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

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.

Introduction

The Swedish mathematician Gösta Mittag-Leffler discovered a special function in 1903 (see [12, 13]) defined as

$$ E_{\eta }(u) = \sum_{m=0}^{\infty } \frac{u^{m}}{\Gamma (\eta m + 1) m!}, \quad \bigl(\eta , u \in \mathbb{C} ; \Re (\eta )>0 \bigr), $$
(1.1)

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:

$$ E_{\eta ,\kappa }(u) = \sum_{m = 0}^{\infty } \frac{u^{m}}{\Gamma (\eta m + \kappa ) m!},\quad \bigl(\eta , \kappa \in \mathbb{C}; \Re (\eta )> 0, \Re ( \kappa )> 0 \bigr). $$
(1.2)

Subsequently, the generalized form of series (1.1) and (1.2) was studied by Prabhakar [16] in 1971:

$$ E_{\eta ,\kappa }^{\sigma }(u) = \sum _{m=0}^{\infty } \frac{u^{m} (\sigma )_{m}}{\Gamma (\eta m + \kappa ) m!}, \quad \bigl(\eta , \kappa , \sigma \in \mathbb{C}; \Re (\eta )>0, \Re (\kappa )>0, \Re ( \sigma )>0 \bigr), $$
(1.3)

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, 810, 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

$$ e_{\eta ,\kappa }(u;q) = \sum_{m = 0}^{\infty } \frac{u^{m}}{\Gamma _{q}(\eta m + \kappa )}, \quad \bigl( \vert u \vert < (1-q)^{-\eta } \bigr), $$
(1.4)

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:

$$\begin{aligned}& E_{\eta ,\kappa }^{\sigma }(u;q) = \sum _{m=0}^{\infty } \frac{(q^{\sigma };q)_{m}}{(q;q)_{m}} \frac{u^{m}}{\Gamma _{q}(\eta m + \kappa )}, \\& \bigl(\eta , \kappa , \sigma \in \mathbb{C}; \Re (\eta )>0, \Re (\kappa )>0, \Re (\sigma )>0 , \vert q \vert < 1 \bigr). \end{aligned}$$
(1.5)

Prelude

In the theory of q-series (see [6]), for complex λ and \(0< q<1\), the q-shifted factorial is defined as follows:

$$ (\lambda ;q)_{m} = \textstyle\begin{cases} 1 ; &m=0, \\ (1 - \lambda ) (1 - \lambda q) \cdots (1 - \lambda q^{m - 1}) ;& m \in \mathbb{N}, \end{cases} $$
(2.1)

which is equivalent to

$$ (\lambda ;q)_{m} = \frac{(\lambda ;q)_{\infty }}{(\lambda q^{m};q)_{\infty }} $$
(2.2)

and its extension naturally is

$$ (\lambda ;q)_{\eta } = \frac{(\lambda ;q)_{\infty }}{(\lambda q^{\eta };q)_{\infty }}, \quad \eta \in \mathbb{C}, $$
(2.3)

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

$$ (s-t)^{(m)} = \textstyle\begin{cases} 1 ;& m = 0, \\ \prod_{i = 0}^{m-1} (s-tq^{i}) ;&m\neq 0 \end{cases} $$
(2.4)

and connected by the following relationship:

$$ (s-t)^{(m)} = s^{m}(t/s;q)_{m} \quad (s\neq 0). $$

Obviously, its expansion for \(\tau \in \mathbb{R}\) is as follows:

$$ (s-t)^{(m)} = s^{m}\frac{(t/s;q)_{\infty }}{(q^{m}t/s;q)_{\infty }}, \quad \quad (s;q)_{\tau }=\frac{(s;q)_{\infty }}{(s q^{\tau };q)_{\infty }}. $$
(2.5)

Note that

$$ (s-t)^{(\tau )} = s^{\tau }(t/s;q)_{\tau }. $$

The q-analogue of binomial coefficient is defined for \(s,t >0\) as

$$ {\binom{s}{t}}_{q} = \frac{[s]_{q}!}{[t]_{q}![s-t]_{q}!} = \frac{(q;q)_{s}}{(q;q)_{t}(q;q)_{s - t}} = {\binom{s}{s-t}}_{q}. $$
(2.6)

The definition can be generalized in the following way. For arbitrary complex τ, we have

$$ {\binom{\tau }{m}}_{q} = \frac{(q^{-\tau };q)_{m}}{(q;q)_{m}} (-1)^{m} q^{ \tau m - \binom{m}{2}} = \frac{\Gamma _{q}(\tau + 1)}{\Gamma _{q}(m + 1)\Gamma _{q}(\tau - m + 1)}, $$
(2.7)

where \(\Gamma _{q}(u)\) is the q-gamma function.

The q-gamma and q-beta functions [6] are defined by

$$ \Gamma _{q}(u) = \frac{(q;q)_{\infty }}{(q^{u};q)_{\infty }}(1 - q)^{1-u} $$
(2.8)

for \(u\in \mathbb{R} \setminus \{0,-1,-2,-3,\ldots \}; \vert q \vert <1\).

Clearly,

$$ \Gamma _{q}(u + 1) = [u]_{q} \Gamma _{q}(u) $$
(2.9)

and

$$\begin{aligned}& B_{q}(\eta , \kappa ) = \frac{\Gamma _{q}(\eta )\Gamma _{q}(\kappa )}{\Gamma _{q}(\eta + \kappa )} = \int _{0}^{1} u^{\eta -1} \frac{(qu;q)_{\infty }}{(q^{\kappa }u;q)_{\infty }} \,d_{q}u = \int _{0}^{1}u^{ \eta - 1} (uq;q)_{\kappa - 1} \,d_{q}u, \\& \bigl(\Re (\eta ),\Re (\kappa ) >0 \bigr). \end{aligned}$$
(2.10)

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:

$$ D_{q}f(u) = \frac{f(u) - f(uq)}{u(1 - q)} \quad (u\neq 0, q\neq 1), (D_{q}f) (0) = \lim_{u\rightarrow 0}(D_{q}f) (u) $$
(2.11)

and

$$ \int _{0}^{u} f(t) \,d(t;q) = u (1 - q) \sum _{m = 0}^{\infty } q^{m} f \bigl(u q^{m} \bigr). $$
(2.12)

Generalized q-Mittag-Leffler function and its properties

In this section, we generalize definition (1.5) by introducing the following relation for \((q^{c}, q)_{m}\):

$$ \frac{(q^{c};q)_{m}}{(q^{\sigma };q)_{m}} = \frac{B_{q}(\sigma + m, c - \sigma )}{B_{q}(\sigma , c - \sigma )}. $$
(3.1)

Now, we define the generalization of Mittag-Leffler function (1.5) using the above relation as follows:

$$\begin{aligned}& E_{\eta ,\kappa }^{(\sigma ;c)}(u;q ) = \sum _{m=0}^{\infty } \frac{B_{q}(\sigma + m, c - \sigma )}{B_{q}(\sigma , c - \sigma )} \frac{(q^{c};q)_{m}}{(q;q)_{m}} \frac{u^{m}}{\Gamma _{q}(\eta m + \kappa )} \\& \bigl(\Re (c)>\Re (\sigma )>0, \vert q \vert < 1 \bigr), \end{aligned}$$
(3.2)

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.

  1. (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).

  2. (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).

  3. (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.

  4. (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).

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

$$\begin{aligned} \lim_{m\rightarrow \infty } \biggl\vert \frac{s_{m+1}}{s_{m}} \biggr\vert &= \biggl\vert \frac{B_{q}(\sigma +m+1,c-\sigma )}{B_{q}(\sigma +m,c-\sigma )} \biggr\vert \biggl\vert \frac{(q^{c},q)_{m+1}}{(q^{c},q)_{m}} \biggr\vert \biggl\vert \frac{(q,q)_{m}}{(q,q)_{m+1}} \biggr\vert \biggl\vert \frac{\Gamma (\eta m + \kappa )}{\Gamma (\eta m + \eta + \kappa )} u \biggr\vert \\ &= \lim_{m\rightarrow \infty } \biggl\vert \frac{(q^{c+m},q)_{\infty }}{(q^{c+m+1},q)_{\infty }} \frac{(q^{\sigma +m},q)_{\infty }}{(q^{\sigma +m+1},q)_{\infty }} \frac{(q^{\eta m+\kappa },q)_{\infty }}{(q^{\eta m+\kappa +\eta },q)_{\infty }} \frac{(q^{m+1},q)_{\infty }}{(q^{m},q)_{\infty }} (1 - q)^{-\eta } u \biggr\vert \\ &= \lim_{m\rightarrow \infty } \biggl\vert \bigl(1 - q^{c+m} \bigr) \bigl(1 - q^{ \sigma +m} \bigr) \bigl(1 - q^{\eta m + \kappa } \bigr)^{\eta } \frac{(1 - q)^{-\eta }}{(1 - q^{m})} u \biggr\vert \\ &= \bigl\vert (1 - q)^{-\eta } \bigr\vert \vert u \vert \quad \text{for } 0< \vert q \vert < 1. \end{aligned}$$
(4.1)

 □

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

$$ E_{\eta ,\kappa }^{(\sigma ;c)}(u;q ) = E_{\eta ,\kappa }^{(\sigma + 1;c + 1)}(u;q ) - u q^{c} E_{\eta ,\eta + \kappa }^{(\sigma + 1;c + 1)}(u;q ). $$

Proof

Using definition (3.2), we obtain

$$\begin{aligned} E_{\eta ,\kappa }^{(\sigma ;c)}(u;q ) &= \sum_{m=0}^{\infty } \frac{B_{q}(\sigma + m, c - \sigma )}{B_{q}(\sigma , c - \sigma )} \frac{(q^{c};q)_{m}}{(q;q)_{m}} \frac{u^{m}}{\Gamma _{q}(\eta m + \kappa )}, \\ &= \frac{1}{\Gamma (\kappa )}+ \sum_{m=1}^{\infty } \frac{B_{q}(\sigma + m, c - \sigma )}{B_{q}(\sigma , c - \sigma )} \frac{(1-q^{c})(q^{c+1};q)_{m-1}}{(q;q)_{m}} \frac{u^{m}}{\Gamma _{q}(\eta m + \kappa )}. \end{aligned}$$

Since \((1-q^{c}) = (1-q^{c+m})-q^{c}(1-q^{m})\), the above equation reduces to

$$\begin{aligned} E_{\eta ,\kappa }^{(\sigma ;c)}(u;q ) = {}&\frac{1}{\Gamma (\kappa )}+ \sum _{m=1}^{\infty } \frac{B_{q}(\sigma + m, c - \sigma )}{B_{q}(\sigma , c - \sigma )} \frac{(1-q^{c+m})(q^{c+1};q)_{m-1}}{(q;q)_{m}} \frac{u^{m}}{\Gamma _{q}(\eta m + \kappa )} \\ &{}- q^{c} \sum_{m=1}^{\infty } \frac{B_{q}(\sigma + m, c - \sigma )}{B_{q}(\sigma , c - \sigma )} \frac{(1-q^{m})(q^{c+1};q)_{m-1}}{(q;q)_{m}} \frac{u^{m}}{\Gamma _{q}(\eta m + \kappa )}. \end{aligned}$$

On replacing m with \(m+1\) in the second summation, it becomes

$$\begin{aligned} E_{\eta ,\kappa }^{(\sigma ;c)}(u;q ) &= \frac{1}{\Gamma (\kappa )}+ \sum _{m=1}^{\infty } \frac{B_{q}(\sigma + m, c - \sigma )}{B_{q}(\sigma , c - \sigma )} \frac{(q^{c+1};q)_{m}}{(q;q)_{m}} \frac{u^{m}}{\Gamma _{q}(\eta m + \kappa )} \\ &\quad{} - q^{c} \sum_{m=1}^{\infty } \frac{B_{q}(\sigma + m + 1, c - \sigma )}{B_{q}(\sigma , c - \sigma )} \frac{ (q^{c+1};q)_{m}}{(q;q)_{m}} \frac{u^{m + 1}}{\Gamma _{q}(\eta m + \eta + \kappa )}, \end{aligned}$$

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.

Theorem 6.1

(Integral representation)

For the generalized q-Mittag-Leffler function, we have

$$ E_{\eta ,\kappa }^{(\sigma ;c)}(u;q ) = \frac{1}{B_{q}(\sigma , c-\sigma )} \int _{0}^{1}t^{\sigma - 1} \frac{(tq;q)_{\infty }}{(tq^{c-\sigma };q)_{\infty }} E_{\eta ,\kappa }^{(c)}(tu;q) \,d_{q}t, $$
(6.1)

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:

$$\begin{aligned} E_{\eta ,\kappa }^{(\sigma ;c)}(u;q ) ={}& \sum_{m=0}^{\infty } \biggl\{ \int _{0}^{1}t^{\sigma +m-1} \frac{(tq;q)_{\infty }}{(tq^{c-\sigma };q)_{\infty }} \,d_{q}t \biggr\} \frac{1}{B_{q}(\sigma , c-\sigma )} \\ &{}\times \frac{(q^{c};q)_{m}}{\Gamma _{q}(\eta m + \kappa )} \frac{u^{m}}{(q;q)_{m}} \\ = {}&\frac{1}{B_{q}(\sigma , c-\sigma )}\sum_{m=0}^{\infty } \biggl\{ \int _{0}^{1}t^{\sigma -1} \frac{(tq;q)_{\infty }}{(tq^{c-\sigma };q)_{\infty }} \,d_{q}t \biggl( \frac{(q^{c};q)_{m}}{(q;q)_{m}} \frac{{tu}^{m}}{\Gamma _{q}(\eta m + \kappa )} \biggr) \biggr\} , \end{aligned}$$

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

$$ D_{q}^{m} \bigl[u^{\kappa - 1} E_{\eta , \kappa }^{(\sigma ;c)} \bigl(\lambda u^{ \eta };q \bigr) \bigr] = u^{\kappa - m - 1} E_{\eta , \kappa - m}^{(\sigma ;c)} \bigl( \lambda u^{\eta };q \bigr). $$
(6.2)

Proof

By considering the function

$$ f(u) = u^{\kappa -1} E_{\eta , \kappa }^{(\sigma ;c)} \bigl(\lambda u^{\eta };q \bigr). $$

In view of (2.11) and using definition (3.2), we obtain

$$\begin{aligned}& \begin{aligned} D_{q} \bigl[u^{\kappa - 1} E_{\eta , \kappa }^{(\sigma ;c)} \bigl(\lambda u^{\eta } \bigr) \bigr] &= \sum_{m=0}^{\infty } \frac{B_{q}(\sigma + m + 1, c - \sigma )}{B_{q}(\sigma , c - \sigma )} \frac{ (q^{c };q)_{m}}{(q;q)_{m}} \\ &\quad {}\times \frac{{\lambda ^{m}}(1-q^{\eta m+\kappa -1})}{1-q} \frac{u^{\eta m + \kappa -2}}{\Gamma _{q}(\eta m + \kappa )}. \end{aligned} \end{aligned}$$

Since, according to the functional equation (2.9), the right-hand side of the above expression can be written as

$$ \sum_{m=0}^{\infty } \frac{B_{q}(\sigma + m + 1, c - \sigma )}{B_{q}(\sigma , c - \sigma )} \frac{ (q^{c };q)_{m}}{(q;q)_{m}} \frac{\lambda ^{m} u^{\eta m + \kappa -2}}{\Gamma _{q}(\eta m + \kappa - 1)}=u^{ \kappa -2} E_{\eta , \kappa -1}^{(\sigma ;c)} \bigl(\lambda u^{\eta };q \bigr). $$

Conclusively, we obtain

$$ D_{q} \bigl[u^{\kappa - 1} E_{\eta , \kappa }^{(\sigma ;c)} \bigl( \lambda u^{\eta };q \bigr) \bigr] = u^{\kappa - 2} E_{\eta ,\kappa -1}^{(\sigma ;c)} \bigl(\lambda u^{\eta };q \bigr). $$

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

$$\begin{aligned}& \int _{0}^{1} u^{\xi - 1}(1 - qu)_{(\zeta - 1)} E_{\eta ,\kappa }^{( \sigma ;c)} \bigl(xu^{\rho };q \bigr) \,d_{q}u \\& \quad = \sum_{m=0}^{\infty } \frac{B_{q}(\sigma + m, c - \sigma )(q^{c};q)_{m}}{B_{q}(\sigma , c - \sigma )(q;q)_{m}} \frac{x^{m} \Gamma _{q}(\xi + \rho m)\Gamma _{q}(\xi )}{\Gamma _{q}(\eta m + \kappa )\Gamma _{q}(\xi + \zeta + \rho m )}. \end{aligned}$$
(6.3)

In particular,

$$ \int _{0}^{1} u^{\xi - 1}(1 - qu)_{(\zeta - 1)} E_{\eta ,\kappa }^{( \sigma ;c)} \bigl(xu^{\rho };q \bigr) \,d_{q}u = \Gamma _{q}(\zeta ) E_{\eta , \kappa +\zeta }^{(\sigma ;c)}(x;q ) . $$
(6.4)

Proof

By using definition (3.2), the left-hand side of equation (6.3) can be written as

$$ \int _{0}^{1} u^{\xi - 1}(1 - qu)_{(\zeta - 1)}\sum_{m=0}^{ \infty } \frac{B_{q}(\sigma + m, c - \sigma )(q^{c};q)_{m}}{B_{q}(\sigma , c - \sigma )(q;q)_{m}} \frac{u^{\rho m} x^{m}}{\Gamma _{q}(\eta m + \kappa )} \,d_{q}u. $$

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:

$$\begin{aligned} _{q}L_{s} \bigl[E_{\eta ,\kappa }^{(\sigma ;c)} \bigl(xu^{\rho };q \bigr) \bigr] = {}&\frac{1}{s} \sum _{m=0}^{\infty } \frac{B_{q}(\sigma + m, c - \sigma )(q^{c};q)_{m}}{B_{q}(\sigma , c - \sigma )(q;q)_{m}} \frac{ \Gamma _{q}(1 + \rho m)}{\Gamma _{q}(\eta m + \kappa )} \\ &{}\times \biggl(\frac{(1-q)^{\rho } x}{s^{\rho }} \biggr)^{m} \end{aligned}$$
(6.5)

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]:

$$ _{q}L_{s} \bigl\{ f(u) \bigr\} = \frac{1}{(1-q)} \int _{0}^{s^{-1}} E_{q}^{qsu}f(u) \,d_{q}u. $$
(6.6)

The q-extension of the exponential function [6] is given by

$$ E_{q}^{u} = {_{0}\phi _{0}} (-,-; q, -u) = \sum_{m = 0}^{ \infty } \frac{q^{\binom{{m}}{{2}}} u^{m}}{(q;q)_{m}} = (-u;q)_{\infty } $$
(6.7)

and

$$ e_{q}^{u} = {_{1}\phi _{0}}(0,-; q, -u) = \sum_{m = 0}^{ \infty } \frac{u^{m}}{(q;q)_{m}} = \frac{1}{(u;q)_{\infty }},\quad \vert u \vert < 1. $$
(6.8)

By using the above q-exponential series and the q-integral equation (2.12), we can write equation (6.6) as

$$ _{q}L_{s} \bigl\{ f(u) \bigr\} = \frac{(q;q)_{\infty }}{s} \sum_{j=0}^{\infty } \frac{q^{j} f(s^{-1}q^{j})}{(q;q)_{j}}. $$
(6.9)

Using definition (3.2) and the definition of q-Laplace transform, we obtain

$$\begin{aligned}& \begin{aligned} {}_{q}L_{s} \bigl[E_{\eta ,\kappa }^{(\sigma ;c)} \bigl(xu^{\rho };q \bigr) \bigr]={}& \frac{(q;q)_{\infty }}{s}\sum _{j=0}^{\infty }\frac{q^{j}}{(q;q)_{j}} \\ &{}\times \sum_{m=0}^{\infty } \frac{B_{q}(\sigma + m, c - \sigma )}{B_{q}(\sigma , c - \sigma )} \frac{(q^{\sigma };q)_{m}}{(q;q)_{m}} \frac{[u(s^{-1} q^{j})^{\sigma }]^{m}}{ \Gamma _{q}(\eta m + \kappa )}. \end{aligned} \end{aligned}$$

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). □

Kober-type fractional q-calculus operators

Agarwal [4] established Kober-type fractional q-integral operator in the following manner:

$$ \bigl(I_{q}^{\nu ,\mu } f \bigr) (u) = \frac{u^{-\nu -\mu }}{\Gamma _{q}(u)} \int _{0}^{u}(u-tq)_{ \mu -1} t^{\nu }f(t)\,d_{q}t, $$
(7.1)

where \(\Re (\mu )>0\). Also, Garg et al. [5] introduced Kober fractional q-derivative operator given by

$$ \bigl(D_{q}^{\nu ,\mu } f \bigr) (u) = \prod _{i = 0}^{m} \bigl([\nu + j]_{q} + uq^{ \nu + j} D_{q} \bigr) \bigl(I_{q}^{\nu +\mu , m - \mu } f \bigr) (u), $$
(7.2)

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:

$$\begin{aligned}& I_{q}^{\nu , \mu } \bigl\{ u^{m} \bigr\} = \frac{\Gamma _{q}(\nu + m + 1)}{\Gamma _{q}(\nu +\mu + m + 1)}u^{m}, \end{aligned}$$
(7.3)
$$\begin{aligned}& D_{q}^{\nu , \mu } \bigl\{ u^{m} \bigr\} = \frac{\Gamma _{q}(\nu +\mu + m + 1)}{\Gamma _{q}(\nu + m + 1)}u^{m}. \end{aligned}$$
(7.4)

Theorem 7.1

The following assumption holds true:

$$\begin{aligned} I_{q}^{\nu ,\mu } \bigl\{ E_{\eta ,\kappa }^{(\sigma ;c)}(u;q) \bigr\} &= \sum_{m=0}^{\infty } \frac{B_{q}(\sigma + m, c - \sigma )}{B_{q}(\sigma , c -\sigma )} \frac{(q^{c};q)_{m}}{(q;q)_{m}} \\ &\quad {} \times \frac{\Gamma _{q}(\nu + m + 1)}{\Gamma _{q}(\nu + \mu + m + 1)} \frac{u^{m}}{\Gamma _{q}(\eta m + \kappa )}, \end{aligned}$$
(7.5)

particularly,

$$ I_{q}^{\nu ,\mu } E_{\eta ,\kappa }^{(\nu +\mu ;1)}(u;q) = \frac{\Gamma _{q}(\nu + 1)}{\Gamma _{q}(\nu + \mu + 1)}E_{\eta , \kappa }^{(\nu + 1;1)}(u;q), $$
(7.6)

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:

$$\begin{aligned} D_{q}^{\nu ,\mu } \bigl\{ E_{\eta ,\kappa }^{(\sigma ;c)}(u;q ) \bigr\} = {}&\sum_{m=0}^{\infty } \frac{B_{q}(\sigma + m, c -\sigma )}{B_{q}(\sigma , c - \sigma )} \frac{(q^{c};q)_{m}}{(q;q)_{m}} \\ &{} \times \frac{\Gamma _{q}(\nu + \mu + m + 1)}{\Gamma _{q}(\nu + m + 1)} \frac{u^{m}}{\Gamma _{q}(\eta m + \kappa )}, \end{aligned}$$
(7.7)

particularly,

$$ D_{q}^{\nu ,\mu } E_{\eta ,\kappa }^{(\nu +1;1)}(u;q) = \frac{\Gamma _{q}(\nu + \mu + 1)}{\Gamma _{q}(\nu + 1)}E_{\eta , \kappa }^{\nu + \mu }(u;q) $$
(7.8)

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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MSC

  • 33C05
  • 33C45
  • 33C47
  • 33C90

Keywords

  • q-gamma functions
  • q-beta functions
  • Mittag-Leffler function