The extended Mittag-Leffler function and its properties

Abstract

In this paper, we present the extended Mittag-Leffler functions by using the extended Beta functions (Chaudhry et al. in Appl. Math. Comput. 159:589-602, 2004) and obtain some integral representations of them. The Mellin transform of these functions is given in terms of generalized Wright hypergeometric functions. Furthermore, we show that the extended fractional derivative (Özarslan and Özergin in Math. Comput. Model. 52:1825Model. 52: -1833Model. 52: , 2010) of the usual Mittag-Leffler function gives the extended Mittag-Leffler function. Finally, we present some relationships between these functions and the Laguerre polynomials and Whittaker functions.

Extension of the fractional derivative operator of the Riemann-Liouville

“…If we set p = q in (5.1), we get the interesting known result given byÖzarslan and Yilmaz [11,Theorem 9].…”

A Class of Extended Mittag–Leffler Functions and Their Properties Related to Integral Transforms and Fractional Calculus

“…Again, the sequence κ = 1 ( ∈ N) yields the known definition of Özarslan and Yilmaz [26] (with c = 1):…”

A new Generalization of Extended Beta and Hypergeometric Functions

“…The latest development and properties of such extension is found in the recent work of various researchers (see e.g., [1,2,3,4,7,9,12,11,13,14]). …”

“…Very recently Shadab et al [17] introduced a new and modified extension of beta function defined by 12) where (ς 1 ) > 0, (ς 2 ) > 0 and E λ . is Mittag-Leffler function defined by…”