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http://dx.doi.org/10.4134/CKMS.c170216

A NEW EXTENSION OF THE MITTAG-LEFFLER FUNCTION  

Arshad, Muhammad (Department of Mathematics International Islamic University)
Choi, Junesang (Department of Mathematics Dongguk University)
Mubeen, Shahid (Department of Mathematics University of Sargodha)
Nisar, Kottakkaran Sooppy (Department of Mathematics College of Arts and Science at Wadi Al-dawaser Prince Sattam bin Abdulaziz University)
Rahman, Gauhar (Department of Mathematics International Islamic University)
Publication Information
Communications of the Korean Mathematical Society / v.33, no.2, 2018 , pp. 549-560 More about this Journal
Abstract
Since Mittag-Leffler introduced the so-called Mittag-Leffler function in 1903, due to its usefulness and diverse applications, a variety and large number of its extensions (and generalizations) and variants have been presented and investigated. In this sequel, we aim to introduce a new extension of the Mittag-Leffler function by using a known extended beta function. Then we investigate ceratin useful properties and formulas associated with the extended Mittag-Leffler function such as integral representation, Mellin transform, recurrence relation, and derivative formulas. We also introduce an extended Riemann-Liouville fractional derivative to present a fractional derivative formula for a known extended Mittag-Leffler function, the result of which is expressed in terms of the new extended Mittag-Leffler functions.
Keywords
gamma function; beta function; extended beta function; Mittag-Leffler function; extended Mittag-Leffler functions; Fox-Wright function; generalized hypergeometric function; Mellin transform; Riemann-Liouville fractional derivative; extended Riemann-Liouville fractional derivative;
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