Note on the Fractional Mittag-Leffler Functions by Applying the Modified Riemann-Liouville Derivatives

abstract: In this article, the fractional derivatives in the sense of the modified Riemann-Liouville derivative is employed for constructing some results related to Mittag-Leffler functions and established a number of important relationships between the Mittag-Leffler functions and the Wright function.


Introduction
It is well known that with the classical Riemann-Liouville definition of fractional derivative [2,5,15], the fractional derivative of a constant is not zero. The most useful alternative which has been proposed to cope with this feature is known Caputo derivative [6], but in this derivative fractional derivative would be defined for differentiable functions only. A modification of the Riemann-Liouville has been defined to deal with non-differentiable functions [3,4,9,21,16,23] and it is given as: ; m ≤ α < m + 1.
The other important function which is a generalization of series is represented by: The functions (1.1) and (1.2) play important role in fractional calculus, also we note that when β = 1 in Another form which is generalization of (1.1) and (1.2) was introduced by Prabhakar [22] such as: where (δ) k , the Pochhammer symbol, is defined by There are some special cases of (1.3) such as: The second functions will be discussed is Wright function, which is defined as This function plays an important role in the solution of a linear partial differential equation. Furthermore, there is an interesting link between the Wright function and the Mittag-Leffler function. Hence, some useful relationships between those functions have been obtained in this work.

Main Result
Now, we point out some formulas which do not hold for the classical Riemann-Liouville definition, but apply with the modified Riemann-Liouville definition.
Then we obtain the following relation Also, the following formula is given Remark 2.2. 1. Since 2. When δ = 1 in formula (2.1), then we obtain 3. When δ = 1 and β = 1 in formula (2.1) and 1 − α −→ 0 + , then we have the following intersting formula A. Kiliçman and W. Saleh Also, we can show this formula by another method such as The following figures show some modified Riemann-Liouville derivative of order closed to zero for E α (x α ).   Corollary 2.3. Let α > 0,β > 0 and for λ ∈ R, then the following formula holds Proof. We can write Note on the Fractional Mittag-Leffler Functions

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Let u = λ 1 α x 1 α and by applying the fractional derivative properties, we get In the following figures there are some modified Riemann-Liouville derivative of order closed to zero for E α (x).
Kiryakova introduced and studied the multi-index Mittag-Leffler function as their typical representatives, including many interesting special cases that have already proven their usefulness in FC and its applications [12]. Definition 2.13. Assume that n > 1 is an integer, η 1 , ..., η n > 0 and β 1 , ..., β n are arbitrary real numbers. The multi-idex Mittag-Leffler function is given as The same function was given by Lunchko [17], called by him Mittag-Leffler function of vector index.
Futhermore, the Wright generalized hypergeometric function mWn is defined as Note on the Fractional Mittag-Leffler Functions

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The mWn function is special case of the Fox H-function In particlar, when A i = B j = 1, ∀i, j, then Meijer's G-function is obtained For more detils see [10,11,13,14].
There are some interested properties related to multi-Mittag-Leffler function which were proven in [12]: In the same paper, the author showed Wright function as a case of multi-Mittag-Leffler function with n = 2: (α,1),(β,1) (x).
which is the result.
As expected when α = 1 η i and n = 1,the last formula turns to be the formula (2.7) when λ = 1.
and by appling (2.17), the following formula is given: We would like to mention that if α = 1 η i and n = 1 in formula (2.18), then (2.8) is obtained.
Hence, we get a very useful relation Note on the Fractional Mittag-Leffler Functions
Moreover, since cos The  The next step we study D β cos α (t α ) and D β cos α (t).
Theorem 2.16. The fractional derivative of hyperbolic function of order m is given as when v − α −→ 0 + , then Proof. Since hyperbolic function of order m is defined as then by using formula (2.6) we get the result.
Theorem 2.17. The fractional derivative of Mellin-Ross function, is given by The proof is directed by using formula (2.2).

Conclusion
In this note, some useful formulas have been established by using modified Riemann-Liouville definition of fractional derivative. These formulas can be used to solve some linear fractional differential equations which are useful in several physical problems.