Some results on the generalized Mittag-Leffler function operator
© Prajapati et al.; licensee Springer 2013
Received: 25 October 2012
Accepted: 13 January 2013
Published: 29 January 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.
where (; , , , ).
where ; , , ; .
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)
where (, , ); , andis the H-function defined by (1.12).
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.
where (, , ); andis the H-function defined by (1.12).
where, , , , andis the H-function defined by (1.12).
3 Laplace transform of the operator (1.8)
whereand the integral is convergent.
and is the Laplace transform of . □
For , Theorem 3.1 reduces to the following corollary.
where (, , ); and the integral operator is the one discussed by Prabhakardefined by (1.10).
where, , , ; andis the Laplace transform ofdefined by (3.2).
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