Further results on the generalized Mittag-Leffler function operator

Ram K Saxena,Ranjan K Jana,Ajay K Shukla,Jignesh P Chauhan

doi:10.1186/s13660-015-0589-4

Abstract

The present paper deals with the study of a generalized Mittag-Leffler function operator. This paper is based on the generalized Mittag-Leffler function introduced and studied by Saxena and Nishimoto (J. Fract. Calc. 37:43-52, 2010). Laplace and Mellin transforms of this new operator are investigated. The results are useful where the Mittag-Leffler function occurs naturally. The boundedness and composition properties of this operator are established. The importance of the derived results further lies in the fact that the results of the generalized Mittag-Leffler function defined by Prabhakar (Yokohama Math. J. 19:7-15, 1971), Shukla and Prajapati (J. Math. Anal. Appl. 336:797-811, 2007), and the multiindex Mittag-Leffler function due to Kiryakova (Fract. Calc. Appl. Anal. 2:445-462, 1999; J. Comput. Appl. Math. 118:214-259, 2000; J. Fract. Calc. 40:29-41, 2011) readily follow as a special case of our findings. Further the results obtained are of general nature and include the results given earlier by Prajapati et al. (J. Inequal. Appl. 2013:33, 2013) and Srivastava and Tomovski (Appl. Math. Comput. 211:198-210, 2009). Some special cases of the established results are also given as corollaries.

Highlights

The Swedish mathematician Gösta Mittag-Leffler in the year introduced the function [, ] ∞ zn Eα(z) =α ∈ C; Re(α) > . ( ) n=In the year, Wiman [ ] introduced a generalization of ( ) in the form Eα,β (z) =(αn + β) α, β ∈ C; Re(α) >, Re(β) >
(γ, κ); ω – pα φ (p), where α, β, γ, ω ∈ C; Re(p) >, Re(α) > max[, Re(κ) – ], Re(β) >, Re(γ ) >, when γ tends to zero and β is replaced by α, by virtue of the limit formula ( ), the result ( ) reduces to the Riemann-Liouville fractional integral and we arrive at an important result given in Samko et al [ ]
5 Applications we present the solution of the following differential equation associated with a Hilfer derivative

Summary

Introduction

The Swedish mathematician Gösta Mittag-Leffler in the year introduced the function [ , ]. A further generalization of the Mittag-Leffler function defined by Shukla and Prajapati [ ] was given by Saxena et al [ ] in the year as. If we set γ = κ = in ( ), it reduces to the following multiindex Mittag-Leffler function studied by Kiryakova [ – ]:. M); Re(κ) > , Re( mj= αj) > max[ , κ – ], the function defined by ( ) is represented by the Mellin-Barnes type integral as follows:. The Riemann-Liouville fractional integral operator of f (t) is defined as [ , – ]. The Riemann-Liouville fractional derivative operator of f (t) is defined by [ , – ]. We take γ = κ = , ( ) yields the following integral operator associated with the multiindex Mittag-Leffler function defined by Kiryakova [ ]:.

Therefore we have

Conclusion