Skip to main content

Theory and Modern Applications

Certain fractional formulas of the extended k-hypergeometric functions


In this article, we aim to investigate various formulae for the \((p,k)\)-analogues of Gauss hypergeometric functions, including the integral transforms and the operators of fractional calculus. All the outcomes presented here are of general attractiveness and can yield a number of previous works as special cases.

1 Overture

Throughout this work, \(\mathbb{N}:=\{1,2,3,\ldots\}\) denotes the set of positive integers, \(\mathbb{N}_{0}=\mathbb{N}\cup \{0\}\), \(\mathbb{Z}^{-}:=\{-1,-2,-3,\ldots\}\) denotes the set of negative integers, \(\mathbb{Z}^{-}_{0}=\mathbb{Z}^{-}\cup \{0\}\), \(\mathbb{R}^{+}\) denotes the set of positive real numbers, and \(\mathbb{C}\) denotes the set of complex numbers.

In 1813, Gauss first summarized his investigations of hypergeometric functions, which has been of great significance in the mathematical modeling of physical phenomena and other applications. Recently, various developments and expansions of the hypergeometric functions have been proposed and discussed (for example, see [112]).

In [13], Diaz and Pariguan introduced an interesting extensions of the gamma, beta, Pochhammer, and hypergeometric functions as follows.

Definition 1.1

For \(k\in \mathbb{R}^{+}\), the k-gamma function \(\Gamma ^{k}(y)\) is defined by

$$\begin{aligned} \begin{aligned} &\Gamma ^{k}(y)=& \int ^{\infty }_{0} u^{y-1} e^{- \frac{u^{k}}{k}} \,du, \end{aligned} \end{aligned}$$

where \(y\in \mathbb{C}\setminus k\mathbb{Z}^{-}\). We note that \(\Gamma ^{k}(y)\to \Gamma (y)\) for \(k\to 1\) where \(\Gamma (y)\) is the classical Euler’s gamma function and \((y)_{m,k}\) is the k-Pochhammer symbol given by

$$\begin{aligned} (y)_{n,k}=\frac{\Gamma ^{k}{(y+nk)}}{\Gamma ^{k}{(y)}}= \textstyle\begin{cases} y(y+k)\ldots(y+(n-1)k), & n\in \mathbb{N}, y\in \mathbb{C}, \\ 1, & n=0, k\in \mathbb{R}^{+}, y\in \mathbb{C}\setminus \{0\}, \end{cases}\displaystyle \end{aligned}$$

the relation between the \(\Gamma ^{k}(y)\) and the usual gamma function \(\Gamma (y)\) follows easily as

$$\begin{aligned} \Gamma ^{k}(y)=k^{\frac{y}{k}-1}\Gamma \biggl(\frac{y}{k} \biggr)\quad \text{or}\quad \Gamma (w)=k^{1-w} \Gamma ^{k}(kw). \end{aligned}$$

Definition 1.2

The k-beta function \(\mathbf{B}^{k}(s,t)\) is defined by

$$\begin{aligned} \mathbf{B}^{k}(s,t)= \textstyle\begin{cases} \frac{1}{k}\int ^{1}_{0} y^{\frac{s}{k}-1} (1-y)^{\frac{t}{k}-1} \,dy, & (k\in \mathbb{R}^{+},\min \{ \operatorname{Re}(s), \operatorname{Re}(t)\}>0), \\ \frac{\Gamma ^{k}(s)\Gamma ^{k}(t)}{\Gamma ^{k}(s+t)}, & (k \in \mathbb{R}^{+}, s,t\in \mathbb{C}\setminus \mathbb{Z}^{-}_{0}). \end{cases}\displaystyle \end{aligned}$$

Clearly, the case \(k=1\) in (1.3) reduces to the known beta function \(\mathbb{B}(s,t)\).

$$\begin{aligned} \mathbb{B}(s,t)= \int {1}_{0} y^{s-1} (1-y)^{t-1}\,dy. \end{aligned}$$

Also, the relation between the k-beta function \(\mathbf{B}^{k}(s,t)\) and the original beta function \(\mathbb{B}(s,t)\) is

$$\begin{aligned} \mathbf{B}^{k}(s,t)=\frac{1}{k}\mathbb{B}\biggl( \frac{s}{k},\frac{t}{k}\biggr). \end{aligned}$$

Definition 1.3

Let \(k\in \mathbb{R}^{+}\) and \(\alpha _{1},\alpha _{2},y\in \mathbb{C}\) and \(\alpha _{3}\in \mathbb{C}\setminus \mathbb{Z}^{-}_{0}\), then k-hypergeometric series is defined by the form

$$\begin{aligned} \begin{aligned} _{2}H^{k}_{1} \left [{\textstyle\begin{array}{c} (\alpha _{1};k),(\alpha _{2};k) \\ (\alpha _{3};k);\end{array}\displaystyle }\ y\right ]&= \sum^{\infty }_{n=0} \frac{(\alpha _{1})_{n,k} (\alpha _{2})_{n,k}}{(\alpha _{3})_{n,k}}. \frac{y^{n}}{n!},\quad \vert y \vert < \frac{1}{k}, \end{aligned} \end{aligned}$$

where \((\alpha _{1})_{n,k}\) is the k-Pochhammer symbol given in (1.2).

Indeed, in their special case when \(k=1\), Eq. (1.4) is reduced to the Gauss hypergeometric function \(_{2}H_{1}(\cdot)\). The \(_{2}H_{1}(\cdot)\) is the special case of the generalized hypergeometric functions \(_{m}H_{n}(\cdot)\) of m numerator and n denominator parameters defined by (see, e.g., [14, Sect. 1.5]):

$$\begin{aligned} \begin{aligned} _{m}H_{n} \left [{\textstyle\begin{array}{c} \alpha _{1}\ldots\alpha _{m}; \\ \delta _{1}\ldots\delta _{n};\end{array}\displaystyle }\ y\right ]&= \sum^{\infty }_{j=0} \frac{(\alpha _{1})_{j}\ldots(\alpha _{m})_{j}}{(\delta _{1})_{j}\ldots(\delta _{n})_{j}}. \frac{y^{j}}{j!}, \end{aligned} \end{aligned}$$


$$\begin{aligned} (\alpha _{1})_{n}= \frac{\Gamma {(\alpha _{1}+n)}}{\Gamma {(\alpha _{1})}}= \textstyle\begin{cases} \alpha _{1}(\alpha _{1}+1)\ldots(\alpha _{1}+n-1), & i\in \mathbb{N}, \alpha _{1}\in \mathbb{C}, \\ 1, & i=0; \alpha _{1}\in \mathbb{C}\setminus \{0\}, \end{cases}\displaystyle \end{aligned}$$

is the usual Pochhammer symbol (or the shifted factorial) and \(\Gamma (\cdot)\) is the standard gamma function (see, e.g., [14, Sect. 1.1])).

Currently, several different outcomes concerning the k-analogue of special functions have been archived, the interested reader may refer to the monographs by many researchers (see, e.g., [1522] and the references cited therein).

Recently, Abdalla and Hidan [23] and Hidan et al. [24] introduced and studied several properties of the following \((p,k)\)-analogues of Gauss hypergeometric functions:

$$\begin{aligned} \begin{aligned} _{2}\mathbf{H}^{(p,k)}_{1} \left [{\textstyle\begin{array}{c} (\alpha _{1};k),(\alpha _{2};k) \\ (\alpha _{3};k)\end{array}\displaystyle };w \right ]&= \sum^{\infty }_{j=0} \frac{(\alpha _{1})_{j,k} (\alpha _{2})_{j,k}}{(\alpha _{3})_{j,k}}. \frac{w^{j}}{(pj)!}, \end{aligned} \end{aligned}$$

which is an entire function for \(p>1\), where \(k\in \mathbb{R}^{+}\) and \(\alpha _{1},\alpha _{2},w\in \mathbb{C}\) and \(\alpha _{3}\in \mathbb{C}\setminus \mathbb{Z}^{-}_{0}\), and \((\alpha )_{j,k}\) is the k-Pochhammer symbol defined in (1.2).

Remark 1.1

Among the important special cases of \(_{2}\mathbf{H}^{(p,k)}_{1}\) are equations of type (1.4) and (1.5). Further, at \(k=1\), we obtain the p-extended Gauss hypergeometric functions in the following form (cf. [25]):

$$\begin{aligned} \begin{aligned} _{2}\mathbf{H}^{(p)}_{1} \left [{\textstyle\begin{array}{c} \alpha _{1},\alpha _{2} \\ \alpha _{3}\end{array}\displaystyle };w \right ]&= \sum^{\infty }_{j=0} \frac{(\alpha _{1})_{j} (\alpha _{2})_{j}}{(\alpha _{3})_{j}}. \frac{w^{j}}{(pj)!},\quad p\in \mathbb{N}, \end{aligned} \end{aligned}$$

which is also an entire function for \(p>1\).

The purpose of this work is to continue the investigation of new formulae like integral transforms and fractional calculus operators on the \((p,k)\)-analogues of Gauss hypergeometric functions \(_{2}\mathbf{H}^{(p,k)}_{1}\). In Sects. 2 and 3, respectively, we introduce several integral transforms and image formulae for the \((p,k)\)-analogues of Gauss hypergeometric functions \(_{2}\mathbf{H}^{(p,k)}_{1}\) by applying a certain integral transform (like Laplace transform and fractional Fourier transform) and diverse fractional operators. Also, some special cases and significance of our main outcomes are considered.

2 Integral transforms

In this section, we prove two theorems, which exhibit the connection between integral transforms like the Laplace transform and the fractional Fourier transform for the \(_{2}\mathbf{H}^{(p,k)}_{1}\) given in (1.7). We recall the Laplace transform and the fractional Fourier transform, respectively.

Definition 2.1

(Laplace transform)

Let \(f (\xi )\) be a function of \(\xi > 0\). Then the Laplace transform of \(f(\xi )\) is defined by

$$\begin{aligned} \begin{aligned} \mathbf{F}(s)=\mathcal{L} \bigl\{ f (\xi ):s \bigr\} = \int ^{ \infty }_{0} e^{-s \xi } f(\xi ) \,d\xi ,\quad \Re (s)>0, \end{aligned} \end{aligned}$$

provided that the improper integral exists, \(e^{-\lambda \xi } \) is the kernel of the transformation, and the function \(f(\xi )\) is called the inverse Laplace transform of \(\mathbf{F}(\lambda )\) (see [26, Chap. 3]).

Definition 2.2

(Fractional Fourier transform)

Assume that φ is a function belonging to Lizorkin space \(\psi (R)\). The fractional Fourier transform (FFT) of order \(\beta , 0<\beta \leq 1\), is defined as (cf. [26, 27])

$$\begin{aligned} \begin{aligned} \varphi _{\beta }(\omega )= \mathfrak{F}_{\beta }[\varphi ]( \omega )= \int _{R}e^{i\omega ^{\frac{1}{\beta }}\xi } \varphi (\xi )\,d\xi ,\quad i= \sqrt{-1}. \end{aligned} \end{aligned}$$

Remark 2.1

It may by observed that when \(\omega >0\) it reduces to the FFT introduced by Luchko, Martinez, and Trujillo (see, e.g., [27, pp. 225–240] for details). The relationship between the two Fourier transforms, the classical and FFT, is given by the following relation:

$$\begin{aligned} \begin{aligned} \mathfrak{F}_{\beta }[\varphi ](\omega )= \mathfrak{F}[ \varphi ](s) \quad \text{for } s=\omega ^{\frac{1}{\beta }}, 0< \beta \leq 1. \end{aligned} \end{aligned}$$

Theorem 2.1

The Laplace transform for the \(_{2}\mathbf{H}^{(p,k)}_{1}\) given by (1.7) is in the form

$$\begin{aligned} \begin{aligned} &\mathcal{L} \left\{\xi ^{\frac{\delta }{k}-1}_{2} \mathbf{H}^{(p,k)}_{1}\left [{\textstyle\begin{array}{c} (\alpha _{1},k)(\alpha _{2},k) \\ (\alpha _{3},k);\end{array}\displaystyle }\quad u \xi \right ] \right\} \\ &\quad =\frac{k\Gamma ^{k}(\delta )}{(ks)^{\frac{\delta }{k}}}_{3} \mathbf{H}^{(p,k)}_{1} \left [{\textstyle\begin{array}{c} (\alpha _{1},k)(\alpha _{2},k)(\delta ,k) \\ (\alpha _{3},k);\end{array}\displaystyle }\quad \frac{u}{ks}\right ], \end{aligned} \end{aligned}$$

(\(\alpha _{1},\alpha _{2},u,\xi \in \mathbb{C}\), \(\alpha _{3} \in \mathbb{C}\setminus \mathbb{Z}^{-}_{0}\), \(\operatorname{Re}(\alpha _{1})>0\), \(\operatorname{Re}( \alpha _{2})>0\), \(\operatorname{Re}(s)>0\), \(|\frac{u}{ks}|<1\), \(k\in \mathbb{R}^{+}\) and \(p\in \mathbb{N} \)).


Taking the left-hand side of Eq. (2.3) by \(\mathfrak{I}\) and upon using (1.7), we have

$$\begin{aligned} \begin{aligned} \mathfrak{I}&= \int ^{\infty }_{0}e^{-s\xi }\xi ^{ \frac{\delta }{k}-1}\sum^{\infty }_{n=0} \frac{(\alpha _{1})_{n,k}(\alpha _{2})_{n,k}(u\xi )^{n}}{(\alpha _{3})_{n,k}(pn)!}\,d \xi \\ &=\sum^{\infty }_{n=0} \frac{(\alpha _{1})_{n,k}(\alpha _{2})_{n,k}}{(\alpha _{3})_{n,k}} \frac{u^{n}}{(pn)!} \biggl\{ \int ^{\infty }_{0}e^{-s\xi }\xi ^{ \frac{\delta }{k}+n-1}\,d\xi \biggr\} . \end{aligned} \end{aligned}$$

Putting \(s\xi =\frac{v^{k}}{k}\), we have

$$\begin{aligned} \begin{aligned} \mathfrak{I}&=\sum^{\infty }_{n=0} \frac{(\alpha _{1})_{n,k}(\alpha _{2})_{n,k}}{(\alpha _{3})_{n,k}} \frac{u^{n}}{(np)!} \int ^{\infty }_{0}e^{\frac{-v^{k}}{k}} \biggl( \frac{v^{k}}{ks} \biggr)^{\frac{\delta }{k}+n-1}\frac{v^{k-1}}{s}\,dv \\ &=\sum^{\infty }_{n=0} \frac{(\alpha _{1})_{n,k}(\alpha _{2})_{n,k}}{(\alpha _{3})_{n,k}} \frac{u^{n}}{(np)!} \int ^{\infty }_{0} \frac{e^{\frac{-v^{k}}{k}}v^{\delta +nk-1}}{k^{\frac{\delta }{k}+n-1}s^{\frac{\delta }{k}+n}} \\ &=\frac{k\Gamma ^{k}(\delta )}{(ks)^{\frac{\delta }{k}}}\sum^{\infty }_{n=0} \biggl(\frac{u}{ks} \biggr)^{n} \frac{(\alpha _{1})_{n,k}(\alpha _{2})_{n,k}(\delta )_{n,k}}{(\alpha _{3})_{n,k}} \frac{1}{(pn)!}. \end{aligned} \end{aligned}$$

We thus obtain the required result. □

Theorem 2.2

For \(\alpha _{1},\alpha _{2},w\in \mathbb{C}\), \(\alpha _{3} \in \mathbb{C} \setminus \mathbb{Z}^{-}_{0}\), \(\operatorname{Re}(\alpha _{1})>0\), \(\operatorname{Re}(\alpha _{2})>0\), \(k\in \mathbb{R}^{+}\), \(p\in \mathbb{N}\), and \(0<\beta \leq 1\), the following fractional Fourier transform (FFT) holds true:

$$\begin{aligned} \begin{aligned} &\mathfrak{F}_{\beta } \left\{_{2} \mathbf{H}^{(p,k)}_{1} \left [{\textstyle\begin{array}{c} (\alpha _{1},k)(\alpha _{2},k) \\ (\alpha _{3},k)\end{array}\displaystyle };w \right ] \right) \\ &\quad =\sum^{\infty }_{n=0} \frac{(\alpha _{1})_{n,k}(\alpha _{2})_{n,k}}{(\alpha _{3})_{n,k}}( \omega )^{-(\frac{n+1}{\beta })}(-1)^{n}(i)^{(n-1)} \\ &\qquad {}\times \frac{n!}{(pn)!}. \end{aligned} \end{aligned}$$


For convenience, let the left-hand side of (2.4) be denoted by T. Applying the fractional Fourier transform (2.2) to (1.7) when \(w<0\), we observe that

$$\begin{aligned} \begin{aligned} \mathrm{T}&= \int _{R} e^{i\omega ^{\frac{1}{\beta }}w} \sum ^{\infty }_{n=0} \frac{(\alpha _{1})_{n,k}(\alpha _{2})_{n,k}}{(\alpha _{3})_{n,k}(np)!}w^{n} \,dw \\ &=\sum^{\infty }_{n=0} \frac{(\alpha _{1})_{n,k}(\alpha _{2})_{n,k}}{(\alpha _{3})_{n,k}} \int ^{0}_{-\infty } e^{i\omega ^{\frac{1}{\beta }}w} w^{n}\,dw. \end{aligned} \end{aligned}$$

Letting \(-t=i\omega ^{\frac{1}{\beta }}w\), we obtain

$$\begin{aligned} \mathrm{T}&=\sum^{\infty }_{n=0} \frac{(\alpha _{1})_{n,k}(\alpha _{2})_{n,k}}{(\alpha _{3})_{n,k}(np)!} \int ^{\infty }_{0}e^{-t} \biggl( \frac{-t}{i\omega ^{\frac{1}{\beta }}} \biggr)^{n} \biggl(\frac{dt}{i\omega ^{\frac{1}{\beta }}} \biggr) \\ & =\sum^{\infty }_{n=0} \frac{(\alpha _{1})_{n,k}(\alpha _{2})_{n,k}}{(\alpha _{3})_{n,k}(np)!}(-1)^{n} (i)^{-(n+1)}(\omega )^{\frac{-(n+1)}{\beta }}\Gamma (n+1), \end{aligned}$$

which yields our required result (2.4). □

Corollary 2.1

For \(p=1\), the FFT of k-Gauss hypergeometric function of order β is (see [28])

$$\begin{aligned} \begin{aligned} \mathfrak{F}_{\beta } \left[_{2} \mathbf{H}^{(k)}_{1} \left [{\textstyle\begin{array}{c} (\alpha _{1},k)(\alpha _{2},k) \\ (\alpha _{3},k)\end{array}\displaystyle };w \right ] \right) =\sum^{\infty }_{n=0} \frac{(\alpha _{1})_{n,k}(\alpha _{2})_{n,k}}{(\alpha _{3})_{n,k}}(-1)^{n} (i)^{-(n+1)}( \omega )^{\frac{-(n+1)}{\beta }}. \end{aligned} \end{aligned}$$

Further, for \(p=1\) and \(k=1\), we get the FFT of Gauss hypergeometric function

$$\begin{aligned} \begin{aligned} \mathfrak{F}_{\beta } \left[_{2} \mathbf{H}_{1}\left [{\textstyle\begin{array}{c} (\alpha _{1})(\alpha _{2}) \\ (\alpha _{3})\end{array}\displaystyle };w \right ] \right) =\sum^{\infty }_{n=0} \frac{(\alpha _{1})_{n}(\alpha _{2})_{n}}{(\alpha _{3})_{n}}(-1)^{n} (i)^{-(n+1)}(\omega )^{ \frac{-(n+1)}{\beta }}. \end{aligned} \end{aligned}$$

3 k-fraction calculus of the \(_{2}\mathbf{H}^{(p,k)}_{1}\)

Nowadays, computations of images of k-analogues of special functions under operators of k-fractional calculus have found significant importance and applications by many references (for instance, see [1517, 2840]).

The k-Riemann–Liouville fractional integral using k-gamma function is defined in [31] as follows:

$$\begin{aligned} \begin{aligned} \bigl(\mathbf{I}^{\upsilon }_{k} f( \tau )\bigr) (x)= \frac{1}{k\Gamma ^{k}(\upsilon )} \int ^{x}_{0} f(\tau ) (x-\tau )^{ \frac{\upsilon }{k}-1} \,d\tau ,\quad \upsilon ,k\in \mathbb{R}^{+}. \end{aligned} \end{aligned}$$

Therefore, the k-Riemann–Liouville fractional derivative of order υ is introduced in [29, 31] by

$$\begin{aligned} \begin{aligned} \mathbf{D}^{\upsilon }_{k}\bigl\{ f( \eta )\bigr\} =D\bigl(\mathbf{I}^{(1- \upsilon )}_{k} f(\eta )\bigr); \quad 0< \upsilon \leq 1, D=\frac{d}{d\eta }. \end{aligned} \end{aligned}$$

Let \(\alpha ,\beta ,\gamma ,\delta ,\eta \in \mathbb{C}(\operatorname{Re}(\eta )>0)\) and \(x>0\), then the generalized fractional calculus operator (the Marichev–Saigo–Maeda operator) is defined by (see [33, 41, 42])

( I 0 , x α , β , γ , δ , η ) f ( x ) = x α Γ ( η ) 0 x ( x t ) η 1 t β × F 3 [ α β γ δ η ; 1 t x , 1 x t ] f ( t ) d t ,

where \(F_{3}\) denotes the Appell third function, known also as Horn’s \(F_{3}\)-function for \((\max \{|z| < 1, |w| \}< 1)\) defined by the series

F 3 [ α β γ δ η ; z , w ] = m , n = 0 ( α ) m ( β ) n ( γ ) m ( δ ) n ( η ) m + n m ! n ! z m w n ,

which reduces to the Gauss hypergeometric function as follows:

H 1 2 [ α , β η ; w ] = F [ α β γ δ η ; z , 0 ] = F 3 [ α 0 γ δ η ; z , w ] = F 3 [ α β γ 0 η ; z , w ] .

The following image formula, which is required in the sequel, can be easily derived from the direct application of the fractional integral operator (3.3), (see, e.g., [41, 42]):

$$\begin{aligned} \begin{aligned} & \bigl(\mathbf{I}^{\alpha ,\beta ,\gamma ,\delta ,\eta }_{0,x} t^{\theta -1} \bigr)(x) \\ &\quad = \frac{\Gamma (\theta ) \Gamma (\theta +\eta -\alpha -\beta -\gamma ) \Gamma (\theta +\delta -\beta )}{\Gamma (\theta +\delta ) \Gamma (\theta +\eta -\alpha -\beta ) \Gamma (\theta +\eta -\beta -\gamma )} x^{\theta +\eta - \alpha -\beta -1}, \end{aligned} \end{aligned}$$

where \(\operatorname{Re}(\eta )>0\), \(\operatorname{Re}(\theta )>\max \{0,\operatorname{Re}(\alpha +\beta +\gamma -\eta ),\operatorname{Re}(\beta -\delta )\}\).

Here, we aim at establishing certain new image formulas for the \((p,k)\)-analogues of Gauss hypergeometric functions by applying the k-fractional derivative by (3.2) and left-sided operator of Marichev–Saigo–Maeda fractional integral defined by (3.3). On account of the general nature of the hypergeometric functions, a number of known formulas can easily be found as special cases of our main outcomes.

Theorem 3.1

For \(\alpha _{1},\alpha _{2},\nu ,u\in \mathbb{C}\), \(\alpha _{3} \in \mathbb{C}\setminus \mathbb{Z}^{-}_{0}\), \(\operatorname{Re}(\alpha _{1})>0\), \(\operatorname{Re}(\alpha _{2})>0\), \(k\in \mathbb{R}^{+}\), \(p\in \mathbb{N}\), and \(0<\Re (\nu )\leq 1\), we have

$$\begin{aligned} \begin{aligned} &\mathbf{D}^{\nu }_{k} \left \{u^{\frac{\delta }{k}}_{2} \mathbf{H}^{(p,k)}_{1}\left [{\textstyle\begin{array}{c} (\alpha _{1},k)(\alpha _{2},k) \\ (\alpha _{3},k)\end{array}\displaystyle };u \right ] \right\} \\ &\quad =\frac{\lambda \Gamma ^{k}(\lambda )}{k\Gamma ^{k}(1-\nu +\delta )}u^{ \frac{1-\nu +\delta }{k}-1}_{3} \mathbf{H}^{(p,k)}_{2}\left [{\textstyle\begin{array}{c} (\alpha _{1},k) (\alpha _{2},k) (\delta +k,k) \\ (\alpha _{3},k) (1-\nu +\delta ,k)\end{array}\displaystyle };u \right ]. \end{aligned} \end{aligned}$$


From (1.7) and (3.2), we observe that

$$\begin{aligned} \begin{aligned} &\textbf{D}^{\nu }_{k} \left[u^{\frac{\delta }{k}}_{2} \mathbf{H}^{(p,k)}_{1} \left [{ \textstyle\begin{array}{c} (\alpha _{1},k)(\alpha _{2},k) \\ (\alpha _{3},k) \end{array}\displaystyle };u\right ] \right]\\ &\quad = \frac{d}{du} \left[\mathbf{I}^{1-\nu }_{k}u^{ \frac{\delta }{k}}_{2} \mathbf{H}^{(p,k)}_{1}\left [{ \textstyle\begin{array}{c} (\alpha _{1},k)(\alpha _{2},k) \\ (\alpha _{3},k) \end{array}\displaystyle };t\right ] \right]\,dt \\ &\quad =\frac{d}{du}\frac{1}{k\Gamma ^{k}(1-\nu )} \int ^{u}_{0}(u-t)^{ \frac{1-\nu }{k}-1}t^{\frac{\delta }{k}}_{2} \mathbf{H}^{(p,k)}_{1} \left [{ \textstyle\begin{array}{c} (\alpha _{1},k)(\alpha _{2},k) \\ (\alpha _{3},k) \end{array}\displaystyle };t\right ] ]\,dt. \end{aligned} \end{aligned}$$

Putting \(t=ux\) in the above equation and after simple computations, we arrive at

$$\begin{aligned} \begin{aligned} &\textbf{D}^{\nu }_{k} \left[u^{\frac{\delta }{k}}_{2} \mathbf{H}^{(p,k)}_{1} \left [{ \textstyle\begin{array}{c} (\alpha _{1},k)(\alpha _{2},k) \\ (\alpha _{3},k) \end{array}\displaystyle };u\right ] \right] \\ &\quad =\frac{1}{k\Gamma ^{k}(1-\nu )}\sum^{\infty }_{n=0} \frac{(\alpha _{1})_{n,k}(\alpha _{2})_{n,k}}{(\alpha _{3})_{n,k}(np)!} \frac{d}{du} \int ^{1}_{0}(u-ux)^{\frac{1-\nu }{k}-1}(ux)^{n+ \frac{\delta }{k}}udx \\ &\quad =\frac{1}{k\Gamma ^{k}(1-\nu )}\sum^{\infty }_{n=0} \frac{(\alpha _{1})_{n,k}(\alpha _{2})_{n,k}}{(\alpha _{3})_{n,k}(np)!} \frac{d}{du} \int ^{1}_{0}(1-x)^{\frac{1-\nu }{k}-1}(x)^{n+ \frac{\delta }{k}}u^{\frac{1-\nu +\delta +nk}{k}} \,dx \\ &\quad =\frac{1}{k\Gamma ^{k}(1-\nu )}\sum^{\infty }_{n=0} \frac{(\alpha _{1})_{n,k}(\alpha _{2})_{n,k}}{(\alpha _{3})_{n,k}(np)!} \frac{d}{du}u^{\frac{1-\nu +\delta +nk}{k}}\mathbb{B} \biggl( \frac{1-\nu }{k},n+\frac{\delta }{k}+1 \biggr) \\ &\quad =\frac{1}{k\Gamma ^{k}(1-\nu )}\sum^{\infty }_{n=0} \biggl[ \frac{(\alpha _{1})_{n,k}(\alpha _{2})_{n,k}}{(\alpha _{3})_{n,k}(np)!}\biggl( \frac{1-\nu +\delta +nk}{k}\biggr)u^{\frac{1-\nu +\delta +nk}{k}-1} \frac{\Gamma (\frac{1-\nu }{k})\Gamma (\frac{nk+\delta +k}{k})}{(\frac{1-\nu +\lambda +nk+}{k})} \biggr] \\ &\quad =\sum^{\infty }_{n=0} \frac{(\alpha _{1})_{n,k}(\alpha _{2})_{n,k}}{(\alpha _{3})_{n,k}} \frac{\Gamma ^{k}(nk+\delta +k)}{k\Gamma ^{k}(1-\nu +\delta +nk)} \frac{u^{\frac{1-\nu +\delta +nk}{k}-1}}{(np)!} \\ &\quad = u^{\frac{1-\nu +\delta }{k}-1} \frac{\delta \Gamma ^{k}(\delta )}{k\Gamma ^{k}(1-\nu +\delta )}\sum^{ \infty }_{n=0} \frac{(\alpha _{1})_{n,k}(\alpha _{2})_{n,k}(\delta +k)_{n,k}}{(\alpha _{3})_{n,k}(1-\nu +\delta )_{n,k}} \frac{u^{n}}{(np)!}. \end{aligned} \end{aligned}$$

This completes the proof of Theorem 3.1. □

Theorem 3.2

Assume that \(\alpha ,\beta ,\gamma ,\delta ,\eta ,\vartheta , \alpha _{1},\alpha _{2} \in \mathbb{C}\), \(\alpha _{3} \in \mathbb{C}\setminus \mathbb{Z}^{-}_{0}\), \(x>0\), \(k\in \mathbb{R}^{+}\), and \(p\in \mathbb{N}\) such that \(\Re {(\frac{\vartheta }{k})}>\max \{0,\Re (\beta -\delta ),\Re ( \alpha +\beta +\gamma -\eta ) \}\), then we have

( I 0 , x α , β , γ , δ , η w ϑ k 1 2 H 1 ( p , k ) [ ( α 1 ; k ) , ( α 2 ; k ) ( α 3 ; k ) ; w ] ) ( x ) = k η x α β + η + ϑ k 1 Γ k ( ϑ ) Γ k ( ϑ k β + k δ ) Γ k ( ϑ k α k β k γ + k η ) Γ k ( ϑ + k δ ) Γ k ( ϑ k α k β + k η ) Γ k ( ϑ k β k γ + k η ) × 5 H 4 ( p , k ) [ ( α 1 ; k ) ( α 2 ; k ) ( ϑ ; k ) ( ϑ k β + k δ ; k ) ( ϑ k α k β k γ + k η ; k ) ( α 3 ; k ) ( ϑ + k δ ; k ) ( ϑ k α k β + k η ; k ) ( ϑ k β k γ + k η ; k ) ; x ] .


We indicate the left-hand side of(3.6) by ϒ, and invoking to Eqs. (3.3) and (1.7), we find

$$\begin{aligned} \begin{aligned} \Upsilon ={}& \Biggl(\mathbf{I}^{\alpha ,\beta ,\gamma , \delta ,\eta }_{0,x} w^{\frac{\vartheta }{k}-1} \sum^{\infty }_{n=0} \frac{(\alpha _{1})_{n,k} (\alpha _{2})_{n,k}}{(\alpha _{3})_{n,k}} \frac{w^{n}}{(pn)!} \Biggr) (x) \\ ={}&\sum^{\infty }_{n=0} \frac{(\alpha _{1})_{n,k} (\alpha _{2})_{n,k}}{(\alpha _{3})_{n,k}}\frac{1}{(pn)!} \bigl( \mathbf{I}^{\alpha ,\beta ,\gamma ,\delta ,\eta }_{0,x} w^{ \frac{\vartheta }{k}+n-1} \bigr) (x) \\ ={}&\sum^{\infty }_{n=0} \frac{(\alpha _{1})_{n,k} (\alpha _{2})_{n,k}}{(\alpha _{3})_{n,k}} \frac{\Gamma (\frac{\vartheta }{k}+n)}{(pn)! \Gamma (\delta +\frac{\vartheta }{k}+n)} \\ &{}\times \frac{\Gamma (-\beta +\delta +\frac{\vartheta +nk}{k}) \Gamma (-\alpha -\beta -\gamma +\eta +\frac{\vartheta +nk}{k})}{\Gamma (-\alpha -\beta +\eta +\frac{\vartheta +nk}{k}) \Gamma (-\beta -\gamma +\eta +\frac{\vartheta +nk}{k})} x^{-\alpha -\beta +\eta +\frac{\vartheta +nk}{k}-1}. \end{aligned} \end{aligned}$$

Upon using (3.4) and after a simplification, we get the following expression:

$$\begin{aligned} \begin{aligned} \Upsilon ={}&\sum^{\infty }_{n=0} \frac{(\alpha _{1})_{n,k} (\alpha _{2})_{n,k}}{(\alpha _{3})_{n,k}}\frac{x^{n} k^{\eta }}{(pn)!} \frac{(\vartheta )_{n,k} \Gamma ^{k}(\vartheta )}{(\vartheta +k\delta )_{n,k} \Gamma ^{k}(\vartheta +k \delta )} \\ &{}\times \frac{(\vartheta -k\beta +k\delta )_{n,k} \Gamma ^{k}(-k\beta +k \delta +\vartheta )}{(\vartheta -k\alpha -k\beta +k\eta )_{n,k} \Gamma ^{k}(\vartheta -k \alpha -k\beta +k\eta )} \\ &{}\times\frac{(\vartheta -k \alpha -k \beta +k\eta )_{n,k}}{(\vartheta -k \beta -k \gamma +k\eta )_{n,k}}. \frac{\Gamma ^{k}(\vartheta -k \alpha -k \beta -k \gamma +k\eta ) }{\Gamma ^{k}(\vartheta -k \beta -k \gamma +k\eta )} x^{-\alpha -\beta +\eta +\frac{\vartheta }{k}-1}, \end{aligned} \end{aligned}$$

whose last summation, in view of (1.2), is easily seen to arrive at the expression in (3.6). This completes the proof of Theorem 3.2. □

4 Conclusion

Recently, the applications and importance of integral transforms and fractional calculus operators involving a variety of special functions have received more attention in various fields like mathematical analysis, survival analysis, physics, statistics, and engineering. In fact, this manuscript is a continuation of the recent authors’ articles [23, 24], where we have introduced the \((p,k)\)-analogues of hypergeometric functions and their various properties and applications. In this line of research, we have derived integral transforms and image formulas for the \((p,k)\)-analogues of hypergeometric functions. We also have considered that by setting \(p\to 1\), the various outcomes considered in this manuscript reduce to the corresponding outcomes (see [28, 31, 36]). Also, for \(k\to 1\), we obtain many interesting new outcomes for the p-extended hypergeometric functions. Further, if we take both \(k\to 1\) and \(p\to 1\), then the obtained results reduce to the results analogous to the usual hypergeometric functions. This approach allows the related research work to be reported in further articles. Additionally, all the outcomes presented here are expected to find some applications in control theory and to the solutions of fractional-order systems, for instance, see [4347] and the references cited therein.

Availability of data and materials

Not applicable.


  1. Agarwal, P., Agarwal, R.P., Ruzhansky, M.: Special Functions and Analysis of Differential Equations. Chapman and Hall, London (2020)

    Book  MATH  Google Scholar 

  2. Agarwal, P., Baleanu, D., Chen, Y., Momani, S., Tenreiro Machado, J.A. (eds.): Fractional Calculus: ICFDA 2018 1st edn. Springer Proceedings in Mathematics Statistics, vol. 303. Springer, Singapore (2019).

    Book  Google Scholar 

  3. Agarwal, P., Dragomir, S., Jleli, M., Samet, B.: Advances in Mathematical Inequalities and Applications. Trends in Mathematics. Birkhäuser, Basel (2019)

    MATH  Google Scholar 

  4. Abdalla, M.: Special matrix functions: characteristics, achievements and future directions. Linear Multilinear Algebra 68, 1–28 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  5. Abdalla, M.: Fractional operators for the Wright hypergeometric matrix functions. Adv. Differ. Equ. 2020, Article ID 246 (2020)

    Article  MathSciNet  Google Scholar 

  6. Abdalla, M., Idris, S.A., Mekawy, I.: Some results on the extended hypergeometric matrix functions and related functions. J. Math. 2021, Article ID 2046726 (2021)

    Article  MathSciNet  Google Scholar 

  7. Abdalla, M., Bakhet, A.: Extended Gauss hypergeometric matrix functions. Iran. J. Sci. Technol. Trans. A, Sci. 42, 1465–1470 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Goyal, R., Momani, S., Agarwal, P., Rassias, M.: An extension of beta function by using Wimans function. Axioms 10, 187 (2021)

    Article  Google Scholar 

  9. Agarwal, P., Chand, M., Purohit, S.D.: A note on generating functions involving the generalized Gauss hypergeometric functions. Nat. Acad. Sci. Lett. 37, 457–459 (2014)

    Article  MathSciNet  Google Scholar 

  10. Hidan, M., Abdalla, M.: A note on the Appell hypergeometric matrix function \(F_{2}\). Math. Probl. Eng. 2020, Article ID 6058987 (2020)

    Article  Google Scholar 

  11. Vivas-Cortez, M., Ali, M.A., Budak, H., Kalsoom, H., Agarwal, P.: Some new HermiteHadamard and related inequalities for convex functions via \((p; q)\)-integra. Entropy 23(7), 828 (2021)

    Article  Google Scholar 

  12. Agarwal, P., Restrepo, J.E.: An extension by means of ω-weighted classes of the generalized Riemann–Liouville k-fractional integral inequalities. J. Math. Inequal. 14, 35–46 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  13. Diaz, R., Pariguan, E.: On hypergeometric functions and k-Pochhammer symbol. Divulg. Mat. 15, 179–192 (2007)

    MathSciNet  MATH  Google Scholar 

  14. Srivastava, H.M., Choi, J.: Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier, Amsterdam (2012)

    MATH  Google Scholar 

  15. Ali, R.S., Mubeen, S., Ahmad, M.M.: A class of fractional integral operators with multi-index Mittag-Leffler k-function and Bessel k-function of first kind. J. Math. Comput. Sci. 22, 266–281 (2021)

    Google Scholar 

  16. Agarwal, P., Chand, M., Choi, J., Singh, G.: Certain fractional integrals and image formulas of generalized k-Bessel function. Commun. Korean Math. Soc. 33, 423–436 (2018)

    MathSciNet  MATH  Google Scholar 

  17. Agarwal, P., Ntouyas, S., Jain, S., Chand, M., Singh, G.: Fractional kinetic equations involving generalized k-Bessel function via Sumudu transform. Alex. Eng. J. 57, 1937–1942 (2018)

    Article  Google Scholar 

  18. Abdalla, M., Hidan, M.: Investigation of the k-analogue of Gauss hypergeometric functions constructed by the Hadamard product. Symmetry 13, 714 (2021)

    Article  Google Scholar 

  19. Mondal, S., Akel, M.: Differential equation and inequalities of the generalized k-Bessel functions. J. Inequal. Appl. 2018, 175 (2018)

    Article  MathSciNet  Google Scholar 

  20. Li, S., Dong, Y.: K-hypergeometric series solutions to one type of non-homogeneous k-hypergeometric equations. Symmetry 11, 262 (2019)

    Article  MATH  Google Scholar 

  21. Yilmazer, R., Ali, K.: Discrete fractional solutions to the k-hypergeometric differential equation. Math. Methods Appl. Sci. 44, 7614–7621 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  22. Amsalu, H., Shimelis, B., Suthar, D.L.: Pathway fractional integral formulas involving S-function in the kerne. Math. Probl. Eng. 2020, Article ID 4236823 (2020)

    Article  MATH  Google Scholar 

  23. Abdalla, M., Hidan, M., Boulaaras, S., Cherif, B.: Investigation of extended k-hypergeometric functions and associated fractional integrals. Math. Probl. Eng. 2021, Article ID 9924265 (2021)

    Article  MathSciNet  Google Scholar 

  24. Hidan, M., Boulaaras, S., Cherif, B., Abdalla, M.: Further results on the \((p;k)\)-analogue of hypergeometric functions associated with fractional calculus operators. Math. Probl. Eng. 2021, Article ID 5535962 (2021)

    MathSciNet  Google Scholar 

  25. Mohamed, M.T.: Certain Functions and Sets of Polynomials and their Convergence Properties. PhD. Dissertation, Assiut University, Egypt (2003)

  26. Debnath, L., Bhatta, D.: Integral Transforms and Their Applications, 3rd edn. Chapman and Hall, London (2015)

    MATH  Google Scholar 

  27. Kochubei, A., Luchko, Y.: Handbook of Fractional Calculus with Applications. Volume 1, Basic Theory. De Gruyter, Berlin (2019)

    Google Scholar 

  28. Gupta, V., Bhatt, M.: Some results of k-hypergeometric functions associated with integral transforms and fractional calculus. Int. J. Math. Sci. Comput. 5, 106–109 (2015)

    MathSciNet  Google Scholar 

  29. Mubeen, S., Habibullah, G.M.: k-fractional integrals and application. Int. J. Contemp. Math. Sci. 7, 89–94 (2012)

    MathSciNet  MATH  Google Scholar 

  30. Naz, S., Naeem, M.N.: On the generalization of k-fractional Hilfer-Katugampola derivative with Cauchy problem. Turk. J. Math. 45, 110–124 (2021)

    Article  MathSciNet  Google Scholar 

  31. Rahman, G., Mubeen, S., Nisar, K.: On generalized k-fractional derivative operator. AIMS Math. 5, 1936–1945 (2020)

    Article  MathSciNet  Google Scholar 

  32. Azam, M., Farid, G., Rehman, M.: Study of generalized type k-fractional derivatives. Adv. Differ. Equ. 2017, 249 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kiryakova, V.: Unified approach to fractional calculus images of special functions a survey. Mathematics 8, Article ID 136 (2020).

    Article  Google Scholar 

  34. Zayed, M., Abul-Ez, M., Abdalla, M., Saad, N.: On the fractional order Rodrigues formula for the shifted Legendre-type matrix polynomials. Mathematics 8, Article ID 136 (2020)

    Article  Google Scholar 

  35. Zayed, M., Hidan, M., Abdalla, M., Abul-Ez, M.: Fractional order of Legendre-type matrix polynomials. Adv. Differ. Equ. 2020, Article ID 506 (2020)

    Article  MathSciNet  Google Scholar 

  36. Mittal, E., Joshi, S.: Note on a k-generalised fractional derivative. Discrete Contin. Dyn. Syst., Ser. S 13, 797–804 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  37. Gunerhan, H., Celik, E.: Analytical and approximate solutions of fractional partial differential algebraic equations. Appl. Math. Nonlinear Sci. 5, 109–120 (2020)

    Article  MathSciNet  Google Scholar 

  38. Īlhan, E., Kıymaz, Ī.O.: A generalization of truncated M-fractional derivative and applications to fractional differential equations. Appl. Math. Nonlinear Sci. 5, 171–188 (2020)

    Article  MathSciNet  Google Scholar 

  39. Yokus, A., Gulbahar, S.: Numerical solutions with linearization techniques of the fractional Harry Dym equation. Appl. Math. Nonlinear Sci. 4, 35–42 (2019)

    Article  MathSciNet  Google Scholar 

  40. Ziane, D., Cherif, M.H., Cattani, C., Belghaba, K.: Yang-Laplace decomposition method for nonlinear system of local fractional partial differential equations. Appl. Math. Nonlinear Sci. 4, 489–502 (2019)

    Article  MathSciNet  Google Scholar 

  41. Saigo, M., Maeda, N.: More generalization of fractional calculus, transform methods and special functions. In: Varna, vol. 96, p. 386–400. Bulgarian Academy of Sciences, Sofia (1998)

    Google Scholar 

  42. Kabra, S., Nagar, H., Nisar, K.S., Vyas, V.K.: Marichev-Saigo-Maeda fractional operators on generalized function \(G_{\rho ,\eta ,\gamma } [a,z] \). Math. Eng. Sci. Aerosp. 11, 371–379 (2020)

    Google Scholar 

  43. Abdalla, M., Akel, M.: Computation of Fourier transform representations involving the generalized Bessel matrix polynomials. Adv. Differ. Equ. 2021, Article ID 418 (2021)

    Article  MathSciNet  Google Scholar 

  44. Matsubara-heo, S.J.: On Mellin-Barnes integral representations for GKZ hypergeometric functions. Kyushu J. Math. 74, 109–125 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  45. Guefaifia, R., Boulaaras, S.M., El-Sayed, A.A., Abdalla, M., Cherif, B.: On existence of sequences of weak solutions of fractional systems with Lipschitz nonlinearity. J. Funct. Spaces 2021, Article ID 5510387 (2021)

    MathSciNet  MATH  Google Scholar 

  46. Kamache, F., Boulaaras, S.M., Guefaifia, R., Chung, N.T., Cherif, B., Abdalla, M.: On existence of multiplicity of weak solutions for a new class of nonlinear fractional boundary value systems via variational approach. J. Funct. Spaces 2021, Article ID 5544740 (2021)

    MathSciNet  Google Scholar 

  47. Boulaaras, S., Choucha, A., Cherif, B., Alharbi, A., Abdalla, M.: Blow up of solutions for a system of two singular nonlocal viscoelastic equations with damping, general source terms and a wide class of relaxation functions. AIMS Math. 6(5), 4664–4676 (2021)

    Article  MathSciNet  Google Scholar 

Download references


The third and fourth authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Research Group Project under Grant No. (R.G.P-1/95/42).


Not applicable.

Author information

Authors and Affiliations



The authors contributed equally in this article. They have all read and approved the final manuscript.

Corresponding author

Correspondence to Salah Boulaaras.

Ethics declarations

Ethics approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abdalla, M., Boulaaras, S., Akel, M. et al. Certain fractional formulas of the extended k-hypergeometric functions. Adv Differ Equ 2021, 450 (2021).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: