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Some results on the generalized Mittag-Leffler function operator
Journal of Inequalities and Applications volume 2013, Article number: 33 (2013)
This paper is devoted to the study of a generalized Mittag-Leffler function operator introduced by Shukla and Prajapati (J. Math. Anal. Appl. 336:797-811, 2007). Laplace and Mellin transforms of this operator are investigated in this paper. Some special cases of the established results are also deduced as corollaries. The results obtained are useful where the Mittag-Leffler function occurs naturally.
MSC:33E12, 26A33, 44A45.
A generalization of (1.1) was given by Wiman  in 1905 in the form
In 1971, in connection with the solution of certain singular integral equations, a further interesting and useful generalization of (1.2) was introduced by Prabhakar  in the form
where is the Pochhammer symbol defined by
A generalization of (1.3) is given by Shukla and Prajapati  in the following form:
where (; , , , ).
The above generalization studied by Shukla and Prajapati is shown in a series of papers [5–8]. A generalization of Mittag-Leffler functions defined by (1.3) and (1.5) is introduced and studied by Srivastava and Tomovski  in the form
where ; , , , and the Pochhammer symbol for is defined by
The object of this paper is to derive the Laplace and Mellin transforms of the following integral operator associated with the generalized Mittag-Leffler function, defined by Shukla and Prajapati  as well as Srivastava and Tomovski , in the next sections:
where ; , , ; .
Shukla and Prajapati  have shown that the Mellin-Barnes integral for the function defined by (1.5) is given by
As , then by virtue of the limit formula,
reduces to the familiar Reimann-Liouville fractional integral
In the following, we will use the representation of a high transcendental function in terms of the so-called H-function defined as 
and an empty product is always interpreted as unity; with , , , or ℂ (; ) such that
The contour L is the path of integration in the complex ξ-plane running from to for some real number γ.
2 Mellin transform of the operator (1.8)
Theorem 2.1 It is shown here that
where (, , ); , andis the H-function defined by (1.12).
Mellin transform is defined as
Therefore, we have
Interchanging the order of integration, which is permissible under the conditions given in Theorem 2.1, we find that
If we consider in the r.h.s. of (2.2), we get
To evaluate the u-integral, we express the Mittag-Leffler function in terms of its Mellin-Barnes contour integral by means of the formula (1.9), then the above expression transforms into the form
If the u-integral is evaluated with the help of the formula
then after some simplification, it is seen that the right-hand side of above equation (2.4) simplifies to
which, on being interpreted by the definition of H-function (1.12), yields the desired result. □
For , Theorem 2.1 reduces to the following corollary.
Corollary 2.1 The following result holds:
where (, , ); andis the H-function defined by (1.12).
Theorem 2.2 It is shown here that
where, , , , andis the H-function defined by (1.12).
3 Laplace transform of the operator (1.8)
Theorem 3.1 The following result holds:
where (); andis the Laplace transform ofdefined by
whereand the integral is convergent.
Proof By virtue of the definitions (1.8) and (3.2), it follows that
Interchanging the order of integration, which is permissible under the conditions given in Theorem 3.1, we find that
If we consider , we obtain
On making use of the series definition (1.5), the above expression becomes
and is the Laplace transform of . □
For , Theorem 3.1 reduces to the following corollary.
Corollary 3.1 The following result holds:
where (, , ); and the integral operator is the one discussed by Prabhakardefined by (1.10).
Theorem 3.2 The following result holds:
where, , , ; andis the Laplace transform ofdefined by (3.2).
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The authors declare that they have no competing interests.
The authors contributed equally and significantly in writing this article. The authors read and approved the final manuscript.
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Prajapati, J., Jana, R., Saxena, R. et al. Some results on the generalized Mittag-Leffler function operator. J Inequal Appl 2013, 33 (2013). https://doi.org/10.1186/1029-242X-2013-33
- generalized Mittag-Leffler function
- Laplace transform
- Mellin transform
- generalized Wright function