International Journal of Mathematics Research.
ISSN 0976-5840 Volume 9, Number 1 (2017), pp. 5-12
© International Research Publication House
http://www.irphouse.com
Certain Type of Pathway Fractional Integral
Operator Associate via General Class of Polynomial
Hemlata Saxena1 and Rajendra Kumar Saxena2
Professor of Mathematics1 and Asst. Professor of Mathematics2
1&2
Career Point University, Kota (Rajasthan) India.
Abstract
In the present paper we consider product of some special functions with a
general class of polynomial via pathway fractional integral operator .This
operator generalizes of the Riemann-Liouville fractional integral operator. Our
results are quite general in nature .Some known and new results are also obtain
here.
Keywords: Pathway fractional integral operator/ M-Series/ New generalized
Mittag-Leffler function/ a general class of polynomials
INTRODUCTION
DEFINITION: Let 𝑓(𝑥) ∈ 𝐿(𝑎, 𝑏) , 𝜂 ∈ 𝐶 , 𝑅( 𝜂) > 0, 𝑎 > 0 and let us take a
“Pathway parameter” 𝛼 < 1.Then the pathway fractional integration operator is
defined by Nair [8]
(𝜂 ,𝛼)
(𝑃0+ 𝑓) (𝑥)
𝜂
(
= 𝑥 ∫0
𝑥
)
𝑎(1−𝛼)
𝜂
[1 −
𝑎(1−𝛼)𝑡 (1−𝛼)
]
𝑥
𝑓(𝑡)𝑑𝑡
......(1.1)
when𝛼 = 0 , 𝑎 = 1 𝑎𝑛𝑑 𝜂 is replaced by 𝜂 − 1 in (1.1) it yields
𝜂
1
𝑥
(𝐼0+ 𝑓)(𝑥) = Г(𝜂) ∫0 (𝑥 − 𝑡) 𝜂−1 𝑓(𝑡)𝑑𝑡
…(1.2)
which is the left – sided Riemann-Liouville fractional integral defined by
Samko et. al.[9].
Hemlata Saxena and Rajendra Kumar Saxena
6
The pathway model is introduced by Mathai [5] and studied further by Mathai and
Houbold [6] [7].
For 𝑅(𝛼) > 0, the pathway model for scalar random variables is represented by the
following probability density function.
𝛽
𝑓(𝑥) = 𝑐|𝑥|𝛾−1 [1 − 𝑎(1 − 𝛼)|𝑥|𝛿 ]1−𝛼
.......(1.3)
𝛾 > 0, 𝛿 > 0, 𝛽 ≥ 0 , {1 − 𝑎(1 − 𝛼)|𝑥|𝛿 } > 0, −∞ < 𝑥 < ∞,
where c is the normalizing constant and 𝛼 is pathway parameter. For real, the
normalizing constant is as follows:
𝛾
𝑐=
𝛾
𝛾
𝛽
1 𝛿[𝑎(1−𝛼)]𝛿 𝛤( 1−𝛼 )
𝛾
𝛽
𝛾
2
𝛤( ) 𝛤(
− )
𝛿
𝛼<1
,
.......(1.4)
1−𝛼
𝛿
=
𝛽
1 𝛿[𝑎(1−𝛼)]𝛿 𝛤(𝛿 + 1−𝛼 + 1)
𝛽
𝛾
2
+ 1)
𝛤( ) 𝛤(
1−𝛼
,
𝛿
1
for 1−𝛼 −
𝛾
𝛿
> 0, 𝛼 > 1
......(1.5)
𝛾
1 𝛿[𝑎𝛽]𝛿
=2
𝛾
𝛿
𝛤( )
,
for 𝛼 → 1
......(1.6)
For 𝛼 < 1, it is a finite range density with [ 1 − 𝑎(1 − 𝛼)|𝑥|𝛿 ] > 0 and (1.3) remains
in the extended generalized type-1 beta family. For 𝛼 < 1, the pathway density in
(1.3) includes the extended type-1 beta density, the triangular density, the uniform
density and many other p.d. f.
when 𝛼 > 1, we write 1 − 𝛼 = −(𝛼 − 1) , then
(𝜂 ,𝛼)
(𝑃0+ 𝑓) (𝑥)
𝜂
(−
= 𝑥 ∫0
𝑥
)
𝑎(𝛼−1)
𝜂
[1 +
𝑎(𝛼−1)𝑡 −(𝛼−1)
]
𝑥
𝑓(𝑥) = 𝑐 |𝑥|𝛾−1 [1 + 𝑎(𝛼 − 1)|𝑥|𝛿 ]
Where
𝛽
𝛼−1
−
𝑓(𝑡)𝑑𝑡
...... (1.7)
𝛼 > 1, 𝛿 > 0, 𝛽 ≥ 0, −∞ < 𝑥 < ∞,
which is 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.
Here, we consider only the case of pathway parameter 𝛼 < 1. For 𝛼 → 1 both (1.3)
and (1.7) take the exponential form, since.
Certain Type of Pathway Fractional Integral Operator Associate via General Class…
7
𝜂
𝑙𝑖𝑚 |𝑥|𝛾−1
𝑐
[1 − 𝑎(1 − 𝛼)|𝑥|𝛿 ]1−𝛼
𝛼→1
𝜂
−
𝑙𝑖𝑚 |𝑥|𝛾−1
=
𝑐
[1 + 𝑎(𝛼 − 1)|𝑥|𝛿 ] 𝛼−1
𝛼→1
= 𝑐|𝑥|𝛾−1 𝑒 −𝛼𝜂|𝑥|
𝛿
.........(1.8)
𝜂
For → 1− , [1 −
𝑎(1−𝛼)𝑡 1−𝛼
]
𝑥
(𝜂 ,1)
(𝑃0+
𝑎𝜂
→ 𝑒 − 𝑥 𝑡 , the operator (1.1) reduces to the following form
𝑎𝜂
∞
𝑓) (𝑥) = 𝑥 𝜂 ∫0 𝑒 − 𝑥 𝑡 𝑓(𝑡)𝑑𝑡
𝑎𝜂
= 𝑥 𝜂 𝐿𝑓 ( 𝑥 )
......(1.9)
It reduces to the Laplace integral transform of f with parameter
𝑎𝜂
𝑥
.
In this paper we will integrate product of M-series, Fox’s H-function and generalized
Mittag-Leffler function by means of pathway model.
The following general class of polynomials introduced by Srivastava [13]
[𝑛⁄ ] (−𝑛)𝑚𝑙
𝑆𝑛𝑚 [𝑥] = ∑𝑙=0𝑚
= 𝜓1 (𝑙)
𝐴
𝑛,𝑙𝑥
𝑙!
𝑙
l =0,1,2
............(1.10)
When m is an arbitrary positive integer and the coefficients 𝐴𝑛,𝑙 (𝑛, 𝑙 ≥ 0) are
arbitrary constants, real or complex
The generalized M-series is defined and studied by Sharma and Jain [11] as follows
′
′
𝛼 ′ , 𝛽′ (𝑧) = ∑∞ (𝑎1 )𝑘…….(𝑎𝜌 )𝑘
𝑘=0 (𝑏 ′ ) …….(𝑏′ )
𝜎 𝑘
1 𝑘
𝜌 𝑀 𝜎
= 𝜓1 (𝑘)
𝑧𝑘
𝛤(𝛼′ 𝑘+𝛽 ′ )
…(1.11)
Where 𝑧, 𝛼 ′ , 𝛽 ′ ∈ 𝐶, 𝑅𝑒(𝛼 ′ ) > 0
Here (𝑎𝑗′ ) , (𝑏𝑗′ ) are known as Pochammer symbols. The series (1.11) is defined
𝑘
𝑘
′
when none of the parameters 𝑏𝑗𝑠
, 𝑗 = 1,2, … 𝜎 is negative integer or zero. The series
in (1.11) is convergent for all z if 𝜌 ≤ 𝜎, it is convergent for |𝑧| < 𝛿 = 𝛼 𝛼 if 𝜌 = 𝜎 +
1 and divergent, if 𝜌 > 𝜎 + 1 . When 𝜌 > 𝜎 + 1 and |𝑧| < 𝛿, the series can
converge on conditions depending on the parameters.
Hemlata Saxena and Rajendra Kumar Saxena
8
The series representation of Fox H- function studied by Fox C [2] as follows:
𝑀,𝑁
𝐻𝑃,𝑄
[𝑧|
where 𝜉 =
𝜉
𝑣
(𝑒𝑝 , 𝐸𝑝 )
∞ (−1) 𝑋(𝜉) 1
∑
] = ∑𝑁
(
)
,
ℎ=1 𝑣=0 𝑣! 𝐸
𝑧
(𝑓𝑄, 𝐹𝑄 )
ℎ
𝑒ℎ −𝑣−1
𝐸ℎ
(ℎ = 1,2, …
and
…(1.12)
, 𝑁)
and
𝑁
∏𝑀
𝑗=1 𝛤(𝑓𝑗 +𝐹𝑗 𝜉) ∏𝑗=1 𝛤(1−𝑒𝑗 +𝐸𝑗 𝜉)
𝑋(𝜉) =
𝑗≠ℎ
𝑄
∏𝑗=𝑚+1 𝛤(1−𝑓𝑗 −𝐹𝑗 𝜉) ∏𝑃
𝑗=𝑁+1 𝛤(𝑒𝑗 +𝐸𝑗 𝜉)
…..(1.13)
Following are the convergence conditions:
𝑄
𝑃
𝑀
𝑇1 = ∑𝑁
𝑖=1 𝐸𝑖 − ∑𝑖=𝑁+1 𝐸𝑖 + ∑𝑖=1 𝐹𝑖 − ∑𝑖=𝑀+1 𝐹𝑖
….(1.14)
𝑞
𝑇2 = ∑𝑛𝑖=1 𝛼𝑖 − ∑𝑞𝑖=𝑛+1 𝛼𝑖 + ∑𝑚
𝑖=1 𝛽𝑖 − ∑𝑖=𝑚+1 𝛽𝑖
…(1.15)
Recently, a new generalization of Mittag-Leffler function was defined by Faraj and
Salim [3] as follows:
(𝛾)
𝑧𝑛
𝛾,𝛿,𝑞
𝑞𝑛
𝐸𝛼,𝛽,𝑝 (𝑧) = ∑∞
𝑛=0 𝛤(𝛼𝑛+𝛽) (𝛿)
𝑝𝑛
…(1.16)
Where 𝑧, 𝛼, 𝛽, 𝛾, 𝛿 ∈ 𝐶; 𝑀𝑖𝑛{ 𝑅𝑒(𝛼), 𝑅𝑒(𝛽), 𝑅𝑒(𝛾), 𝑅𝑒(𝛿)} > 0, 𝑝, 𝑞 > 0
Further, generalization of Mittag- Leffler function was defined by Khan and Ahmed
[4] as follows:
𝜇,𝜌,𝛾,𝑞
𝐸𝛼,𝛽,𝑣,𝜎,𝛿,𝑝 (𝑧) = ∑∞
𝑛=0
(𝜇)𝜌𝑛 (𝛾)𝑞𝑛 𝑧 𝑛
𝛤(𝛼𝑛+𝛽)(𝑣)𝜎𝑛 (𝛿)𝑝𝑛
…(1.17)
Where 𝛼, 𝛽, 𝛾, 𝛿, 𝜇, 𝑣, 𝜌, 𝜎 ∈ 𝐶; 𝑝, 𝑞 > 0 and 𝑞 ≤ 𝑅𝑒(𝛼) + 𝜌𝑝, and 𝑀𝑖𝑛{𝑅𝑒(𝛼),
𝑅𝑒(𝛽), 𝑅𝑒(𝛾), 𝑅𝑒(𝛿), 𝑅𝑒( 𝜇), 𝑅𝑒(𝑣), 𝑅𝑒(𝜌), 𝑅𝑒(𝜎)} > 0 If we take 𝜇 = 𝑣, 𝜌 = 𝜎 in
(1.17) it reduces to eq. (1.16).
Certain Type of Pathway Fractional Integral Operator Associate via General Class…
9
Write generalized hypergeometric function was defined by Srivastava and Manocha
[12] as follows:
𝑝𝜓𝑞 [(𝑎1 , 𝐴1 ), … (𝑎𝑝 , 𝐴𝑝 ); (𝑏1 , 𝐵1 ), … , (𝑏𝑞 , 𝐵𝑞 ); 𝑧]
𝑝
∏
𝑖=1
= ∑∞
𝑛=0 ∏𝑞
𝛤(𝑎𝑖 +𝐴𝑖 𝑛)𝑧 𝑛
𝑗=1 𝛤(𝑏𝑗 +𝐵𝑗 𝑛)𝑛!
′′
𝑀′ ′ −β
𝑆𝑛′
[d t
]
….(1.18)
MAIN RESULTS
Theorem-1 Let 𝜂, 𝛾, 𝛿, 𝑞, 𝑝, 𝜔, 𝜌 ∈ 𝐶, 𝑐, 𝑏 ∈ 𝑅, 𝑅𝑒(𝛽) > 0, 𝑅𝑒(𝛿) > 0, 𝑅𝑒(𝜂) > 0,
𝜂
𝑅𝑒(𝛾) > 0, 𝑅𝑒(𝜔) > 0, 𝑅𝑒 (1 + 1−𝛼) > 0, 𝑅𝑒(𝜌) > 0, 𝛼 < 1, 𝑏 ∈ 𝑅, 𝑐 ∈ 𝑅,
𝑓
𝛼′
1
𝑅𝑒 (𝜔 + 𝛿 𝐹𝑗 ) > 0, |arg 𝑐| < 𝑇1 𝜋, 𝑇1 𝑇2 > 0, 𝜌 ≤ 𝜎, |𝑑| < 𝛼 ′ , 𝛽 ∗ > 0, 𝑗 = 1, … , 𝑄;
2
𝑗
Then
(𝜂,𝛼)
𝑃0+
{𝑡 𝜔−1
′
∗
′′
′ (𝑒𝑝 , 𝐸𝑝 )
𝛼 ′, 𝛽′
𝛾,𝛿,𝑞
𝑀,𝑁
[𝑑𝑡 −𝛽 ]. 𝑆𝑛𝑀′ [d′ t −β ] 𝐻𝑃,𝑄
[𝑐𝑡 𝛿 |
] . 𝐸𝛼,𝛽,𝑝 (𝑏𝑡𝜌 )}
𝑀
(𝑓
𝐹
)
𝜌
𝑄, 𝑄
𝜎
= 𝜓1 (𝑘) 𝜓2 (𝑙)
∗
𝜂
𝑑 𝑘 [𝑑′ ]𝑙 𝑥 𝜂+𝜔−𝛽 𝑘 (𝛽 ′′ )𝑙 𝛤 (1 + 1 − 𝛼 ) 𝛤(𝛿)
𝛤(𝛾)[𝑎(1 − 𝛼)]𝜔−𝛽
∗𝑘
− (𝛽 ′′ )𝑙
(𝛾, 𝑞) (𝜔 − 𝛽 ∗ 𝑘, (𝛽 ′′ )𝑙 − 𝛿ƺ, 𝜌)
𝑏𝑥 𝜌
𝜂
].
|
2𝜓3 [
− 𝛿𝜉 − 𝛽 ∗ 𝑘, (𝛽 ′′ )𝑙, 𝜌)
[𝑎(1 − 𝛼)]𝜌 (𝛽, 𝛼)(𝛿, 𝜌) (1 + 𝜔 +
1−𝛼
′
(𝑒𝑝 , 𝐸𝑝 )
𝑐𝑥 𝛿
𝑀,𝑁
𝐻𝑃,𝑄
[
]
………..(2.1)
′|
𝛿
[𝑎(1−𝛼)]
(𝑓𝑄, 𝐹𝑄 )
Proof: The theorem -1 can be evaluated by using the definitions
(1.1),(1.10),(1.11),(1.12) and (1.16) then by interchange the order of integrations and
summations, evaluate the inner integral by making use of beta function formula, we
arrive at the desired result (2.1).
Theorem-2 Let 𝜂, 𝛾, 𝛿, 𝑞, 𝑝, 𝛽, 𝑇1 , 𝑇2 > 0, 𝜇, 𝜌, 𝛾, 𝜗, 𝛽, 𝑣, 𝜎, 𝛿 ∈ 𝐶 , 𝑅𝑒(𝜂) > 0, 𝑅𝑒(𝛾)
𝑓
𝜂
> 0, 𝑅𝑒(𝛽) > 0, 𝑅𝑒 (1 + 1−𝛼) > 0, 𝑏, 𝑐 ∈ 𝑅, 𝛼 < 1, 𝑅𝑒 (𝜔 + 𝛿 𝐹𝑗 ) > 0, |arg 𝑐| <
1
𝑇 𝜋, 𝜌
2 1
≤ 𝜎 𝑎𝑛𝑑 |𝑑| < 𝛼
′𝛼
′
𝑗
∗
, 𝛽 > 0, 𝑗 = 1, … , 𝑄
𝑎𝑛𝑑 min(𝑅𝑒(𝜗), 𝑅𝑒(𝛽), 𝑅𝑒(𝛾), 𝑅𝑒(𝛿), 𝑅𝑒(𝜇), 𝑅𝑒(𝑣), 𝑅𝑒(𝜌), 𝑅𝑒(𝜎)) > 0
Hemlata Saxena and Rajendra Kumar Saxena
10
Then
(𝜂,𝛼)
𝑃0+
{𝑡𝛽−1
= 𝜓1 (𝑘)
∗
′ (𝑒𝑝 , 𝐸𝑝 )
𝛼 ′, 𝛽′
𝜇,𝜌,𝛾,𝑞
𝑀,𝑁
𝑀′ ′ −β′′
[𝑑𝑡 −𝛽 ]. 𝑆𝑛′
[d t
] 𝐻𝑃,𝑄
[𝑐𝑡 𝛿 |
] . 𝐸𝜗,𝛽,𝑣,𝜎,𝛿,𝑝 (𝑏𝑡 𝜗 )}
(𝑓𝑄, 𝐹𝑄 )
𝜌 𝑀 𝜎
𝑑𝑘 𝑥 𝜂+𝛽−𝛽
∗ 𝑘−𝛽 ′′ 𝑙
𝜂
𝛤 (1 + 1 − 𝛼) 𝛤(𝑣)𝛤(𝛿)
[𝑎(1 − 𝛼)]𝛽−𝛽∗𝑘−𝛽′′𝑙 𝛤(𝜇)𝛤(𝛾)
. 𝜓2 (𝑙) (𝑑 ′ )𝑙
(𝜇, 𝜌) (𝛾, 𝑞) (𝛽 − 𝛽 ∗ 𝑘 − 𝛽′′𝑙 − 𝛿′ƺ, 𝜗)
𝑏𝑥 𝜗
𝜂
|
].
3𝜓3 [
[𝑎(1 − 𝛼)]𝜗 (𝛽, 𝛼)(𝛿′, 𝜌)(𝜗, 𝜎) (1 + 𝛽 +
− 𝛿 ′ 𝜉 − 𝛽 ∗ 𝑘 − 𝛽′′𝑙, 𝜗)
1−𝛼
′
(𝑒𝑝 , 𝐸𝑝 )
𝑐𝑥 𝛿
𝑀,𝑁
𝐻𝑃,𝑄
[[𝑎(1−𝛼)]𝛿′ |
]
...(2.2)
(𝑓𝑄, 𝐹𝑄 )
Proof: The theorem -2 can be evaluated by using the definitions (1.1),(1.10) (1.11)
and (1.16) then by interchange the order of integrations and summations, evaluate the
inner integral by making use of beta function formula, we arrive at the desired result
(2.2).
SPECIAL CASES:
1. If we take 𝛿 = 𝑝 = 𝑞 = 1, 𝜌 = 𝛽,𝛼 = 𝛽, 𝛽 = 𝜔 and in H- function 𝛿 ′ = 𝛿 in
theorem -1 then we at once arrive at the known result of [1,Theorem-2].
2. If we take 𝛿 = 𝑝 = 1 in theorem -1 then we get the following particular case of
the solution (2.1)
Corollary-1
The following formula holds
(𝜂,𝛼)
𝑃0+
{𝑡 𝜔−1
∗
′ (𝑒𝑝 , 𝐸𝑝 )
𝛼 ′, 𝛽′
𝛾,𝑞
𝑀,𝑁
[𝑑𝑡 −𝛽 ]. 𝐻𝑃,𝑄
[𝑐𝑡 𝛿 |
] . 𝐸𝜌,𝜔 (𝑏𝑡𝜌 )}
(𝑓𝑄, 𝐹𝑄 )
𝜌 𝑀 𝜎
= 𝜓1 (𝑘)
∗
𝜂
)
𝑑𝑘 𝑥 𝜂+𝜔−𝛽 𝑘 𝛤(1+
1−𝛼
∗
𝛤(𝛾)[𝑎(1−𝛼)]𝜔−𝛽 𝑘
(𝜔 − 𝛿𝜉 − 𝛽 ∗ 𝑘, 𝜌)(𝛾, 𝑞)
𝑏𝑥 𝜌
𝜂
|
]
2𝜓2 [
[𝑎(1 − 𝛼)]𝜌 (𝜔, 𝜌) (1 + 𝜔 +
− 𝛿𝜉 − 𝛽 ∗ 𝑘, 𝜌)
1−𝛼
′
𝑀,𝑁
𝐻𝑃,𝑄
(𝑒𝑝 , 𝐸𝑝 )
𝑐𝑥 𝛿
[
|
]
′
[𝑎(1 − 𝛼)]𝛿 (𝑓𝑄, 𝐹𝑄 )
Certain Type of Pathway Fractional Integral Operator Associate via General Class…
11
Where 𝜂, 𝛾, 𝑞, 𝜔, 𝜌 ∈ 𝐶, 𝑐, 𝑏 ∈ 𝑅, 𝑅𝑒(𝛽) > 0, 𝑅𝑒(𝛿) > 0, 𝑅𝑒(𝜂) > 0, 𝑅𝑒(𝛾) >
𝑓𝑗
𝜂
0, 𝑅𝑒(𝜔) > 0, 𝑅𝑒 (1 + 1−𝛼) > 0, 𝑅𝑒(𝜌) > 0, 𝛼 < 1, 𝑏 ∈ 𝑅, 𝑐 ∈ 𝑅, 𝑅𝑒 (𝜔 + 𝛿 𝐹 ) >
0, |arg 𝑐| <
1
𝑇 𝜋, 𝑇1 𝑇2
2 1
> 0, 𝜌 ≤ 𝜎, |𝑑| < 𝛼
′𝛼
′
𝑗
∗
, 𝛽 > 0, 𝑗 = 1, … , 𝑄;
3. If we take 𝜇 = 𝑣, 𝜌 = 𝜎, 𝛿 = 𝑝 = 𝑞 = 1 and 𝜗 → 𝛽, 𝛽 → 𝜔 in H function
𝛿 ′ = 𝛿 in theorem-2 then we at once arrive at the known result of
[1, Theorem-1] .
4. If we take 𝜇 = 𝑣, 𝜌 = 𝜎 then we at once arrive at the theorem-1.
5. Making 𝛽 ∗ , 𝛿 ′ → 0 and 𝛿 = 𝑝 = 𝑞 = 1, 𝜌 = 𝛽 in the result (2.1) and
𝛽 ∗ , 𝛿 ′ → 0, 𝜇 = 𝑣, 𝜌 = 𝜎, 𝛿 = 𝑝 = 𝑞 = 1 in result (2.2) then we at once arrive
at the known result of Nair in[8].
6. If we take 𝑛′ → 0 in theorem1 and theorem2 we get the result recently
established by H. Saxena[10]
7. 𝛽 ′ , 𝛿 ′ 𝑎𝑛𝑑 𝜂′ → 0 and 𝛿 = 𝑝 = 𝑞 = 1, 𝜑 = 𝛽 𝑖𝑛 (2.1) and 𝛽 ∗ , 𝛿 ′ , 𝜂′ → 0,
𝜇 = 𝜗 𝜌 = 𝜎, 𝛿 = 𝜌 = 𝑞 = 1 𝑖𝑛 (2.2) then we at once arrive at the know
results of Nair in [8]
REFERENCES
[1]
Chaurasia V. B. L. and Singh J., 2012, “Pathway Fractional Integral operator
Associated with certain special functions”, Global J. Sci. Front. Res., 12(9)
(Ver.1.0).
[2]
Fox, C., 1961, The G and H-functions as symmetrical Fourier kernels, Trans.
Amer. Math soc., 98, pp. 395-429.
[3]
Fraj, A, Salim, T., Sadek, S. and Ismail,J., 2013, “ Generalized Mittag-Leffler
function associated with Weyl Fractional Calculus Operators”, J. of
Mathematics Hindawi Publishing Corporation , ID 8217621,5 pages.
[4]
Khan, M.A and Ahmed, S., 2013, “On some properties of the generalized
Mittag-Leffler function”, Springer plus, 2:337.
[5]
Mathai, A.M., 2005, “A pathway to matrix variate gamma and normal
densities”, Linear Algebra and its Applications, 396, pp. 317-328.
[6]
Mathai, A.M. and Haubold, H.J., 2008, “On generalized distribution and
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[7]
Mathai, A.M. and Haubold, H.J., 2007, “Pathway models, superstatisties,
trellis statistics and a generalized measure of entropy”, physica, A 375, pp.
110-122.
12
Hemlata Saxena and Rajendra Kumar Saxena
[8]
Nair, Seema S, 2009, “Pathway fractional integration operator”, Fract.
Cal.Appl. Anal., 12(3), 237-259.
[9]
Samko, S.G., Kilbas, A.A. and Marichev, O.I., 1993, “Fractional integral and
Derivatives”. Theory and Applications, Gordenand Breach, Switzerland.
[10] Saxena H., Saxena, R.K., 2015, “On Certain type fractional integration of
special functions via Pathway Operator”, Global Journal Of Science Frontier
Research (F) Vol.(15) issue 8 Version I .
[11] Sharma, Manoj and Jain, Renu, 2009, “A note on a generalized M-series as a
special function of fractional calculus”, Fract. Cal. Appl. Anal. 12(4), pp.
449-452.
[12] Srivastava, H.M. and Manocha, H.L., 1984, “A treatise on generating
functions,” John Wiley and Sons, Eillis Horwood, New York, Chichester.
[13] Srivastava, H.M., 1972, “A Contour integral involving Fox’s H-function”,
Indian J-Math, 14, pp. 1-6.