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 [15–17, 28–40]).
The k-Riemann–Liouville fractional integral using k-gamma function is defined in [31] as follows:
$$\begin{aligned} \begin{aligned} \bigl(\mathbf{I}^{\upsilon }_{k} f( \tau )\bigr) (x)= \frac{1}{k\Gamma ^{k}(\upsilon )} \int ^{x}_{0} f(\tau ) (x-\tau )^{ \frac{\upsilon }{k}-1} \,d\tau ,\quad \upsilon ,k\in \mathbb{R}^{+}. \end{aligned} \end{aligned}$$
(3.1)
Therefore, the k-Riemann–Liouville fractional derivative of order υ is introduced in [29, 31] by
$$\begin{aligned} \begin{aligned} \mathbf{D}^{\upsilon }_{k}\bigl\{ f( \eta )\bigr\} =D\bigl(\mathbf{I}^{(1- \upsilon )}_{k} f(\eta )\bigr); \quad 0< \upsilon \leq 1, D=\frac{d}{d\eta }. \end{aligned} \end{aligned}$$
(3.2)
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])
(3.3)
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
which reduces to the Gauss hypergeometric function as follows:
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]):
$$\begin{aligned} \begin{aligned} & \bigl(\mathbf{I}^{\alpha ,\beta ,\gamma ,\delta ,\eta }_{0,x} t^{\theta -1} \bigr)(x) \\ &\quad = \frac{\Gamma (\theta ) \Gamma (\theta +\eta -\alpha -\beta -\gamma ) \Gamma (\theta +\delta -\beta )}{\Gamma (\theta +\delta ) \Gamma (\theta +\eta -\alpha -\beta ) \Gamma (\theta +\eta -\beta -\gamma )} x^{\theta +\eta - \alpha -\beta -1}, \end{aligned} \end{aligned}$$
(3.4)
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
$$\begin{aligned} \begin{aligned} &\mathbf{D}^{\nu }_{k} \left \{u^{\frac{\delta }{k}}_{2} \mathbf{H}^{(p,k)}_{1}\left [{\textstyle\begin{array}{c} (\alpha _{1},k)(\alpha _{2},k) \\ (\alpha _{3},k)\end{array}\displaystyle };u \right ] \right\} \\ &\quad =\frac{\lambda \Gamma ^{k}(\lambda )}{k\Gamma ^{k}(1-\nu +\delta )}u^{ \frac{1-\nu +\delta }{k}-1}_{3} \mathbf{H}^{(p,k)}_{2}\left [{\textstyle\begin{array}{c} (\alpha _{1},k) (\alpha _{2},k) (\delta +k,k) \\ (\alpha _{3},k) (1-\nu +\delta ,k)\end{array}\displaystyle };u \right ]. \end{aligned} \end{aligned}$$
(3.5)
Proof
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
(3.6)
Proof
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. □