Theory and Modern Applications

# 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.

## 1 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 . The special function defined in (1.1) is called the Mittag-Leffler function.

For the very first time, in 1905, Wiman  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  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 .

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  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 . For the Kilbas–Saigo q-analogue of the Mittag-Leffler function, we refer to .

Recently, Sharma and Jain  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)

## 2 Prelude

In the theory of q-series (see ), 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  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  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)

## 3 Generalized q-Mittag-Leffler function and its properties

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

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

## 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

\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)

□

## 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

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

### 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). □

## 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

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

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

The q-extension of the exponential function  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). □

## 7 Kober-type fractional q-calculus operators

Agarwal  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.  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  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). □

Not applicable.

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None.

## Funding

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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