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1.   Introduction

The classical beta function

and its relation with well known gamma function is given by

The Gauss hypergeometric, confluent hypergeometric and Appell's functions which are respectively defined by(see ^[27])

and

The Appell's series or bivariate hypergeometric series is defined by

for all $\delta_1, \delta_2, \delta_3, \delta_4\in \mathbb{C}, \delta_4\neq 0, -1, -2, -3, \cdots, \quad |x|, |y| < 1 < 1$.

The integral representation of hypergeometric, confluent hypergeometric and Appell's functions are respectively defined by

and

The $\mathtt{k}$-gamma function, $\mathtt{k}$-beta function and the $\mathtt{k}$-Pochhammer symbol introduced and studied by Diaz and Pariguan ^[5]. The integral representation of $\mathtt{k}$-gamma function and $\mathtt{k}$-beta function respectively given by

Here, we recall the following relations (see ^[5]).

where $(z)_{n, \mathtt{k}} = (z)(z+\mathtt{k})(z+2\mathtt{k})\cdots(z+(n-1)\mathtt{k}); \quad (z)_{0, \mathtt{k}} = 1$ and $\mathtt{k} > 0$

and

These studies were followed by Mansour ^[16], Kokologiannaki ^[13], Krasniqi ^[14] and Merovci ^[17]. In 2012, Mubeen and Habibullah ^[18] defined the $\mathtt{k}$-hypergeometric function as

where $\delta_1, \delta_2, \delta_3\in\mathbb{C}$ and $\delta_3\neq0, -1, -2, \cdots$ and its integral representation is given by

The $\mathtt{k}$-Riemann-Liouville (R-L) fractional integral using $\mathtt{k}$-gamma function introduced in ^[19]:

Later on Mubeen and Iqbal ^[11] established the improved version of Gruss type inequalities by utilizing $k$-fractional integrals. In ^[1], Agarwal et al. presented certain Hermite-Hadamard type inequalities for generalized $k$-fractional integrals. Set et al. ^[29] presented an integral identity and generalized Hermite–Hadamard type inequalities for Riemann–Liouville fractional integral. Mubeen et al. ^[24] established integral inequalities of Ostrowski type for $k$-fractional Riemann–Liouville integrals. Recently, many researchers have introduced generalized version of $k$-fractional integrals and investigated a large bulk of various inequalities via the said fractional integrals. The interesting readers are referred to see the work of ^[9,10,26,30]. Farid et al. ^[7] introduced Hadamard $k$-fractional integrals. In ^[8] introduced Hadamard-type inequalities for $k$-fractional Riemann-Liouville integrals. In ^[12,31], the authors established certain inequalities by utilizing Hadamard-type inequalities for $k$-fractional Riemann-Liouville integrals. In ^[25], Nisar et al. established certain Gronwall type inequalities associated with Riemann-Liouville $k$- and Hadamard $k$-fractional derivatives and their applications. In ^[25], they presented dependence solutions of certain $k$-fractional differential equations of arbitrary real order with initial conditions. Recently, Samraiz et al. ^[28] defined an extension of Hadamard $k$-fractional derivative and proved its various properties.

The solution of some integral equations involving confluent $\mathtt{k}$-hypergeometric functions and $\mathtt{k}$-analogue of Kummer's first formula are given in ^[22,23]. While the $\mathtt{k}$-hypergeometric and confluent $\mathtt{k}$-hypergeometric differential equations are introduced in ^[20]. In 2015, Mubeen et al. ^[21] introduced $\mathtt{k}$-Appell hypergeometric function as

for all $\delta_1, \delta_2, \delta_3, \delta_4\in \mathbb{C}, \delta_4\neq 0, -1, -2, -3, \cdots, \quad \max\{|z_{1}|, |z_{2}|\} < \frac{1}{\mathtt{k}}$ and $\mathtt{k} > 0$. Also, Mubeen et al. defined its integral representation as

2.   Extension of fractional derivative operator

In this section, we recall the following definition of fractional derivatives from and give a new extension called Riemann-Liouville $\mathtt{k}$-fractional derivative.

Definition 2.1. The well-known R-L fractional derivative of order $\mu$ is defined by

For the case $m-1 < \Re(\mu) < m$ where $m = 1, 2, \cdots$, it follows

For further study and applications, we refer the readers to the work of ^{[2,3,4,15,32]}. In the following, we define Riemann-Liouville $\mathtt{k}$-fractional derivative of order $\mu$ as

Definition 2.2.

For the case $m-1 < \Re(\mu) < m$ where $m = 1, 2, \cdots$, it follows

Note that for $\mathtt{k} = 1$, definition 2.2 reduces to the classical R-L fractional derivative operator given in definition 2.1.

Now, we are ready to prove some theorems by using the new definition 2.2.

Theorem 1. The following formula holds true,

Proof. From (2.3), we have

Substituting $t = uz$ in (2.6), we get

Applying definition (1.9) to the above equation, we get the desired result.

Theorem 2. Let $\Re(\mu) > 0$ and suppose that the function $f(z)$ is analytic at the origin with its Maclaurin expansion given by $f(z) = \sum_{n = 0}^\infty a_n z^n$ where $|z| < \rho$ for some $\rho\in \mathbb{R^+}$. Then

Proof. Using the series expansion of the function $f(z)$ in (2.3) gives

As the series is uniformly convergent on any closed disk centered at the origin with its radius smaller then $\rho$, therefore the series so does on the line segment from $0$ to a fixed $z$ for $|z| < \rho$. Thus it guarantee terms by terms integration as follows

which is the required proof.

Theorem 3. The following result holds true:

where $\Re(\mu) > \Re(\eta) > 0$ and $|z| < 1$.

Proof. By direct calculation, we have

Substituting $t = zu$ in the above equation, we get

Applying (1.14) and after simplification we get the required proof.

Theorem 4. The following result holds true:

where $\Re(\mu) > \Re(\eta) > 0$, $\Re(\alpha) > 0$, $\Re(\beta) > 0$, $\max\{|az|, |bz|\} < \frac{1}{\mathtt{k}}$.

Proof. To prove (2.9), we use the power series expansion

Now, applying Theorem 1, we obtain

In view of (1.16), we get

Theorem 5. The following Mellin transform formula holds true:

where $\Re(\eta) > -1$, $\Re(\mu) < 0$, $\Re(s) > 0$.

Proof. Applying the Mellin transform on definition (2.3), we have

Interchanging the order of integrations in above equation, we get

which completes the proof.

Theorem 6. The following Mellin transform formula holds true:

where $\Re(\alpha) > 0$, $\Re(\mu) < 0$, $\Re(s) > 0$, and $|z| < 1$.

Proof. Using the power series for $(1-\mathtt{k}z)^{-\frac{\alpha}{\mathtt{k}}}$ and applying Theorem 5 with $\eta = n\mathtt{k}$, we can write

which is the required proof.

Theorem 7. The following result holds true:

where $\gamma, \delta, \mu\in\mathbb{C}$, $\Re(p) > 0$, $\Re(q) > 0$, $\Re(\mu) > \Re(\eta) > 0$ and $E_{\mathtt{k}, \gamma, \delta}^{\mu}(z)$ is $\mathtt{k}$-Mittag-Leffler function (see ^[6]) defined as:

Proof. Using (2.13), the left-hand side of (2.12) can be written as

By Theorem 2, we have

In view of Theorem 1, we get the required proof.

Theorem 8. The following result holds true:

where $\Re(p) > 0$, $\Re(q) > 0$, $\Re(\mu) > \Re(\eta) > 0$ and $_m\Psi_n(z)$ is the Fox-Wright function defined by (see ^[15], pages 56–58)

Proof. Applying Theorem 1 and followed the same procedure used in Theorem 7, we get the desired result.

3.   Concluding remarks

Recently, many researchers have introduced various generalizations of fractional integrals and derivatives. In this line, we have established a $k$-fractional derivative and its various properties. If we letting $\mathtt{k}\rightarrow1$ then all the results established in this paper will reduce to the results related to the classical Reimann-Liouville fractional derivative operator.

Acknowledgements

The author K.S. Nisar thanks to Deanship of Scientific Research (DSR), Prince Sattam bin Abdulaziz University for providing facilities and support.

Conflict of interest

The authors declare no conflict of interest.