GENERALIZED k-MITTAG LEFFLER FUNCTION AND ITS
COMPOSITION WITH PATHWAY INTEGRAL OPERATORS
K. S NISAR, S. D. PUROHIT, M. S. ABOUZAID, MAYSAA AL QURASHI, AND D.
BALEANU
Abstract. Our purpose in this paper is to consider a more generalized form
of the Mittag Leffler function. For this extended Mittag Leffler function, we
obtain some composition formulas with pathway fractional integral operators.
We also point out some important special cases of the main results.
1. Introduction
Mittag Leffler functions are important in studying solutions of fractional differential equations, and they are associated with a wide range of problems in many
areas of mathematics and physics. These considerations have led various workers in
the field of special functions for exploring the possible extensions and applications
for the Mittag Leffler function. A useful generalization of the Mittag Leffler called
as k-Mittag Leffler function has been introduced and studied in [1]. Here we aim at
introducing a more generalized k-Mittag Leffler function and also presenting certain
image formulas under pathway fractional integral formulas for the newly defined
function.
Throughout this paper, let C, R, N be the sets of complex numbers, real numbers,
positive integers respectively, and N0 := N ∪ {0}.
Let α, β, γ ∈ C, k ∈ R, {< (α) , < (β) , < (γ)} > 0 and q ∈ (0, 1) ∪ N, then the
generalized k-Mittag Leffler function is defined by Gehlot [1] as:
(1.1)
γ,q
Ek,α,β
(z) =
∞
X
zn
,
Γ (αn + β) n!
n=0 k
(γ)nq,k
where (γ)nq,k is k-Pochhammer symbol given by
(1.2)
(x)n,k = x (x + k) (x + 2k) .... (x + (n − 1) k) (x ∈ C, k ∈ R, n ∈ N).
The integral form of the generalized k-Gamma function is given by
Z ∞
−tk
(1.3)
Γk (z) =
e k tz−1 dt,
0
2000 Mathematics Subject Classification. Primary 05C38, 15A15; Secondary 05A15, 15A18.
Key words and phrases. Mittag Leffler functions, Pathway Integral Operator.
1
2 K. S NISAR, S. D. PUROHIT, M. S. ABOUZAID, MAYSAA AL QURASHI, AND D. BALEANU
where k ∈ R, z ∈ C, < (z) > 0 and
(1.4)
Γk (x + k) = xΓk (x) ,
γ
Γk (γ) =
−1
γ,q
Ek,α,β,δ
(z) =
∞
X
Γ
γ .
k
In this paper, we introduce a more generalized k-Mittag Leffler function as under:
(1.5)
(k) k
zn
,
Γ (αn + β) (δ)n
n=0 k
(γ)nq,k
where α, β, γ, δ ∈ C, k ∈ R, {< (α) , < (β) , < (γ) , < (δ)} > 0 and q ∈ (0, 1) ∪ N.
Particular cases:
(i) For δ = 1, we get the generalized k-Mittag Leffler function (1.1) (see [1]).
γ
(ii) Again, if δ = q = 1 then (1.5) reduces to Ek,α,β
(z) (see [3]).
Recently, Nair [9] introduce a pathway fractional integral operator by using the
pathway idea of Mathai [8] and developed further by Mathai and Haubold [10],
[11]), and it is defined as follows:
Let f (x) ∈ L (a, b) , η ∈ C, < (η) > 0, a > 0 and the pathway parameter α < 1
as (cf. [12]), then
(1.6)
(η,α)
x
[ a(1−α)
Z ]
P0+ f (x) = xη
1−
a (1 − α) t
x
η
(1−α)
f (t) dt.
0
For a real scalar α, the pathway model for scalar random variables is represented
by the following probability density function (p.d.f.):
γ−1
f (x) = c |x|
h
δ
β
i (1−α)
1 − a (1 − α) |x|
,
h
i
δ
provided that −∞ < x < ∞, δ > 0, β ≥ 0, 1 − a (1 − α) |x| > 0, and γ > 0 where
c is the normalizing constant and α is called the pathway parameter.
Futher, for real α, the normalizing constant as follows:
γ
γ
β
δ
δ
[a
(1
−
α)]
Γ
+
+
1
δ
1−α
1
, for α < 1
c =
β
γ
2
Γ δ Γ 1−α + 1
γ
β
δ
δ
[a
(1
−
α)]
Γ
α−1
1
1
γ
, for
=
− > 0, α > 1
2 Γ γΓ β − γ
1−α
δ
(1.7)
δ
α−1
δ
γ
δ
=
1 (aβ)
, α → 1.
2 Γ γδ
h
i
δ
Note that for α < 1 it is a finite range density with 1 − a (1 − α) |x| > 0
and (1.7) remains in the extended generalized type-1 beta family. The pathway
density in (1.7), for α < 1, includes the extended type-1 beta density, the triangular
density, the uniform density and many other p.d.f.
GENERALIZED k-MITTAG LEFFLER FUNCTION AND ITS PROPERTIES
3
For instance, α > 1, writing (1 − α) = − (α − 1) in (1.6) gives
(1.8)
(η,α)
P0+ f
η
x
]
[ −a(1−α)
Z
a (α − 1) t
1+
x
(x) = x
η
−(α−1)
f (t) dt,
0
and
(1.9)
γ−1
f (x) = c |x|
h
δ
1 + a (α − 1) |x|
β
i− (α−1)
,
provided that −∞ < x < ∞, δ > 0, β ≥ 0, and α > 1 which is the extended generalized type-2 beta model for real x. It includes the type-2 beta density, the F
density, the Student-t density, the Cauchy density and many more.
Moreover, when α → 1− , the operator (1.6) reduces to the Laplace integral
transform, and when α = 0, a = 1 and η replacing by η − 1, the operator (1.6)
reduces to the Riemann-Liouville fractional integral operator. For more details on
the pathway model and its particular cases, the reader is referred to the recent
papers of Mathai and Haubold [10], [11] and Nair [9].
It is observed that the pathway fractional integral operator (1.6), can lead to
other interesting examples of fractional calculus operators, related to some probability density functions and applications in statistics. This has led various workers
in the field of fractional calculus for exploring the possible extensions of the known
results. For example, the composition of the integral transform operator (1.6) with
the product of generalized Bessel function of the first kind is given in [12]. Recently
Nisar et. al studied the pathway fractional integral operator associated with Struve
function of first kind [17]. The results provided in [12] are extensions of the results
given by Agarwal and Purohit [16] and Nair [9]. The purpose of this work is to investigate the composition formula of integral transform operator due to Nair, with
the more generalized k-Mittag Leffler function introduced in presiding section.
2. Pathway Fractional Integration of Generalized k-Mittag Leffler
function.
In this section, we derive the pathway integral representation of generalized kMittag-Leffler function, which is defined in (1.5). The results given in this section
are based on the preliminary assertions giving by composition formula of pathway
fractional integral (1.6) with a power function. The results are given in the following
theorems.
Theorem 1. Let ρ, β, γ, δ, ξ ∈ C , k ∈ R, {< (ρ) , < (β) < (γ) , < (δ) , < (ξ)} >
ξ
0, η > 0, < 1 + 1−α
> 0, α < 1, k, w ∈ R and q ∈ (0, 1) ∪ N. Then the following
formula holds true:
ξ
h β
ρ i
Γ 1 + 1−α
ξ
β
(ξ,α)
γ,q
P0+
wt k (x) = xξ+ k k (1+ 1−α )
t k −1 Ek,ρ,β,δ
β
[a (1 − α)] k
" kρ #
x
γ,q
×Ek,ρ,β+k 1+ ξ ,δ w
(2.1)
.
( 1−α )
a (1 − α)
4 K. S NISAR, S. D. PUROHIT, M. S. ABOUZAID, MAYSAA AL QURASHI, AND D. BALEANU
Proof. By applying (1.5) and (1.6), we have
(ξ,α)
P0+
h
t
β
k −1
γ,q
Ek,α,β,δ
wt
ρ
k
x
]
Z [ a(1−α)
ξ
β
a (1 − α) t 1−α
t k −1 1 −
x
0
ρ
γ,q
×Ek,α,β,δ
wt k dt.
i
ξ
= x
For convenience, we denote the right hand integral of the above term by I1 , then
x
]
ξ
Z [ a(1−α)
β
a (1 − α) t 1−α
ξ
−1
k
I1 = x
t
1−
x
0
ρ n
∞
X
k
(γ)nq,k
wt
,
×
(δ)n
Γ
(ρn
+
β)
n=0 k
and by interchanging the order of integration and summation, we have
∞
X
(γ)nq,k
n
(w)
I1 = xξ
Γ
(ρn
+
β)
(δ)n
n=0 k
x
]
ξ
Z [ a(1−α)
a (1 − α) t 1−α β + ρ −1
1−
t k n dt.
x
0
Now, by evaluating the inner integral using beta function formula, we get
kρ n+ βk
∞
X
(γ)nq,k
wn
x
I1 = xξ
Γ (ρn + β) (δ)n a (1 − α)
n=0 k
ξ
Γ 1 + 1−α
Γ kρ n + βk
.
× ξ
Γ kρ n + βk + 1 + 1−α
Using (1.4.), we obtain
I1
kρ n
x
β
∞ (γ)nq,k w a(1−α)
X
xξ+ k
ρ β
=
β
[a (1 − α)] k n=0 Γ kρ n + βk k k n+ k −1 (δ)n
ξ
Γ 1 + 1−α
Γ kρ n + βk
× ξ
Γ kρ n + βk + 1 + 1−α
β
ξ+
ξ
xx k Γ 1 + 1−α
=
β
[a (1 − α)] k
kρ n
x
(γ)
w
∞
nq,k
a(1−α)
X
ρ β
×
.
β
ξ
ρ
n+ k −1 (δ)
n=0 Γ k n + k + 1 + 1−α k k
n
Again, on applying (1.4.), we get
ξ
β
" ξ
kρ #
xξ+ k k (1+ 1−α ) Γ 1 + 1−α
x
γ,q
I1 =
Ek,α,β+k 1+ ξ ,δ w
,
β
( 1−α )
a (1 − α)
[a (1 − α)] k
which completes the proof of Theorem 1.
GENERALIZED k-MITTAG LEFFLER FUNCTION AND ITS PROPERTIES
5
Corollary 1. If we put δ = 1, Theorem 1 reduces to the result given in [2]:
ξ
ρ i
h β
Γ
1
+
ξ
β
1−α
(ξ,α)
γ,q
P0+
t k −1 Ek,ρ,β
wt k (x) = xξ+ k k (1+ 1−α )
β
[a (1 − α)] k
" kρ #
x
γ,q
.
×Ek,α,β+k 1+ ξ ,1 w
( 1−α )
a (1 − α)
Corollary 2. If we put δ = q = 1, Theorem 1 reduces to the result of [3]:
ξ
h β
ρ i
Γ 1 + 1−α
ξ
β
(ξ,α)
γ,q
P0+
t k −1 Ek,ρ,β
wt k (x) = xξ+ k k (1+ 1−α )
β
[a (1 − α)] k
" kρ #
x
γ
×Ek,α,β+k 1+ ξ
.
w
( 1−α )
a (1 − α)
Corollary 3. if δ = q = 1 and k = 1, we obtain the result of Nair [9]:
ξ
h
i
Γ 1 + 1−α
(ξ,α)
γ
tβ−1 Eρ,β
(wtρ ) (x) = xξ+β
P0+
β
[a (1 − α)]
ρ x
γ
.
×Eρ,β+1
w
ξ
)
(1+ 1−α
a (1 − α)
Theorem 2. Let ρ,β,γ,
δ, ξ ∈ C , k ∈ R, {< (ρ) , < (β) , < (γ) , < (δ) , < (ξ)} >
ξ
0, η > 0, < 1 + 1−α > 0, α < 1, k, w ∈ R and q ∈ (0, 1) ∪ N. Then the pathway
fractional integral representation of (1.5) is given by
ξ
ρ i
h β
Γ
1
−
ξ
β
α−1
(ξ,α)
γ,q
P0+
t k −1 Ek,ρ,β,δ
wt k (x) = xξ+ k k (1− 1−α )
β
[−a (1 − α)] k
" kρ #
x
γ,q
(2.2)
×Ek,α,β+k 1− ξ ,δ w
.
( α−1 )
−a (1 − α)
Proof. By applying (1.8) and (1.9), we have
(ξ,α)
P0+
h
t
β1 −1
γ,q
Ek,α,β,δ
wt
ρ
k
i
x
]
Z [ −a(1−α)
ξ
β
a (α − 1) t −(α−1)
t k −1 1 +
x
0
ρ
γ,q
×Ek,α,β,δ
wt k dt.
ξ
(x) = x
For convenience, we denote the right hand integral of the above term by I2 , then
I2
x
]
Z [ −a(α−1)
ξ
a (α − 1) t −(α−1)
= x
t
1+
x
0
ρ n
∞
X
(γ)nq,k
wt k
×
.
Γ (ρn + β) (δ)n
n=0 k
ξ
β
k −1
6 K. S NISAR, S. D. PUROHIT, M. S. ABOUZAID, MAYSAA AL QURASHI, AND D. BALEANU
Now, on interchanging the order of integration and summation, we have
∞
X
(γ)nq,k
wn
I2 = xξ
Γ (ρn + β) (δ)n
n=0 k
x
]
ξ
Z [ −a(α−1)
a (α − 1) t −(α−1) β + ρ n−1
1+
tk k
dt.
×
x
0
By putting
(1.4), we get
−a(α−1)t
x
I2
= u and evaluating the inner integral by beta function using
kρ n
x
β
∞ (γ)nq,k w −a(α−1)
X
xξ+ k
ρ β
=
β
[−a (1 − α)] k n=0 Γ kρ n + βk k k n+ k −1 (δ)n
ξ
Γ 1 − 1−α
Γ kρ n + βk
× ξ
Γ kρ n + βk + 1 − α−1
ξ
Γ
1
−
β
α−1
= xξ+ k
β
[−a (α − 1)] k
kρ n
x
(γ)
w
∞
nq,k
−a(α−1)
X
ρ β
×
.
ρ
β
ξ
n+ k −1 (δ)
n=0 Γ k n + k + 1 − α−1 k k
n
Again, on applying (1.4.), we arrive at the following desired result:
ξ
β
" ξ
kρ #
xξ+ k k (1− α−1 ) Γ 1 − α−1
x
γ,q
.
I2 =
Ek,α,β+k
w
ξ
β
),δ
(1− α−1
−a (α − 1)
[−a (α − 1)] k
Corollary 4. Let δ = 1, then Theorem 2 reduces to the result given in [2]:
h β
ρ i
(ξ,α)
γ,q
P0+
t k −1 Ek,ρ,β
wt k (x)
ξ
β
" ξ
kρ #
xξ+ k k (1− α−1 ) Γ 1 − α−1
x
γ,q
=
Ek,α,β+k 1− ξ
w
.
β
( α−1 )
−a (α − 1)
[−a (α − 1)] k
Corollary 5. If δ = 1 and k = q = 1, we get the well known results of [9]:
h β
i
(ξ,α)
γ,1
P0+
t k −1 E1,ρ,β
(wtρ ) (x)
ξ
ρ xξ+β Γ 1 − α−1
x
γ
=
E
w
.
ξ
β
1,α,β+(1− α−1
)
−a (α − 1)
[−a (α − 1)]
3. Conclusion
In this paper, we have introduced a more generalized Mittag Leffler function. For
this extended Mittag Leffler function, we have presented two pathway fractional
integral formulas (PFIF). The result obtained in the present paper provides an
GENERALIZED k-MITTAG LEFFLER FUNCTION AND ITS PROPERTIES
7
extension of the known results, as mentioned earlier. We conclude our paper with
the remark that, the function introduced and results deduced above are significant
and can lead to yield numerous other integral formulas involving various Mittag
Leffler type functions.
Competing Interests All the authors are not having any competing interests in
the manuscript.
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8 K. S NISAR, S. D. PUROHIT, M. S. ABOUZAID, MAYSAA AL QURASHI, AND D. BALEANU
K. S Nisar, Department of Mathematics, College of Arts and Science, Prince Sattam
bin Abdulaziz University, Wadi Al-Dawaser, Saudi Arabia
E-mail address: ksnisar1@gmail.com
S. D. Purohit, Department of HEAS (Mathematics), Rajasthan Technical University,
Kota 324010, Rajasthan, India
E-mail address: sunil a purohit@yahoo.com
M. S. Abouzaid, Department of Mathematics, College of Arts and Science, Prince
Sattam bin Abdulaziz University, Wadi Al-Dawaser, Saudi Arabia
E-mail address: moheb abouzaid@hotmail.com
Maysaa Al Qurashi, Department of Mathematics, King Saud University, P.O. Box
22452, Riyadh 11495, Saudi Arabia
E-mail address: maysaa@ksu.edu.sa
D. Baleanu, Department of Mathematics and Computer Sciences, Cakaya University, Balgat 06530, Ankara, Turkey. Institute of Space Sceinces, Magurele-Bucharest,
Romania
E-mail address: dumitru@cankaya.edu.tr