Abstract
The aim of this paper is to unify the extended Mittag-Leffler function and generalized Q function and define a unified Mittag-Leffler function. Both the extended Mittag-Leffler function and generalized Q function can be obtained from the unified Mittag-Leffler function. The Laplace, Euler beta, and Whittaker transformations are applied for this function, and generalized formulas are obtained. These formulas reproduce integral transformations of various deduced Mittag-Leffler functions and Q function. Also, the convergence of this unified Mittag-Leffler function is proved, and an associated fractional integral operator is constructed.
Highlights
Definition 3. e Whittaker transform is defined by the following improper integral: Mathematical Problems in Engineering
We studied the Laplace, Euler beta, and Whittaker transformation of the unified Mittag-Leffler function and obtained the compact formulas which reproduce integral transformations of Mittag-Leffler function and generalized Q function
We extended the Mittag-Leffler function and generalized Q function simultaneously
Summary
We define a Mittag-Leffler function (will be called the unified Mittag-Leffler function) which unifies the functions given in (7) and (8) as follows: Mλα,,ρβ,,θc,,kδ,,nμ,]. The Laplace transform of the unified MittagLeffler function can be given as follows: LMλα,,ρβ,,θc,,kδ,,nμ,] t; a, b, c, p s− 1Mλα,,ρβ,,θc,,kδ,,1μ,,n]s− 1; a, b, c, p. For ai l, p 0, and R(ρ) > 0, the Laplace transform of the unified Mittag-Leffler function will become. E Euler beta transformation of the unified Mittag-Leffler function is given in the theorem. By the definition of beta transform of an integrable function, we have the following: t; a, b, c, p; m, n tm− 1(1 −. For ai l, p 0, and R(ρ) > 0, the Euler beta transform of unified Mittag-Leffler function will become. E Whittaker transformation of the unified Mittag-Leffler function is given in the theorem.
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