A new generalization of Mittag-Leffler function via q-calculus

The present paper deals with a new different generalization of the Mittag-Leffler function through q-calculus. We then investigate its remarkable properties like convergence, recurrence relation, integral representation, q-derivative formula, q-Laplace transformation, and image formula under q-derivative operator. In addition to this, we consider some specific cases to give the utilization of our main results.

The Mittag-Leffler function plays a vital role in the solution of fractional order differential and integral equations. It has recently become a subject of rich interest in the field of fractional calculus and its applications. Nowadays some mathematicians consider the classical Mittag-Leffler function as the queen function in fractional calculus. An enormous amount of research in the theory of Mittag-Leffler functions has been published in the literature. For a detailed account of the various generalizations, properties, and applications of the Mittag-Leffler function, readers may refer to the literature (see [3, 8-10, 14, 15, 18, 20]).
The q-calculus is the q-extension of the ordinary calculus. The theory of q-calculus operators has been recently applied in the areas of ordinary fractional calculus, optimal control problem, in finding solutions of the q-difference and q-integral equations, and q-transform analysis.
In 2009, Mansoor [11] proposed a new form of q-analogue of the Mittag-Leffler function given as For other analogues of the Mittag-Leffler functions on the quantum time scale by means of the linear Caputo q-fractional initial value problems and of better imitation to the theory of time scales, we refer the reader to Definition 10 and Remark 11 in [1]. For the Kilbas-Saigo q-analogue of the Mittag-Leffler function, we refer to [2].

Prelude
In the theory of q-series (see [6]), for complex λ and 0 < q < 1, the q-shifted factorial is defined as follows: which is equivalent to and its extension naturally is where the principal value of q η is taken.
For s, t ∈ R, the q-analogue of the exponent (st) m is and connected by the following relationship: Obviously, its expansion for τ ∈ R is as follows: The q-analogue of binomial coefficient is defined for s, t > 0 as The definition can be generalized in the following way. For arbitrary complex τ , we have where q (u) is the q-gamma function. The q-gamma and q-beta functions [6] are defined by Also, the q-difference operator and q-integration of a function f (u) defined on a subset of C are given by [6] respectively:

Generalized q-Mittag-Leffler function and its properties
In this section, we generalize definition (1.5) by introducing the following relation for (q c , q) m : Now, we define the generalization of Mittag-Leffler function (1.5) using the above relation as follows: where B q (·) is the q-analogue of beta function. We enumerate the relations as particular cases of q-analogue of the generalized Mittag-Leffler function with other special functions as given below.
On replacing m with m + 1 in the second summation, it becomes , which leads to the required result (5.1).

Some elementary properties of the generalized q-Mittag-Leffler function
We begin with the following theorem, which shows the integral representation of the generalized q-Mittag-Leffler function.
Proof By the definition of q-analogue of beta function, we can rewrite equation (3.2) as follows: , which leads to the required result (6.1).
Proof By considering the function In view of (2.11) and using definition (3.2), we obtain .
Iterating the above result m -1 times, we obtain the required result (6.2).
Proof The q-Laplace transform of a suitable function is given by means of the following q-integral [7]: The q-extension of the exponential function [6] is given by and e u q = 1 φ 0 (0, -; q, -u) = ∞ m=0 u m (q; q) m = 1 (u; q) ∞ , |u| < 1. (6.8) By using the above q-exponential series and the q-integral equation (2.12), we can write equation (6.6) as Using definition (3.2) and the definition of q-Laplace transform, we obtain .
On interchanging the order of summation and writing the j series as 1 φ 0 , which can be summed up as 1 (q 1+ρm ;q) ∞ , and after some simplifications, we obtain the required result (6.5).