Decomposition of Generalized Mittag-Leffler
Function and Its Properties
Jyotindra C. Prajapati1, Ajay Kumar Shukla2
1Department of Mathematical Sciences, Faculty of Applied Sciences, Charotar University of Science and Technology, Anand, India 2Department of Mathematics, S.V. National Institute of Technology, Surat, India
Email: {jyotindra18, ajayshukla2}@rediffmail.com
Received August 16, 2011; revised September 26, 2011; accepted October 10, 2011
ABSTRACT
The principal aim of the paper is devoted to the study of some special properties of the function ,
for ,qE z
1n
. Authors defined the decomposition of the function ,
,
q
E z in the form of truncated power series as Equations (1.7),
(1.8) and their various properties including integral representation, derivative, inequalities and their several special cases are obtained. Some new results are also established for the function ,
,
q E z
E z
.
Keywords: Generalized Mittag-Leffler Function; Beta Function; Gamma Function; Integral Representation
1. Introduction
In 1903, the Swedish mathematician Gosta Mittag-Lef- fler introduced the function (Gorenflo et al. [1])
defined as,
0 1
n
z n
n
E z
0
, (1.1) where z is a complex variable and is a gamma
function, . The Mittag-Leffler function is the di-rect generalisation of the exponential function to which it reduces for 1. For 0 1, it interpolates be-tween the pure exponential and a hypergeometric func- tion 1
1z. Mittag-Leffler function naturally occurs as
the solution of the fractional order differential equations and (or) fractional order integral equations.
Wiman [2] studied the generalisation of E
z
, that is given by
,
0
( )
, ; Re 0,
n
E z
,
Re 0 .
n
z n
, ( )
E z
(1.2)
which is known as Wiman’s function.
Prabhakar [3] investigated the function as
,
0
, , ; Re 0, Re
n
n
E z
Gorenflo et al. ([4]), Kilbas and Saigo [5], Kilbas et al.
[6], Miller [7], Saigo and Kilbas [8] have studied several properties and applications of (1.1)-(1.3).
Recently, Shukla and Prajapati [9] introduced the function ,
,
q
E z , which is defined for , , ;
Re
0, Re
0 and
Re 0, q 0,1
by:
, !
0, Re 0 .
n z
n n
(1.3)
, ,
0 !
n qn
q n
z
E z
n n
, (1.4)
where qn qn
denotes the generalized Poch- hammer symbol which, in particular, reduces to
1
1
q qn
r n
r q
q
if q1
q
.
For , (1.4) reduces to (1.1) which was studied by Mittag-Leffler in terms of its applications to the theory of entire functions.
Incidentally, (1.4) is generalization of the exponential function z
e , the confluent hypergeometric function
, ;z
and the functions which are defined in (1.1)- (1.3).
Ikehata and Siltanen [10] defined truncated power se-ries of E
z as,
1 1 1
0 ! 0 1 1
nm nm j N N n
nN
nm j m m j
n n
z z
E z
m
1
1 1
nm j nm j j
n
z
, ,q E z
1
0 ! 0
nm n
m m
n
z
E z
m
. (1.6)Whereas in [10], they have used the functions (1.5) and (1.6) in the study of electrical impedance tomography.
In this paper, we have defined the decomposition of the function in the form of truncated power series
, , 1
, 0
1 1 0 1
( ) ! ( )
nm N
qm q nN
m n
N n
m j
z
E z
m m
m
1
nm j qm
nm j n
z
(1.7)
and as a special case for 1
n
:
, 1,
0 1 0 1
!
nm qm q
m n
n
m j
z E z
m m
m
1
nm j qm
nm j n
z
(1.8)
where n2, N 1; , ; Re
0, Re
0 and q
0,1 .(1.5) and (1.6) are special cases of (1.7) and (1.8), given by
1
nN
n n
E z
1,1, 1
,1
nN
E z and 1,11
1
,1
n n
E z E z
. (1.9)
In what follows are known formulae, those used studying properties of the functions
, ,q
E z , 1, ,
,
q nN n
E z
and
, 1
,
q n
E z
1
, 1 d ,
0, Re 0.
b
z z
a b
10
t z
z e
Re
z 0
.
The Beta function (Rainville [11]) is defined by,
1 1 0 where Re
a
B a b
z (1.10)
The Gamma function (Rainville [11]) is defined by, d
t t, where . (1.11)
The relation between Beta and Gamma functions is
where ReB a b
, ,
0, Re 0.
a b a b
a b
(1.12)
The Error function (Rainville [11]) is defined by,
20
exp d
z
t t
. (1.13)The complementary Error function (R defined by,
2 π
erf z
ainville [11]) is
1
exp
2 dπ
c
z
erf z erfc z 2
t t. (1.14)
Legendre’s duplication formula (Rainville [11]) is
2 1
1π 2z 2 z z z
. (1.15) The Laplace transform (Sneddon [12]) of the function
2
z is defined as, Ikehata and Siltanen [10] used following inequality,
f
0
d sz
L f z
e f z z, where Re
s 0. (1.16)1 Re
N z
1 ! 1 !
m z e
z
, where N1. (1.17)2. Main Results
THEOREM 1. Integral representations of the function
m N m N
, 1,
q n
E z
is given by
,
1
, 1
1,
1 0
1
d , 1
n
z n
q n n n k
k
n E z u u u
k n
(2.1)where n2, ,
, ,
1q 1 q n
E z E, z
; Re
0, Re
0 and
0,1q .
Proof. Consider the integral,
11, d .
z
n n n k
E z u u u
Substituting u zw
,q
0
and using (1.4), we get
0
d !
m
1 0
( ) nm 1 n
qmz w n k m
zw
Using (1.10), the above equation reduces to
w
m m
.
0
1,1
nm n k
m
k B m
and use of (1.12), yields
!
qmz n
m m n
, 1
1,
0
d
1
. 1
z
q n n n k
nm n k qm m
E z u u u
k
z n
nm n k
m
n
n
0i.e.
, 1d .
nm n k
n n k
u u u
Taking s
1,
0 1 1 0
z
qm q n
m
z n
E z
nm n k k
m
n n
ummation over k1 to k n 1 and using (1.8), we arrive at
n 1 z
n n
n k 1d .u u u
This comp
, ,
, , ,
1 1, 1,
, 1
0 1 1
q q n q
k n
E z E z n E z
k n
letes the proof of the theorem.
THEOREM 2. If n2 ;Re
0, Re
0 and q
0,1 , then
1 1 1
1 , 1
1,
1 1
d d
n n
q n k q n
k k
n
z E z z
k m
z
n k
n n
! !
k m
n n
q m n
z z
m n
(2.2).4) in left-hand side of (2.2), we obtain
Proof. Applying (1
2
2
1 1
1 , 1,
1 1
d
d 1 ! 1 !
n n n
qm qm
2
1 !
m m m
n
q n
m m m n
n
z z z
z E z
m m m
z
m m m
n n n
m n
qm
. (2.3)
Setting m n k in the first summation and replacing m by in second, right-hand side of (2.3) reduces to
1 1
. !
m m
n n
m n
z z
m n
pletes the
Theorem
1 1
1 1 ! 0
n n
q n k q
k n k m m
n k
n n
This com proof of the theorem.
Remark on 2. If q 1, then Theorem 2 leads a particular case of (2.3)
1 1
1 1
, ,
n n
n n
E z
1
1 1
1
d d
k
n n
k
z
z E z z
k z
n
, (2.4)
w
here n2, , ;Re
0, Re
0 and q
0,1 .; Re
0, Re
0 and q
0,1THEOREM 3.If n2, , .
1
, ,
1 1,
, 0 Re
1
n
q q n
k n
E z E z
k n
. in (2 e get
k
z
(2.5)
Proof. Substituting u zw .1), w
1
, , , 1
1 1, 1,
, 1
0
1 d
1
n k
q q n q n n n k
k n
z
E z E z n E z w w w
k n
.Therefore,
1
n
1 ,
1 11,
, 1
0
Re d
1
n k n
n q n n k
k n
z
E z n E z w w
k n
., ,
The simplification of the above inequality gives,
,
1, Re
1
q n
E z
i.e.
1 1
, ,
1 1,
, 1 1
1
1 Re 1
1 1
n k n k
n n
q q n
k k
n
z z
E z E z
k k n k
n n n
1 ,
,
, 1 Re
1
k n
q n k
n
z
E z
k n
.This proves the theorem.
Remark on Theorem 3: It is easy to verify the following inequality,
,
1q 1 1
E z
,
1, .
n q
E z
k n
(2.6) 1
, , , , ,
1 1 1
, , , 0
1
k n
q nN q nN q
k
n n n
z
E z E z E z
THEOREM 4.If n2, , ; Re
0, Re
0 and q
0,1 , then
, , .
1q 1q nN
E z E z T
, ,
n n
, (2.7) where
1,,q
Rezn
.
(2.8)Proof. From (1.7) and (1.8), we have
1
1 1
1
1 1 ! 1
N
n k nN
n
q N qN
k
z z
T E
k nN
N N N
n
1
, , . ,
1 1 1
, , , 0 ! 0 1
nm
N n
qm qm
q q nN q n
m m j
n n n
E z E z E z
m m
m
. 1
nm j
z z
nm j
n
Consider,
1 1 1
, 1
1,
1
1 0
d d
! 1 !
m m
n n n n
z z
n N n
qm qm
q n n n j n j
m N j
z u z u
n n
E z u u u u u
j
m m m m
,using (1.10), the above equation reduces to
1 1 0 0
j j m
n n
1
1 1 !
nm n j n
qm m N j
z
j
m m
n
B 1 j,m 1n
,
Now, by involving (1.12), we have
1 0 1
n
qm m j m
1
nm j
z
nm j
n
.
by which we write
1
1
, , . , ,
1 1 1 1,
1
0 1 0 0
d
! 1 !
m
nm z n n
N n N
qm qm
q q nN q n q n n n j
j m
z n z u
E z E z E z E z u u u
j
m m m m
n
., , , m
n n n
Therefore, further we have,
, , . ,
1 1 1
, , , 0
1 1
1,
1 0 0
!
1 ! 1
nm N
qm
q q nN q n
m
, 1 d .
n n
m
nm n
z n k
n N
qm
q n n n n k
k m
z
E z E z E z
m m
z w
nz
E z z w w w
k m m
n
e have
n
Using the inequality (1.17), w
1,
, Re
q n n
z M
, (2.9)
where
1 1
, , .
1 1
, , 1 1 !
N n q N q q nN
n n
z
E z E z E
N N
1
1 1
1,
, 1
0 0
1
d !
m nm n n k
n N
qm
q n n n n k
m
z w
nz M
1 1
k
E z z w w w
m m
.We see that,
k n
1 1
1,
1 0
1
, 1, 1
1
Re 1 !
1
Re 1 , ! 1
N
n n
n k n
qN q n
k
n k nN n
qN q n
k
z w
n z , 1
d ,
1 ,
n n k
M E z
k N N
n
z k
E z B
k n
N N
n
w w w
N
which on using (1.12), gives Re
0, Re
0, Re
0, and q
0,1 ,then
, 1, Re
1
q n
z E z
k nN
n
. (2.10)
Inequalities (2.9) and (2.10) lead to the proof of the Theorem.
Remark on T q 1
1 1
k nN n qN
k M
N
heorem 4. If then (2.7)
lity reduces to an interesting inequa
1
1 1
1
k n n nN
k
n n
E z E z
exp Re
1
N n
z z
k nN
n
ati [9] obtai
. (2.11)
This inequality contains Mittag-Leffler and Exponen-tial functions.
Recently, Shukla and Prajap ned several properties of the function ,
,q e M
E z . Th ittag-Leffler
function E
z plays an important role in study of theous prop ional calculus (Shukla and Praja-pati [13]).
vari erties fract
THEOREM 5. If , , , ; Re
0,
1
1 , 1 ,
, ,
0 1
d
z
q q
t z t E t t z E z
.(2.12) Proof.Substituting tzu in left-hand side of (2.12)
and then using (1.10) and (1.12), we get the required re-sult.
Special cases of Theorem 5: For Re
0, from(2.12), we obtain several particular cases as listed below:
1 1,1,1 1
01 z d
t
z t e t z E z
, (2.13)
1
1
1,1
2z
z
t cosh t dt z E2,1 z , (2.14) 0
1 1,12, 2
20
1 z sinh( )d
t
t t z E z
t
z , (2.15)
1
2
exp d
z t t z
2 1,11 , 1 0
c
z erf z z E
.(2.16)
; Re
0,
THEOREM 6. If , , Re
0,Re
0 and q
0,1 , then
1
1 , 2
2 , 1
d ( )
z
q
t z t E t t
0
1 , , 2
, 2 , .
q q
z E z E z
(2.17)
Proof. Consider the integral,
, 2 1
z
q z t
2 , 0
E t t
1 d1 t
The above equation reduces to,
2 0
1
2 1
0 0
2 1 !
1
2 ! 1
n qn n
n qn
n
z z
n n
z u
zu
n n
Applying (1.10), (1.12) and further simplifica the above equation becomes,
d .
z u
tion of
2 2
( ) n n
qnz z
1
0 0
( )
2 1 ! 2 1 1 !
qn
n n
z z
n n n n
.i.e.
0 1 !
n n n n qnz
z
.Therefore,
,
, 1
z
q z t q
E t t t z E z
. (2.18)Using T r 1
, 2 1
2 , 0
1 d
1
heorem 5 fo , where the Equation (2.8) leads to
1 ,
, d
q q
t E t t
,
di
, 2 1
2 ,
0 0
1 d
(1 )
z z t z
E t t t
fferentiating above equation with respect to z, we get
1
1 , 2 1
2 ,
0 1 ,
,
1 .
, 2 2 , d
q q
q
z E z t z t E t t
z E z
reby, the theorem is completely proved.
The theorem, that follows now, represents the rela-tionship betwee ion ,
,q
E z and the
expo-nential function. ionship presumably play an im in the study of the fractional diffusion equa-tion, conduction and mass transfer equation.
THEOREM 7.If , , ; Re 0, Re 0,
Re 0 and q
0,1 , thenz
The
n the fu t This relat nc portant role
i.e., heat
2
π
x
x t
Proof. Putting
, 1 ,
4 2 2
, , 1
0 2 2
d
q q
t
e E x x E t
. (2.19)2
n
z in (1.15), we have
2
2 π . (2.20
1 2
2
n n
n
n
)
The left-hand side of (2.19) gives,
2
, 1
4
x q t
n t
2
1
0 0
e d ,
!
x qn
n
4 , 0
d
e E x x x
x x
n n
which leads to
0
2
! 2 2
n qn
n
n
n n
t
.
Use of (2.20) and further simplification of ab tion yields
ove
equa-
2n
t
2 0 π
1 !
2 2
qn n
t
n n
.This completes the proof.
THEOREM 8. If , , ; Re
0, Re
0,
e 0 and
R q
0,1 then
1 , ,
2 , ,
, 1 ,
d d
.
q
q q
x E ax x
x E ax E ax
(2.21)
Proof. The left-hand side of (2.21) reduces to,
21 , ,
0
1 d
d !
n n
qn q
n
a n x
x E ax
x n n
.The right-hand side can be expressed as
2 0
0
!
. !
qn n
qn n
x
n n
a x
n n
1 n n
n a x
n n
In ompleted.
nt notation he binomi heorem is
Funktionen E
x ,” Actavolving (1.4), the proof is c Mathematica, Vol. 29,No. 1,
In a slightly differe , t al t 1905, pp. 191-201. doi:10.1007/BF02403202
[3] T. R. Prabhakar, “A Singular Integral Equation with a Generalized Mittag-Leffler Function in the Kernel,” Yo-kohama Mathem , 1971, pp. 7-15.
[4] R. Gorenflo a ittag-Leffler
Func-
,1 q z
0 !
qn
n n
. (2.22)If q1, then (2.22) reduces to simple bino
pression (Rainville [8]) that is,
n z
atical Journal, Vol. 19
nd F. Mainardi, “On M
mial ex- tion in Fractional Evaluation Processes,” Journal of Computational and Applied Mathematics, Vol. 118, No.
1-2, 2000, pp. 283-299.
doi:10.1016/S0377-0427(00)00294-6
,1
nzn0 !
n n
.1z 1 z
[5] A. A. Kilbas and M. Saigo, “On Mittag-Leffler Type Function, Fractional Calculus Operators and Solution of Integral Equations,” Integral Transforms and Special Functions, Vol. 4, No. 4, 1996, pp. 355-370.
THEOREM 9. If , , ; Re
0, Re
0Re ,
,
0 and q
0,1 then
1 1 ,
,q
t E zt
, 1
q
z
L s
s
doi:10.1080/10652469608819121
[6] A. A. Kilbas, M. Saigo and R. K. Saxena, “Generalised Mittag-Leffler Function and Generalised Fractional
Cal-
. (2.23)
Proof. Applying (2.22) to the left-hand sid 23), we get
e (2.
,
1 1
1 0
1 , ,
1
!
! .
q n
qn
qn
n n
q
z z
L s L s
s
L n
t E zt
3. Acknowledgements
The authors would like to thank the reviewers for their valuable suggestions to improve the quality of paper.
REFERE
[1] R. Gorenflo, A. A. Kilbas and S. V. Rogosin, “On th eralised Mittag-Leffler Type Function,” Integral T
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s
1
n z
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culus Operators,” Integral Transforms and Special Func-tions, Vol. 15, No. 1, 2004, pp. 31-49.
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