In this section, we solve the composition of the Mittag-Leffler with power function to generalized left- and right-sided fractional integral operators and also discuss k-calculus.
Theorem 3
For
\(a, b, c, \rho , \delta \in {\mathbb{C}}\)
with
$$ \Re (a)>0 \quad \textit{and}\quad \Re (\rho +c-b)>0,\qquad \nu >0 , \qquad \lambda > 0 ,\qquad w\in {\mathbb{R}}, $$
we have
$$ \begin{aligned} \bigl(I_{0,u}^{a,b,c}t^{{\rho }-1} E_{\nu ,\rho }^{\delta } \bigl(wt^{{\lambda }} \bigr)\bigr) (u)&= \frac{u ^{-b-1+{\rho }}}{{\varGamma ({\delta })}} \\ &\quad {}\times {}_{3}\varPsi _{3}\left [ \textstyle\begin{array}{c} (c-b+{\rho },{\lambda }),({\rho },{\lambda }),({\delta },1) \\ (a+c+{\rho },{\lambda }),({\rho }-b,{\lambda }),({\rho },{\nu }) \end{array}\displaystyle \middle | wu^{{\lambda }} \right ]. \end{aligned} $$
Proof
Using the power function and (10) in (6), we have
$$\begin{aligned}& \bigl(I_{0,u}^{a,b,c}t^{{\rho }-1}E_{\nu ,\rho }^{\delta } \bigl(wt^{\lambda } \bigr)\bigr) (u)=\frac{u ^{-a-b}}{\varGamma (a)} \int _{0}^{u}(u-t)^{a-1} \\& \hphantom{(I_{0,u}^{a,b,c}t^{{\rho }-1}E_{\nu ,\rho }^{\delta } (wt^{\lambda } ) (u)={}}{}\times {{}_{2}F_{1} \biggl(a+b,-c;a;1- \frac{t}{u} \biggr)}t^{{\rho }-1} \sum _{n=0}^{\infty } \frac{(\delta )_{n}}{\varGamma (\nu n+\rho )n!} \bigl(wt^{ {\lambda }} \bigr)^{n}\,dt \end{aligned}$$
(37)
$$\begin{aligned}& \hphantom{(I_{0,u}^{a,b,c}t^{{\rho }-1}E_{\nu ,\rho }^{\delta } (wt^{\lambda } ) (u)}=\sum_{n=0}^{\infty } \frac{w^{n}(\delta )_{n}}{\varGamma (\nu n+ \rho )n!} \bigl(I_{0,u}^{a,b,c}t^{\rho +\lambda n-1} \bigr) (u). \end{aligned}$$
(38)
Since for \(n= 0,1,2, \ldots \) , \(\Re (\rho +\lambda n)\geq \Re (\rho +c-b)>0 \), using Lemma 1 with ρ replaced by \(\rho + \lambda {n} \) in equation (38), we obtain
$$ \begin{aligned}[b] &\bigl(I_{0,u}^{a,b,c}t^{{\rho }-1}E_{\nu ,\rho }^{\delta } \bigl(wt^{\lambda } \bigr)\bigr) (u) \\ &\quad =\frac{u ^{\rho -b-1}}{\varGamma ({\delta })} \sum_{n=0}^{\infty } \frac{\varGamma ({\delta +n})\varGamma ( {\rho +\lambda n})\varGamma (c-b+{\rho +\lambda n})}{\varGamma (-b+{\rho + \lambda n})\varGamma (a+c+{\rho +\lambda n})\varGamma ({\nu n+\rho })n!} \bigl(wu ^{\lambda } \bigr)^{n}. \end{aligned} $$
(39)
Using (16) in (39), we get
$$ \begin{aligned} \bigl(I_{0,u}^{a,b,c}t^{{\rho }-1} E_{\nu ,\rho }^{\delta } \bigl(wt^{{\lambda }} \bigr)\bigr) (u)&= \frac{u ^{-b-1+{\rho }}}{{\varGamma ({\delta })}} \\ &\quad {}\times {}_{3}\varPsi _{3}\left [ \textstyle\begin{array}{c} (c-b+{\rho },{\lambda }),({\rho },{\lambda }),({\delta },1) \\ (a+c+{\rho },{\lambda }),({\rho }-b,{\lambda }),({\rho },{\nu }) \end{array}\displaystyle \middle | wu^{{\lambda }} \right ]. \end{aligned} $$
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Theorem 4
For
\(a, b, c, \rho , \delta \in {\mathbb{C}}\)
with
$$\begin{aligned}& \Re (a)>0 \quad \textit{and}\quad \Re (a+\rho )>\max \bigl[-\Re (b),-\Re (c) \bigr] ,\qquad \Re (b)\neq \Re (c), \\& \nu >0,\qquad \lambda > 0,\qquad w\in {\mathbb{R}}, \end{aligned}$$
we have
$$ \begin{aligned} \bigl(I_{u,\infty }^{a,b,c}t^{\rho -1}E_{\nu ,\rho }^{\delta } \bigl(wt^{{- \lambda }} \bigr)\bigr) (u)&=\frac{u^{{\rho -b}-1}}{{\varGamma ({\delta })}} \\ &\quad {}\times {}_{3}\varPsi _{3}\left [ \textstyle\begin{array}{c} ({b}-{\rho }+1,{\lambda }),(1+c-{\rho },{\lambda }),({\delta },1) \\ (1-{\rho },{\lambda }),({a}+{b}-{\rho }+c+1,{\lambda }),({\rho }, {\nu }) \end{array}\displaystyle \middle | wu^{{-\lambda }} \right ]. \end{aligned} $$
Proof
Using the power function and (10) in (7), we have
$$ \begin{aligned}[b] \bigl(I_{u,\infty }^{a,b,c}t^{{\rho }-1}E_{\nu ,\rho }^{\delta } \bigl(wt^{{- \lambda }} \bigr)\bigr) (u)&=\frac{1}{\varGamma (a)} \int _{u}^{\infty }(t-u)^{a-1}t^{ {-a-b}} \\ &\quad {}\times {{}_{2}F_{1} \biggl(a+b,-c;a;1- \frac{u}{t} \biggr)}t^{{\rho }-1} \sum _{n=0}^{\infty } \frac{(\delta )_{n}}{{\varGamma }(\nu n+\rho )n!} \bigl(wt ^{{-\lambda }} \bigr)^{n}\,dt \\ & =\sum_{n=0}^{\infty } \frac{w^{n}(\delta )_{n}}{\varGamma (\nu n+ \rho )n!} \bigl(I_{u,\infty }^{a,b,c}t^{{\rho -\lambda n}-1} \bigr) (u). \end{aligned} $$
(40)
Since for \(n= 0,1,2,\ldots \) , \(\Re (\rho -\lambda n-1)\leq \Re (\rho +a-1)>1+ \max [-\Re (b),-\Re (c)] \), using Lemma 2 with ρ replaced by \(\rho -\lambda n\), we reduce equation (40) to
$$ \begin{aligned}[b] & \bigl(I_{u,\infty }^{a,b,c}t^{{\rho }-1}E_{\nu ,\rho }^{\delta } \bigl(wt^{{- \lambda }} \bigr) \bigr) (u) \\ &\quad =\frac{u^{{\rho -b}-1}}{{\varGamma ({\delta })}} \sum_{n=0}^{\infty } \frac{\varGamma ({\delta +n})\varGamma (1-a+ {a+b+\lambda n-\rho })\varGamma (1-a-b+c+{a+b+\lambda n-\rho })}{\varGamma (1-a-b+{a+b+\lambda n-\rho })\varGamma (1+c+{a+b+\lambda n-\rho })\varGamma ({\nu n+\rho })n!} \\ &\qquad {}\times \bigl(wu^{-\lambda } \bigr)^{n}. \end{aligned} $$
(41)
Using (16) in (41), we get
$$ \begin{aligned} \bigl(I_{u,\infty }^{a,b,c}t^{\rho -1}E_{\nu ,\rho }^{\delta } \bigl(wt^{{- \lambda }} \bigr)\bigr) (u)&=\frac{u^{{\rho -b}-1}}{{\varGamma ({\delta })}} \\ &\quad {}\times {}_{3}\varPsi _{3}\left [ \textstyle\begin{array}{c} ({b}-{\rho }+1,{\lambda }),(1+c-{\rho },{\lambda }),({\delta },1) \\ (1-{\rho },{\lambda }),({a}+{b}-{\rho }+c+1,{\lambda }),({\rho }, {\nu }) \end{array}\displaystyle \middle | wu^{{-\lambda }} \right ]. \end{aligned} $$
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Theorem 5
For
\(a, b, c, \rho , \delta \in {\mathbb{C}}\)
with
$$ \Re (a)>0 \quad \textit{and}\quad \Re (\rho +c-b)>0,\qquad \nu >0 ,\qquad \lambda > 0 ,\qquad w\in {\mathbb{R}}, $$
we have
$$ \begin{aligned} \bigl(I_{0,u}^{a,b,c}t^{\frac{\rho }{k}-1} E_{k,\nu ,\rho }^{\delta } \bigl(wt ^{\frac{\lambda }{k}} \bigr)\bigr) (u)&=\frac{k^{1-\frac{\rho }{k}}u^{-b-1+\frac{ \rho }{k}}}{\varGamma (\frac{\delta }{k})} \\ &\quad {}\times {}_{3}\varPsi _{3}\left [ \textstyle\begin{array}{c} (c-b+\frac{\rho }{k},\frac{\lambda }{k}),(\frac{\rho }{k},\frac{ \lambda }{k}),(\frac{\delta }{k},1) \\ (a+c+\frac{\rho }{k},\frac{\lambda }{k}),(\frac{\rho }{k}-b,\frac{ \lambda }{k}),(\frac{\rho }{k},\frac{\nu }{k}) \end{array}\displaystyle \middle | k^{1-\frac{\nu }{k}}wu^{\frac{\lambda }{k}} \right ]. \end{aligned} $$
Proof
Using the power k-function and (11) in (6), we have
$$\begin{aligned}& \bigl(I_{0,u}^{a,b,c}t^{\frac{\rho }{k}-1}E_{k,\nu ,\rho }^{\delta } \bigl(wt ^{\frac{\lambda }{k}} \bigr)\bigr) (u) \\& \quad =\frac{u^{-a-b}}{\varGamma (a)} \int _{0}^{u}(u-t)^{a-1} {{}_{2}F_{1} \biggl(a+b,-c;a;1- \frac{t}{u} \biggr)}t^{\frac{\rho }{k}-1} \sum_{n=0}^{\infty } \frac{(\delta )_{n,k}}{\varGamma _{k}(\nu n+ \rho )n!} \bigl(wt^{\frac{\lambda }{k}} \bigr)^{n} \,dt \end{aligned}$$
(42)
$$\begin{aligned}& \quad =\sum_{n=0}^{\infty } \frac{w^{n}(\delta )_{n,k}}{\varGamma _{k}( \nu n+\rho )n!} \bigl(I_{0,u}^{a,b,c}t^{\frac{\rho +\lambda n}{k}-1} \bigr)_{k}(u). \end{aligned}$$
(43)
Since for \(n= 0\), \(1,2,\ldots\) , \(\Re (\rho +\lambda n)\geq \Re (\rho +c-b)>0 \), using Lemma 1 with ρ replaced by by \(\frac{\rho + \lambda n}{k}\), we reduce equation (43) to
$$ \begin{aligned}[b] &\bigl(I_{0,u}^{a,b,c}t^{\frac{\rho }{k}-1}E_{k,\nu ,\rho }^{\delta } \bigl(wt ^{\lambda } \bigr)\bigr) (u) \\ &\quad =\frac{k^{1-\frac{\rho }{k}}u^{-b-1+\frac{\rho }{k}}}{ \varGamma (\frac{\delta }{k})} \sum_{n=0}^{\infty } \frac{\varGamma (\frac{\delta +nk}{k}) \varGamma (\frac{\rho +\lambda n}{k})\varGamma (c-b+ \frac{\rho +\lambda n}{k})}{\varGamma (-b+\frac{\rho +\lambda n}{k}) \varGamma (a+c+\frac{\rho +\lambda n}{k})\varGamma (\frac{\nu n+\rho }{k})n!} \bigl(wk ^{1-\frac{\nu }{k}}t^{\frac{\lambda }{k}} \bigr)^{n}. \end{aligned} $$
(44)
Using (16) in (44), we get
$$\begin{aligned} \bigl(I_{0,u}^{a,b,c}t^{\frac{\rho }{k}-1} E_{k,\nu ,\rho }^{\delta } \bigl(wt ^{\frac{\lambda }{k}} \bigr)\bigr) (u) =&\frac{k^{1-\frac{\rho }{k}}u^{-b-1+\frac{ \rho }{k}}}{\varGamma (\frac{\delta }{k})} \\ &{}\times {}_{3}\varPsi _{3}\left [ \textstyle\begin{array}{c} (c-b+\frac{\rho }{k},\frac{\lambda }{k}),(\frac{\rho }{k},\frac{ \lambda }{k}),(\frac{\delta }{k},1) \\ (a+c+\frac{\rho }{k},\frac{\lambda }{k}),(\frac{\rho }{k}-b,\frac{ \lambda }{k}),(\frac{\rho }{k},\frac{\nu }{k}) \end{array}\displaystyle \middle | k^{1-\frac{\nu }{k}}wu^{\frac{\lambda }{k}} \right ]. \end{aligned}$$
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Remark 1
If we replace k by one, then we get the result of [3].
Theorem 6
For
\(a, b, c, \rho , \delta \in {\mathbb{C}}\)
with
$$\begin{aligned}& \Re (a)>0 \quad \textit{and} \quad \Re (a+\rho )>\max \bigl[-\Re (b),-\Re (c) \bigr] ,\qquad \Re (b)\neq \Re (c), \\& \nu >0,\qquad \lambda > 0,\qquad w\in {\mathbb{R}}, \end{aligned}$$
we have
$$\begin{aligned}& \bigl(I_{u,\infty }^{a,b,c}t^{\frac{\rho }{k}-1} E_{k,\nu ,\rho }^{\delta } \bigl(wt^{\frac{-\lambda }{k}} \bigr)\bigr) (u) \\& \quad =\frac{k^{1-\frac{\rho }{k}}u^{\frac{ \rho -a-b}{k}+a-1}}{\varGamma (\frac{\delta }{k})} {}_{3}\varPsi _{3} \left [ \textstyle\begin{array}{c} (1+b-\frac{\rho }{k}, \frac{\lambda }{k}),(1+c-\frac{\rho }{k},\frac{ \lambda }{k}),(\frac{\delta }{k},1) \\ (1-\frac{\rho }{k},\frac{\lambda }{k}),(1+a+b+c-\frac{\rho }{k},\frac{ \lambda }{k}),(\frac{\rho }{k},\frac{\nu }{k}) \end{array}\displaystyle \middle | k^{1-\frac{\nu }{k}}wu^{\frac{-\lambda }{k}} \right ]. \end{aligned}$$
(45)
Proof
Using the power k-function and (11) in (7), we have
$$\begin{aligned}& \bigl(I_{u,\infty }^{a,b,c}t^{\frac{\rho }{k}-1}E_{k,\nu ,\rho }^{\delta } \bigl(wt ^{\frac{-\lambda }{k}} \bigr)\bigr) (u) \\& \quad =\frac{1}{\varGamma (a)} \int _{u}^{\infty }(t-u)^{a-1}t ^{-a-b} {{}_{2}F_{1} \biggl(a+b,-c;a;1- \frac{t}{u} \biggr)}t^{\frac{\rho }{k}-1} \sum _{n=0}^{\infty } \frac{(\delta )_{n,k}}{\varGamma _{k}(\nu n+ \rho )n!} \\& \qquad {}\times \bigl(wt^{\frac{-\lambda }{k}} \bigr)^{n} \,dt \end{aligned}$$
(46)
$$\begin{aligned}& \quad =\sum_{n=0}^{\infty } \frac{w^{n}(\delta )_{n,k}}{\varGamma _{k}( \nu n+\rho )n!} \bigl(I_{u,\infty }^{a,b,c}t^{\frac{\rho -\lambda n}{k}-1} \bigr) (u). \end{aligned}$$
(47)
Since for \(n= 0, 1,2,\ldots\) , \(\Re (\rho -\lambda n-1)\leq \Re (\rho +a-1)>1+ \max [-\Re (b),-\Re (c)] \), using Lemma 2 with ρ replaced by \(\frac{\rho -\lambda n}{k}\), we reduce equation (47) to
$$\begin{aligned}& \bigl(I_{u,\infty }^{a,b,c}t^{\frac{\rho }{k}-1}E_{k,\nu ,\rho }^{\delta } \bigl(wt ^{\frac{-\lambda }{k}} \bigr)\bigr) (u) \\& \quad =\frac{k^{1-\frac{\rho +\nu n}{k}}u^{\frac{ \rho }{k}-b-1}}{\varGamma (\frac{\delta }{k})}\sum_{n=0}^{\infty } \frac{\varGamma (\frac{\delta +nk}{k}) \varGamma (1+b-\frac{\rho - \lambda n}{k})\varGamma (1+c-\frac{\rho -\lambda n}{k})}{\varGamma (1- \frac{\rho -\lambda n}{k})\varGamma (1+a+b+c-\frac{ \rho -\lambda n}{k})\varGamma (\frac{\nu n+\rho }{k})n!} \bigl(kwu^{\frac{- \lambda }{k}} \bigr)^{n}. \end{aligned}$$
(48)
Using (16) in (48), we get
$$\begin{aligned}& \bigl(I_{u,\infty }^{a,b,c}t^{\rho -1}E_{\nu ,\rho }^{\delta } \bigl(wt^{\frac{- \lambda }{k}} \bigr)\bigr) (u) \\& \quad =\frac{k^{1-\frac{\rho }{k}}u^{\frac{\rho -a-b}{k}+a-1}}{ \varGamma (\frac{\delta }{k})} {}_{3}\varPsi _{3} \left [ \textstyle\begin{array}{c} (1+b-\frac{\rho }{k}, \frac{\lambda }{k}),(1+c-\frac{\rho }{k},\frac{ \lambda }{k}),(\frac{\delta }{k},1) \\ (1-\frac{\rho }{k},\frac{\lambda }{k}),(1+a+b+c-\frac{\rho }{k},\frac{ \lambda }{k}),(\frac{\rho }{k},\frac{\nu }{k}) \end{array}\displaystyle \middle | k^{1-\frac{\nu }{k}}wu^{\frac{-\lambda }{k}} \right ]. \end{aligned}$$
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Remark 2
If we replace k by one, then we get the result of [4].