R E S E A R C H
Open Access
Some results on the generalized
Mittag-Leffler function operator
JC Prajapati
1, RK Jana
2, RK Saxena
3and AK Shukla
2**Correspondence:
ajayshukla2@rediffmail.com
2Department of Mathematics, S. V.
National Institute of Technology, Surat, India
Full list of author information is available at the end of the article
Abstract
This paper is devoted to the study of a generalized Mittag-Leffler function operator introduced by Shukla and Prajapati (J. Math. Anal. Appl. 336:797-811, 2007). Laplace and Mellin transforms of this operator are investigated in this paper. Some special cases of the established results are also deduced as corollaries. The results obtained are useful where the Mittag-Leffler function occurs naturally.
MSC: 33E12; 26A33; 44A45
Keywords: generalized Mittag-Leffler function; Laplace transform; Mellin transform; H-function; generalized Wright function
1 Introduction
In , the Swedish mathematician Mittag-Leffler [, ] introduced the function
Eα(z) =
∞
n= zn
(αn+ )
α∈C;Re(α) > . (.)
A generalization of (.) was given by Wiman [] in in the form
Eα,β(z) =
∞
n= zn
(αn+β)
α,β∈C;Re(α) > ,Re(β) > . (.)
In , in connection with the solution of certain singular integral equations, a further interesting and useful generalization of (.) was introduced by Prabhakar [] in the form
Eγα,β(z) = ∞
n=
(γ)nzn
(αn+β)n! (.)
α,β,γ ∈C;Re(α) > ,Re(β) > ,Re(γ) > ,
where (γ)nis the Pochhammer symbol defined by
(γ)= ; (γ)n=γ(γ + )· · ·(γ+n– ) =
(γ +n)
(γ) ; γ= . (.)
A generalization of (.) is given by Shukla and Prajapati [] in the following form:
Eγα,,βq(z) =
∞
n=
(γ)qnzn
(αn+β)n!, (.)
where (α,β,γ ∈C;Re(α) > ,Re(β) > ,Re(γ) > ,q∈(, )∪N).
The above generalization studied by Shukla and Prajapati is shown in a series of papers [–]. A generalization of Mittag-Leffler functions defined by (.) and (.) is introduced and studied by Srivastava and Tomovski [] in the form
Eγα,,βκ(z) =
∞
n=
(γ)nκzn
(αn+β)n!, (.)
where α,β,γ,κ ∈C; Re(α) = Re(κ) – > , Re(β) > , Re(γ) > , Re(κ) > and the Pochhammer symbol forλ,μ∈Cis defined by
(λ)μ=
(λ+μ)
(λ) =
⎧ ⎨ ⎩
(μ= ,λ∈C\{});
λ(λ+ )· · ·(λ+n– ), μ=n∈N;λ∈C. (.)
The object of this paper is to derive the Laplace and Mellin transforms of the follow-ing integral operator associated with the generalized Mittag-Leffler function, defined by Shukla and Prajapati [] as well as Srivastava and Tomovski [], in the next sections:
Eγα,,βq,ω;+f(x) =
x
(x–t)β–Eγ,q
α,β w(x–t)
αf(t)dt, (.)
whereα,β,γ ∈C;Re(α) > ,Re(β) > ,Re(γ) > ;q∈(, )∪N.
Shukla and Prajapati [] have shown that the Mellin-Barnes integral for the function defined by (.) is given by
Eγα,,βq(z) = πi(γ)
i∞
–i∞
(–ξ)(γ +qξ)
(αξ+β) (–z)
ξdξ. (.)
Asγ –→, then by virtue of the limit formula,
lim γ–→E
γ α,β(z) =
(β) (.)
reduces to the familiar Reimann-Liouville fractional integral
Iα +f
(x) = (α)
x
(x–t)α–f(t)dt α∈C,Re(α) > . (.)
In the following, we will use the representation of a high transcendental function in terms of the so-called H-function defined as []
Hpm,q,n
z (a,A), . . . , (ap,Ap)
(b,B), . . . , (bq,Bq)
=
πi
L
where
θ(ξ) = [
m
j=(bj+Bjξ)][nj=( –aj–Ajξ)]
[qj=m+( –bj–Bjξ)][
p
j=n+(aj+Ajξ)]
, (.)
and an empty product is always interpreted as unity;m,n,p,q∈Nwith ≤n≤p, ≤ m≤q,Ai,Bj∈R+,ai,bj∈RorC(i= , . . . ,p;j= , . . . ,q) such that
Ai(bj+k)=Bj(ai–l– ) (k,l∈N;i= , . . . ,n;j= , . . . ,m). (.)
The contourLis the path of integration in the complexξ-plane running fromγ–i∞to γ +i∞for some real numberγ.
2 Mellin transform of the operator (1.8)
Theorem . It is shown here that
MEαγ,,βq,ω;+f(x);s= (γ)( –s)
·H,,
–wtα ( –γ,q)
(, ), ( –s–β,α)
Mtβf(t);s, (.)
where(Re(α) > ,Re(β) > ,Re(γ) > );q∈(, )∪N,Re( –s–β) > and H,,(·)is the H-function defined by(.).
Proof Mellin transform is defined as
Mf(x);s=
∞
xs–f(x)dx.
Therefore, we have
MEαγ,,βq,ω;+f
(x);s=
∞
xs–
x
(x–t)β–Eαγ,,βq ω(x–t) α
f(t)dt dx.
Interchanging the order of integration, which is permissible under the conditions given in Theorem ., we find that
MEαγ,,βq,ω;+f
(x);s
=
∞
f(t)dt
∞
t
xs–(x–t)β–Eγα,,βq ω(x–t)αdx. (.)
If we considerx=t+uin the r.h.s. of (.), we get
MEαγ,,βq,ω;+f
(x);s=
∞
f(t)dt
∞
t
(t+u)s–uβ–Eγα,,βq ωu α
du. (.)
trans-forms into the form
MEαγ,,βq,ω;+f
(x);s=
∞
f(t)dt
πi(γ)
i∞
–i∞
(–ξ)(γ+qξ)
(αξ+β) (–ω)
ξ
·
∞
(t+u)s–uβ+αξ–du dξ. (.)
If theu-integral is evaluated with the help of the formula
∞
xν–(x+a)–ρdx=(ν)(ρ–ν)
(ρ) ; Re(ρ) >Re(ν) > , (.)
then after some simplification, it is seen that the right-hand side of above equation (.) simplifies to
πi(γ)( –s)
i∞ –i∞
(–ξ)(γ +qξ)( –s–β–αξ)–ωtαξdξ ·
∞
tβ+s–f(t)dt (.)
which, on being interpreted by the definition of H-function (.), yields the desired
re-sult.
Forq= , Theorem . reduces to the following corollary.
Corollary . The following result holds:
MEαγ,β,ω;+f(x);s= (γ)( –s)
·H,,
–wtα ( –γ, )
(, ), ( –s–β,α)
Mtβf(t);s, (.)
where(Re(α) > ,Re(β) > ,Re(γ) > );Re( –s–β) > and H,,(·)is the H-function defined by(.).
Theorem . It is shown here that
MEαγ,,βκ,ω;+f(x);s= (γ)( –s)
·H,,
–wtα ( –γ,κ)
(, ), ( –s–β,α)
Mtβf(t);s, (.)
whereRe(α) =Re(κ) – > ,Re(β) > ,Re(γ) > ,Re(κ) > ,Re( –s–β) > and H,,(·) is the H-function defined by(.).
3 Laplace transform of the operator (1.8)
Theorem . The following result holds:
LEγα,,βq,ω;+f
(x);p= (γ)p
–β ψ
(γ,q); ω/pα
–;
where(Re(α),Re(β),Re(γ) > );Re(p) >|ω|/Re(α)and F(p)is the Laplace transform of f(t) defined by
Lf(t);p=F(p) =
∞
e–ptf(t)dt, (.)
whereRe(p) > and the integral is convergent.
Proof By virtue of the definitions (.) and (.), it follows that
LEγα,,βq,ω;+f
(x);p=
∞
e–px
x
(x–t)β–Eαγ,,βq ω(x–t) α
f(t)dt dx.
Interchanging the order of integration, which is permissible under the conditions given in Theorem ., we find that
LEγα,,βq,ω;+f(x);p=
∞
f(t)dt
∞
t
e–px(x–t)β–Eγ,q
α,β ω(x–t) αdx.
If we considerx=t+u, we obtain
LEγα,,βq,ω;+f(x);p=
∞
e–ptf(t)dt
∞
e–puuβ–Eγα,,βq ωuαdu.
On making use of the series definition (.), the above expression becomes
LEγα,,βq,ω;+f
(x);p
= ∞
r=
(γ)qrωr
(αr+β)(r!)
∞
e–ptf(t)dt
∞
e–puuβ+αr–du
= ∞
r=
(γ)qrωr pβ+αr(r!)
∞
e–ptf(t)dt
=
(γ)p
–β ψ
(γ,q); ω/pα
–;
F(p),
andF(p) is the Laplace transform off(t).
Forq= , Theorem . reduces to the following corollary.
Corollary . The following result holds:
ŁEαγ,β,ω;+f
(x);p=p–β –apαγF(p), (.)
where(Re(α) > ,Re(β) > ,Re(γ) > );Re(p) > and the integral operator is the one discussed by Prabhakar[]defined by(.).
Theorem . The following result holds:
LEγα,,βκ,ω;+f
(x);p= (γ)p
–β ψ
(γ,κ); ω/pα
–;
whereRe(α) =Re(κ) – > ,Re(β) > ,Re(γ) > ,Re(κ) > ;Re(p) >|ω|/Re(α)and F(p)is the Laplace transform of f(t)defined by(.).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors contributed equally and significantly in writing this article. The authors read and approved the final manuscript.
Author details
1Department of Mathematical Sciences, Charotar University of Science and Technology, Anand, Changa, India. 2Department of Mathematics, S. V. National Institute of Technology, Surat, India.3Department of Mathematics and
Statistics, J. N. V. University, Jodhpur, India.
Received: 25 October 2012 Accepted: 13 January 2013 Published: 29 January 2013 References
1. Mittag-Leffler, GM: Sur la nouvelle function. C. R. Math. Acad. Sci. Paris137, 554-558 (1903)
2. Mittag-Leffler, GM: Sur la representation analytique d’une function monogene (cinquieme note). Acta Math.29, 101-181 (1905)
3. Wiman, A: Uber den Fundamental Salz in der Theorie der Funktionen. Acta Math.29, 191-201 (1905)
4. Prabhakar, TR: A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J. 19, 7-15 (1971)
5. Shukla, AK, Prajapati, JC: On a generalization of Mittag-Leffler function and its properties. J. Math. Anal. Appl.336, 797-811 (2007)
6. Shukla, AK, Prajapati, JC: On a generalization of Mittag-Leffler type function and generalized integral operator. Math. Sci. Res. J.12(12), 283-290 (2008)
7. Shukla, AK, Prajapati, JC: On a recurrence relation of generalized Mittag-Leffler function. Surv. Appl. Math.4, 133-138 (2009)
8. Shukla, AK, Prajapati, JC: Some remarks on generalized Mittag-Leffler function. Proyecciones28(1), 27-34 (2009) 9. Srivastava, HM, Tomovski, Z: Fractional calculus with an integral operator containing a generalized Mittag-Leffler
function in the kernel. Appl. Math. Comput.211, 198-210 (2009)
10. Mathai, AM, Saxena, RK, Haubold, HJ: The H-Function: Theory and Applications. Springer, New York (2010)
doi:10.1186/1029-242X-2013-33
