Caputo MSM fractional differentiation of extended Mittag-Leffler function

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.


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. (1997Chaudhry et al. ( , 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 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 andᵯ 0 and are constants and depend upon the bounded sequence {ҝ n } nϵN 0 . Corresponding to the functionΘ({ҝ n } nϵN 0 ; z), Srivastava et al. (2012) defined extended Gamma function, extended Beta function and extended Gauss hypergeometric function respectively as It is assumed that all the integrals existed.
Corresponding to the extended Beta function It is noted that generalized and extended form of Mittag-Leffler function defined in literature by Desai et al. (2016), Kilbas et al. (2004), Mittag-Leffler (1903), Mittal et al. (2016), Nadir et al. (2014), Özarslan and Yilmaz (2014), and Rahman et al (2017aRahman et al ( , 2017b are special cases of the proposed function defined in (1). Some special cases of this function are descried as: (iii) Another special case of (1) is when ҝ = 1, and = 0 then (1) reduces to Prabhakar's function (Prabhakar, 1971) of three parameters. 1,1 In order to establish our main results, we need definition of Fox-Wright function and the concept of Hadamard products. Pohlen (2009)

Definition: As indicated by
be two power series then the Hadamard product of power series is defined as where and ℎ are the radii of convergence of two series ( ) and ℎ( ) respectively. Therefore, in general, ≥ • ℎ .
It is to be noted that if one of the power series is an analytical function, then the Hadamard product series is also an analytical function.

Caputo-type MSM fractional derivative formula for extended Mittag-leffler function
Here, in this section, our main object is to establish composition of new fractional derivative formulas so called Caputo-type Marichev-Saigo-Maeda fractional operator involving the extended Mittag-leffler function (1) which is defined by Parmar (2015). Some special cases of our main result are considered. We obtain our main goal by applying the Caputo-type MSM fractional derivative given in (8) and (9) Let , / , , / , and the left-sided and rightsided Caputo-type MSM fractional differential operators containing Appell function in their kernel are defined as Here, we discuss lemma, which is essential for the establishment of our main results. This lemma provides the image of power function −1 with Caputo-type Marichev-Saigo-Maeda fractional differentiation. (b) Analogous to above, using right-sided Caputo-type MSM differential operator, we have The last expression can easily be emerged by using Hadamard product rule given in Pohlen (2009) and we get the result.

Lemma
(b) Analogously to the proof of above part, our demonstration of Caputo fractional derivative formula, depends upon the definition of the function (1) and the known result part (b) of the lemma, we have It is to be noted that several further consequences of the main Theorem and Corollaries 3-4 can easily be converted to many other known result by suitable substitutions of the parameters.

Conclusion
In this paper, we obtain the Caputo type MSM fractional derivative of family of Mittag-Leffler function. It is to be noted that said operator transform the required function into a function of higher order. Further, well known operators like Erdelyi-Kober and Saigo's operators are the special case of Marichev-Saigo-Maeda fractional operator.