

α23 ...x

1α2...αp−1p ].

Repeated application of the conditional argument gives the final result as

E[e−tu] = e−tα

, α = α1α2...αp, 0 < α1...αp ≤ 1

which means that u is distributed as a Levy with parameter α1...αp.

From the representation in (19.4.1) we can compute the moments of the natural logarithms of Mittag-Leffler, Levy and gamma variables.

w = uv1α ⇒ lnw = lnu+

1

αln v. (19.4.9)

But from (19.4.3) and (19.4.6) we have the h-th moments of u and v given by

E[uh] =Γ(1 − h

α )

Γ(1 − h),ℜ(h) < α ≤ 1 (19.4.10)

and

E[v1α ]h =

Γ(η + hα )δ

hα

Γ(η),ℜ(η +

h

α) > 0. (19.4.11)

From (19.4.3)

E[wh] =Γ(η + h

α )Γ(1 − hα )δ

hα

Γ(η)Γ(1 − h). (19.4.12)

But for a positive random variable z

E[zh] = E[eln zh

].

Henced

dhE[zh]|h=0 = E[ln zeh ln z ]h=0 = E[ln z].

Therefore from (19.4.9) to (19.4.12) we have the following:

E[lnw] =d

dh

δhα Γ(η + h

α )Γ(1 − hα )

Γ(η)Γ(1 − h)

|h=0

=1

αψ((η) − 1

αψ(1) + ψ(1) +

1

αln δ (19.4.13)

32

by taking the logarithmic derivative, where ψ is a psi function, see, for example Mathai (1993).

E[ln v1α ] =

d

dh

δhα Γ(η + h

α )

Γ(η)

|h=0 =1

αψ(η) +

1

αln δ (19.4.14)

or

E[ln v] = ψ(η) + ln δ

and

E[lnu] =d

dh

Γ(1 − hα )

Γ(1 − h)

|h=0 = − 1

αψ(1) + ψ(1) (14.4.15)

where ψ(1) = −γ where γ is the Euler’s constant.

Note 19.4.2. The relations on the expected values of the logarithms of Mittag-Leffler variable, positiveLevy variable and exponential variable, given on page 1432 of Jayakumar (2003), where η = 1, are notcorrect.

19.5. A Pathway from Mittag-Leffler Distribution to Positive Levy Distribution

Consider the function

f(x) =

∞∑

k=0

(−1)k(η)k

k!Γ(αη + αk)

xαη−1+αk

(a1α )αη+αk

, η > 0, a > 0, 0 < α ≤ 1

=xαη−1

aηEη

α,αη(−xα

a).

Thus x = a1α y where y is a generalized Mittag-Leffler variable. The Laplace transform of f is given by the

following:

Lf(t) =

∞∑

k=0

(−1)k(η)k

k!aη+k

∫ ∞

0

xαη+αk−1e−tx

Γ(αη + αk)dx

=

∞∑

k=0

(−1)k(η)k

k!aη+kt−αη−αk = [1 + atα]−η, |atα| < 1. (19.5.1)

If η is replaced by ηq−1 and a by a(q − 1) with q > 1 then we have a Laplace transform

Lf (t) = [1 + a(q − 1)tα]−η

q−1 , q > 1 (19.5.2)

If q → 1+ then

Lf (t) → e−aηtα

= Lf1(t) (19.5.3)

which is the Laplace transform of a constant multiple of a positive Levy variable with parameter α. Thus qhere creates a pathway of going from the general Mittag-Leffler density f to a positive Levy density f1 withparameter α, the multiplying constant being (aη)

1α . For a discussion of a general rectangular matrix-variate

pathway model see Mathai (2005). The result in (19.5.3) can be put in a more general setting. Consider anarbitrary real random variable y with the Laplace transform, denoted by Ly(t), and given by

Ly(t) = e−φ(t) (19.5.4)

where φ(t) is a function such that φ(txγ) = xφ(t), φ(t) ≥ 0, limt→0 φ(t) = 0 for some real positive γ. Let

33

u = yxγ (19.5.5)

where x and y are independently distributed with y having the Laplace transform in (19.5.4) and x having atwo-parameter gamma density with shape parameter β and scale parameter δ or with the Laplace transform

Lx(t) = (1 + δt)−β . (19.5.6)

Now consider the Laplace transform of u in (19.5.5), denoted by Lu(t). Then

Lu(t) = E[e−tu] = E[e−tyxγ

] = E[E[etyxγ |x]]= E[e−φ(txγ)] = E[e−xφ(t)]

from the assumed property of φ(t)

= [1 + δφ(t)]−β . (19.5.7)

If δ is replaced by δ(q − 1) and β by βq−1 with q > 1 then we get a path through q. That is, when q → 1+,

Lu(t) = [1 + δ(q − 1)φ(t)]−β

q−1 → e−δβφ(t) = e−φ(δγβγt). (19.5.8)

If φ(t) = tα, 0 < α ≤ 1 then

Lu(t) = e−(δβ)γαtα

which means that u goes to a constant multiple of a positive Levy variable with parameter α, the constantbeing (δβ)γ .

19.6. Linnik or α-Laplace Distribution

A Linnik random variable is defined as that real scalar random variable whose characteristic functionis given by

φ(t) =1

1 + |t|α , 0 < α ≤ 2,−∞ < t <∞. (19.6.1)

For α = 2, (19.6.1) corresponds to the characteristic function of a Laplace random variable and hencePillai (1995) called the distribution corresponding to (19.6.1) as the α-Laplace distribution. For positivevariable, (19.6.1) reduces to the characteristic function of a Mittag-Leffler variable. Infinite divisibility,characterizations, other properties and related materials may be seen from the review paper Jayakumar andSuresh (2003) and the many references therein, Pakes (1998) and Mainardi and Pagnini (2008). Multivariategeneralization of Mittag-Leffler and Linnik distributions may be seen from Lim and Teo (2009). Since thesteps for deriving results on Linnik distribution are parallel to those of the Mittag-Leffler variable, furtherdiscussion of Linnik distribution is omitted.

19.7. Multivariable Generalization of Mittag-Leffler, Linnik and Levy Distributions

A multivariate Linnik distribution can be defined in terms of a multivariate Levy vector. Let T ′ =(t1, ..., tp), X

′ = (x1, ..., xp), prime denoting the transpose. A vector variable having positive Levy distribu-tion is given by the characteristic function

E[eiT ′X ] = e−(T ′ΣT )α2 , 0 < α ≤ 2, (19.7.1)

where Σ = Σ′ > 0 is a real positive definite p× p matrix. Consider the representation

u = y1αX (19.7.2)

where the p × 1 vector X , having a multivariable Levy distribution with parameter α, and y a real scalargamma random variable with the parameters δ and β, are independently distributed. Then the characteristicfunction of the random vector variable u is given by the following:

34

E[eiy1α T ′X ] = E[E[eiy

1α T ′X ]|y]

= E[e−y[|T ′ΣT |]α2 ] = [1 + δ|T ′ΣT |α

2 ]−β . (19.7.3)

Then the distribution of u, with the characteristic function in (19.7.3) is called a vector-variable Linnikdistribution. Some properties of this distribution are given in Lim and Teo (2009).

20. Mittag-Leffler Stochastic Processes

The stochastic process x(t), t > 0 having stationary independent increment with x(0) = 0 and x(1)having the Laplace transform

Lx(1)(λ) = (1 + λα)−1, 0 < α ≤ 1, λ > 0,

which is the Laplace transform of a Mittag-Leffler random variable, is called the Mittag-Leffler stochasticprocess. Then the Laplace transform of x(t), denoted by Lx(t)(λ), is given by

Lx(t)(λ) = [(1 + λα)−1]t = [1 + λα]−t. (20.1.1)

The density corresponding to the Laplace transform (20.1.1) or the density of x(t) is then available as thefollowing:

fx(t)(x) =

∞∑

k=0

(−1)k (t)k

k!

xαk+αt−1

Γ(αk + αt)

= xαt−1Etα,αt(−xα), 0 < α ≤ 1, x ≥ 0, t > 0. (20.1.2)

The distribution function of x(t) is given by

Fx(t)(x) =

∫ x

0

fx(t)(y)dy =∞∑

k=0

(−1)k (t)k

k!

xαk+αt

(αk + αt)Γ(αk + αt)

=

∞∑

k=0

(−1)k

k!

Γ(t+ k)

Γ(t)

xαk+αt

Γ(1 + αk + αt), 0 < α ≤ 1, t > 0. (20.1.3)

This form is given by Pillai (1990) and by his students.

20.2. Linear First Order Autoregressive processes

Consider the stochastic process

xn =

en, with probability p, 0 ≤ p ≤ 1en + axn−1 with probability 1 − p, 0 < a ≤ 1.

(20.2.1)

Let the sequence en be independently and identically distributed with Laplace transform Le(λ) and letxn be identically distributed with Laplace transform Lx(λ). From the representation in (20.2.1)

Lxn(λ) = pLe(λ) + (1 − p)Le(λ)Lxn−1(aλ)

Therefore

Le(λ) =Lxn

(λ)

p+ (1 − p)Lxn−1(aλ)=

Lx(λ)

p+ (1 − p)Lx(aλ)(20.2.1)

35

assuming stationarity. When p = 0,

Le(λ) =Lx(λ)

Lx(aλ), 0 < a ≤ 1 (20.2.3)

which defines class L distributions, for all a, 0 < a < 1. When p = 0, (20.2.3) implies that the innovationsequence en belongs to class L distributions. Then (20.2.3) can lead to two autoregressive situations,the first order exponential autoregressive process EAR(1) and the first order Mittag-Leffler autoregressiveprocess MLAR(1).

Concluding remarks

The various Mittag-Leffler functions discussed in this paper will be useful for investigators in variousdisciplines of applied sciences and engineering. The importance of Mittag-Leffler function in physics issteadily increasing. It is simply said that deviations of physical phenomena from exponential behavior couldbe governed by physical laws through Mittag-Leffler functions (power-law). Currently more and more suchphenomena are discovered and studied.

It is particularly important for the disciplines of stochastic systems, dynamical systems theory, anddisordered systems. Eventually, it is believed that all these new research results will lead to the discovery oftruly nonequilibrium statistical mechanics. This is statistical mechanics beyond Boltzmann and Gibbs. Thisnonequilibrium statistical mechanics will focus on entropy production, reaction, diffusion, reaction-diffusion,etc and may be governed by fractional calculus.

Right now, fractional calculus and H-function (Mittag-Leffler function) are very important in researchin physics.

Acknowledgment

The authors would like to thank the Department of Science and Technology, Government of India, New Delhi,for the financial assistance under Project No. SR/S4/MS:287/05 which made this research collaborationpossible.

The authors are grateful to Constantino Tsallis, Enrico Scalas, Francesco Mainardi, Henkel Malte,Zivorad Tomovski, and the Wolfram Demonstration Team for suggestions to improve the paper.

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