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A new generalization of Mittag-Leffler function via q-calculus
Advances in Difference Equations volume 2020, Article number: 695 (2020)
Abstract
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.
1 Introduction
The Swedish mathematician Gösta Mittag-Leffler discovered a special function in 1903 (see [12, 13]) defined as
where \(\Gamma (\cdot )\) is a classical gamma function [17]. The special function defined in (1.1) is called the Mittag-Leffler function.
For the very first time, in 1905, Wiman [21] firstly proposed the generalization of the Mittag-Leffler \(E_{\eta ,\kappa }(u)\) as follows:
Subsequently, the generalized form of series (1.1) and (1.2) was studied by Prabhakar [16] in 1971:
where \((\sigma )_{m} = \frac{\Gamma (\sigma + m )}{\Gamma (\sigma )}\) denotes the Pochhammer symbol [17].
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
where \(\eta > 0 \), \(\kappa \in \mathbb{C}\).
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].
Recently, Sharma and Jain [19] introduced the following q-analogue of the generalized Mittag-Leffler function:
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 \(q^{\eta }\) is taken.
For \(s,t\in \mathbb{R}\), the q-analogue of the exponent \((s-t)^{m}\) is
and connected by the following relationship:
Obviously, its expansion for \(\tau \in \mathbb{R}\) is as follows:
Note that
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 \(\Gamma _{q}(u)\) is the q-gamma function.
The q-gamma and q-beta functions [6] are defined by
for \(u\in \mathbb{R} \setminus \{0,-1,-2,-3,\ldots \}; \vert q \vert <1\).
Clearly,
and
Also, the q-difference operator and q-integration of a function \(f(u)\) defined on a subset of \(\mathbb{C}\) are given by [6] respectively:
and
3 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 \(B_{q}(\cdot )\) 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.
-
(i)
On setting \(c=1\) in (3.2), we obtain
$$ E_{\eta ,\kappa }^{(\sigma ;1)}(u;q ) = \sum _{m=0}^{\infty } \frac{(q^{\sigma };q)_{m}}{(q;q)_{m}} \frac{u^{m}}{\Gamma _{q}(\eta m + \kappa )} = E_{\eta ,\kappa }^{ \sigma }(u;q ), $$(3.3)which is given by equation (1.5).
-
(ii)
Again, on setting \(\sigma = 1\) in (3.2), we obtain
$$ E_{\eta ,\kappa }^{(1;c)}(u;q ) = \sum _{m=0}^{\infty } \frac{u^{m}}{\Gamma _{q}(\eta m + \kappa )} = e_{\eta ,\kappa } (u;q ), $$(3.4)the function \(e_{\eta ,\kappa } (u;q )\) can be termed as q-analogue of the Mittag-Leffler function defined in (1.4).
-
(iii)
On setting \(\eta =\kappa =\sigma =1\), in (3.2), we obtain
$$ E_{1,1}^{(1;c)}(u;q ) = \sum _{m=0}^{\infty } \frac{(q^{c};q)_{m}}{(q;q)_{m}}u^{m} = \frac{(q^{c}u;q)_{\infty }}{(q;q)_{\infty }} = {}_{1}\phi _{0} \bigl(q^{c};-;q,u \bigr), $$(3.5)where the function \({}_{1}\phi _{0}(q^{c};-;q,u) = (1 - u)^{-c}\) can be termed as q-binomial function.
-
(iv)
On setting \(c = c+\sigma \), in (3.2), we obtain q-analogue of the Mittag-Leffler function \(E_{\eta ,\kappa }^{\sigma }(u;q )\) defined in (1.5).
4 Convergence of \(E_{\eta ,\kappa }^{(\sigma ;c)}(u;q) \)
Theorem 4.1
The q-analogue of the generalized Mittag-Leffler function defined by the summation formula (3.2) converges absolutely for \(\vert u \vert <(1 - q)^{-\eta }\) provided that \(0 < q<1\), \(\eta >0\), \(\Re (c)>\Re (\sigma )\), \(c, \sigma \in \mathbb{C}\).
Proof
Writing the summation formula (3.2) as \(E_{\eta ,\kappa }^{(\sigma ;c)}(u;q ) = \sum_{m = 0}^{ \infty }s_{m}\) and by applying the ratio formula, we find
□
5 Recurrence relations
Theorem 5.1
If \(\eta , \kappa , \sigma \in \mathbb{C}\), \(\Re (\eta )>0\), \(\Re (\kappa )>0\), \(\Re (\sigma )>0 \), and \(\sigma \neq c \), then
Proof
Using definition (3.2), we obtain
Since \((1-q^{c}) = (1-q^{c+m})-q^{c}(1-q^{m})\), the above equation reduces to
On replacing m with \(m+1\) in the second summation, it becomes
which leads to the required result (5.1). □
6 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.
Theorem 6.1
(Integral representation)
For the generalized q-Mittag-Leffler function, we have
provided that \(\eta , \kappa , \sigma \in \mathbb{C}\), \(\Re (\eta )>0\), \(\Re (\kappa )>0\), \(\Re (\sigma )>0 \), and \(\sigma \neq c \).
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). □
Theorem 6.2
For \(\eta , \kappa , \sigma \in \mathbb{C}\), \(\Re (\eta )>0\), \(\Re (\kappa )>0\), \(\Re (\sigma )>0\), \(c\neq \sigma \), then for any \(m \in \mathbb{N}\), we have
Proof
By considering the function
In view of (2.11) and using definition (3.2), we obtain
Since, according to the functional equation (2.9), the right-hand side of the above expression can be written as
Conclusively, we obtain
Iterating the above result \(m-1\) times, we obtain the required result (6.2). □
Theorem 6.3
Let \(\xi , \zeta , \sigma , \kappa \in \mathbb{C}\); \(\Re (\xi ), \Re ( \kappa ), \Re (\sigma )> 0\); \(\zeta \neq 0, -1, -2,\ldots \) , then
In particular,
Proof
By using definition (3.2), the left-hand side of equation (6.3) can be written as
Interchanging the order of summation and integration and in view of equation (2.10), we obtain the required result (6.3).
In equation (6.3) replacing \(\eta =\rho \), \(\xi =\kappa \), then in view of equation (3.2), we can clearly obtain (6.4). □
Theorem 6.4
(q-Laplace transform)
The q-analogue of the generalized Laplace transform is defined as follows:
provided that \(\kappa , \sigma , s \in \mathbb{C}\); \(\Re (\beta ), \Re (\kappa ), \Re (s) > 0 \).
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
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 \(_{1}\phi _{0}\), which can be summed up as \(\frac{1}{(q^{1 + \rho m};q)_{\infty }}\), and after some simplifications, we obtain the required result (6.5). □
7 Kober-type fractional q-calculus operators
Agarwal [4] established Kober-type fractional q-integral operator in the following manner:
where \(\Re (\mu )>0\). Also, Garg et al. [5] introduced Kober fractional q-derivative operator given by
where \(m = [\Re (\mu )] + 1\), \(m \in \mathbb{N}\).
The image formulas of the power function \(u^{m}\) under the above operators [5] are given as follows:
Theorem 7.1
The following assumption holds true:
particularly,
provided that if \(\eta , c >0\), \(\kappa , \sigma , u \in \mathbb{C}\); \(\Re (\kappa ), \Re (\sigma )>0\).
Proof
The proof of (7.5) can easily be obtained by making use of definition (3.2) and result (7.3).
Now, on setting \(\sigma = \nu + \mu \) in definition (3.2), we obtain result (7.6). □
Theorem 7.2
The following assumption holds true:
particularly,
provided that if \(\eta , c >0\), \(\kappa , \sigma , u \in \mathbb{C}\); \(\Re (\kappa ),\Re ( \sigma )>0\).
Proof
The proof of (7.7) can easily be obtained by making use of definition (3.2) and result (7.4). Similarly, on setting \(\sigma = \nu + 1 \) in definition (3.2), we obtain result (7.8). □
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The author T. Abdeljawad would like to thank Prince Sultan University for funding this work through the research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM), group number RG-DES-2017-01-17.
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Nadeem, R., Usman, T., Nisar, K.S. et al. A new generalization of Mittag-Leffler function via q-calculus. Adv Differ Equ 2020, 695 (2020). https://doi.org/10.1186/s13662-020-03157-z
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DOI: https://doi.org/10.1186/s13662-020-03157-z