International Journal of Advanced and Applied Sciences, 5(10) 2018, Pages: 28-34
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International Journal of Advanced and Applied Sciences
Journal homepage: http://www.science-gate.com/IJAAS.html
28
Caputo MSM fractional differentiation of extended Mittag-Leffler function
Aneela Nadir *, Adnan Khan
Department of Mathematics, National College of Business Administration and Economics (NCBAandE), Lahore, Pakistan
ARTICLE INFO ABSTRACT
Article history:
Received 28 April 2018
Received in revised form
29 July 2018
Accepted 5 August 2018
Recently, many researchers are interested in the investigation of an extended
form of special functions like Gamma function, Beta function, Gauss
hypergeometric function, Confluent hypergeometric function and Mittag-
Leffler function etc. Here, in this paper, the main objective is to find the
composition of Caputo MSM fractional differential of the extended form of
Mittag-Leffler function in terms of extended Beta function. Further, in this
sequel, some corollaries and consequences are shown that are the special
case of our main findings.
Keywords:
Extended Mittag-Leffler function
Caputo MSM fractional different-ion
Hadmard product
© 2018 The Authors. Published by IASE. This is an open access article under the CC
BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
*Integral and differential operators in fractional
calculus have become research subject in recently
few decades due to ability of having arbitrary order.
For more recent developments in fractional integral
and differential operators, we refer the reader to see
Agarwal and Choi (2016), Choi and Agarwal (2014),
Choi and Agarwal (2015), Gehlot (2013), Gupta and
Parihar (2017), Kilbas et al. (2004), Nadir et al.
(2014), Rahman et al. (2017b), Saxena and Parmar
(2017), Shishkina and Sitnik (2017), Singh (2013),
Srivastava and Agarwal (2013), Srivaastava et al.
(2012), Suthar et al. (2017), and the references cited
therein.
Now a day, a general trend is in the extensions of
special functions like Gamma function, Beta function,
Gauss hypergeometric function and Mittag-Leffler
function etc. due to its diverse applications in many
applied fields. One can consult the papers by
Chaudhry et al. (1997, 2004), Luo and Raina (2013),
Özarslan and Yilmaz (2014), Rahman et al. (2017b),
and Srivastava et al. (2012) containing the
bibliography therein.
Srivastava et al. (2012) defined a function
Θ({ҝ
}
0
;)≔
{
∑ҝ
!
∞
=0
(
||<ℜ
0<ℜ<∞
ҝ
0
≔1
)
ᵯ
0
exp()[1+ (
1
)] (
ℜ()→∞
ᵯ
0
>0;
)
}
* Corresponding Author.
Email Address: aneelanadir@yahoo.com (A. Nadir)
https://doi.org/10.21833/ijaas.2018.10.005
2313-626X/© 2018 The Authors. Published by IASE.
This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/)
where Θ({ҝ
n
}
nϵN
0
;z) is considered to be analytical
within |z|<ℜ,0<ℜ<∞ and {ҝ
n
}
nϵN
0
is a
sequence of Taylor-Maclaurin coefficients and ᵯ
0
and are constants and depend upon the bounded
sequence {ҝ
n
}
nϵN
0
. Corresponding to the
function Θ({ҝ
n
}
nϵN
0
;z), Srivastava et al. (2012)
defined extended Gamma function, extended Beta
function and extended Gauss hypergeometric
function respectively as
Γ
p
{ҝ
n
}
nϵN
0
(z) = ∫ x
z−1
∞
0
Θ({ҝ
n
}
nϵN
0
;−x−
p
x
)dx
(ℜ(z)>0;ℜ(p)≥0)
Β
p
({ҝ
n
}
nϵN
0
)
(α,β;p)=∫ x
α−1
(1
1
0
−x)
β−1
Θ({ҝ
n
}
nϵN
0
;−
p
x(1−x)
)dx.
(min{ℜ(α),ℜ(β)}≥0;ℜ(p)≥0)
ℑ
({ҝ
n
}
nϵN
0
)
(a,b;c;z)
= ∑()
Β
p
({ҝ
n
}
nϵN
0
)
(+,−;)
(,−)
∞
=0
!
(||<1;ℜ()>ℜ()>0;ℜ()≥0) ℑ
It is assumed that all the integrals existed.
Corresponding to the extended Beta function
Β
p
({ҝ
n
}
nϵN
0
)
, Parmar (2015) defined extension of
Mittag-Leffler function
,
({ҝ
n
}
nϵN
0
;)
(;)
=∑
Β
p
({ҝ
n
}
nϵN
0
)
(+,1−;)
(,1−)
∞
=0
(+)
(1)
where
(
,,;ℜ()>0,
ℜ()>0,ℜ()>1;≥0
)