Decomposition of Generalized Mittag-Leffler Function and Its Properties

The principal aim of the paper is devoted to the study of some special properties of the function for  , , E    z 1 n   . Authors defined the decomposition of the function   , , 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   , , E z   

Authors defined the decomposition of the function 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

Introduction
In 1903, the Swedish mathematician Gosta Mittag-Leffler introduced the function (Gorenflo et al. [1]) defined as, where z is a complex variable and is a gamma function,   .The Mittag-Leffler function is the direct generalisation of the exponential function to which it reduces for 1   .For 0 1    , it interpolates between the pure exponential and a hypergeometric function 1 1 z  .Mittag-Leffler function naturally occurs as the solution of the fractional order differential equations and (or) fractional order integral equations.
Recently, Shukla and Prajapati [9] introduced the function , , , which is defined for , and denotes the generalized Pochhammer symbol which, in particular, reduces to 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
Ikehata and Siltanen [10] defined truncated power series of 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 and as a special case for 1 and .

THEOREM 1. Integral representations of the function
Proof.Consider the integral,   0 and using (1.4), we get Using (1.10), the above equation reduces to and use of (1.12), yields   and using (1.8), we arrive at letes the proof of the theorem.
.4) in left-hand side of (2.2), we obtain Setting m n k   in the first summation and replacing m by in second, right-hand side of (2.3) reduces to This com proof of the theorem.

Remark on
2. If 1 q    , then Theorem 2 leads a pa ar case of (2.3) rticul Re The simplification of the above inequality gives, This proves the theorem.
Remark on Theorem 3: It is easy to verify the following inequality, ,  1 1 where
Proof.From (1.7) and (1.8), we have , using (1.10), the above equation reduces to Now, by involving (1.12), we have . by which we write Therefore, further we have, where which on using (1.12), gives   , and Inequalities (2.9) and (2.10) lead to the proof of the Theorem.
Remark on T 1 q This inequality contains Mittag-Leffler and Exponential functions.
Recently, Shukla and Prajap ned several properties of the function plays an important role in study of the ous prop ional calculus (Shukla and Prajapati [13]).vari erties fract (2.12) Proof.Substituting t zu  in left-hand side of (2.12) and then using (1.10) and (1.12), we get the required result.
Special cases of Theorem 5: For ), we obtain several particular cases as listed below: . (2.17) The above equation reduces to, Applying (1.10), (1.12) and further simplifica the above equation becomes, , where the Equation (2.8) leads to fferentiating above equation with respect to z, we get reby, the theorem is completely proved.The theorem, that follows now, represents the relationship betwee ion and the exponential function.
ionship presumably play an im in the study of the fractional diffusion equation, conduction and mass transfer equation.
in (1.15), we have The left-hand side of (2.19) gives, Use of (2.20) and further simplification of ab tion yields .
Proof.The left-hand side of (2.21) reduces to, The right-hand side can be expressed as [4] R. Gorenflo a ittag-Leffler Func-

Acknowledgements
The authors would like to thank the reviewers for their valuable suggestions to improve the quality of paper.