1 Introduction
Nowadays, fractional calculus is used as a mathematical tool for modeling the processes in engineering, mathematical economics, physics, chemistry, biology and other branches of science (see [1‐12]). Motivated by the above mentioned works, here we aim to establish two new fractional integral formulas (of Marichev-Saigo-Maeda type) involving the product of multivariable H-functions and the first class of multivariable polynomials \(S^{m_{1},\ldots,m_{s}}_{n} (x_{1},\ldots,x_{s})\). The so-called Marichev-Saigo-Maeda generalized fractional operators were introduced nearly 40 years ago by Marichev [13] and studied in some recent papers, including the papers by Saigo and Maeda [14] and by Saxena and Saigo [15]. Due to the usefulness and the importance of the Marichev-Saigo-Maeda fractional integral operators, many authors have presented a number of interesting integral formulas involving special functions by using the Marichev-Saigo-Maeda fractional integral operator (see [13, 16‐20]). By virtue of the unified nature of Marichev-Saigo-Maeda fractional integral operators, a large number of new and known results involving Saigo, Riemann-Liouville and Erdélyi-Kober fractional integral operators follow as special cases of our main formulas.
Our main results consist in application of the Marichev-Saigo-Maeda generalized fractional operators: and to the multivariable H-functions (see, e.g., [21, 22]) and the first class of multivariable polynomials \(S^{m_{1},\ldots,m_{s}}_{n} (x_{1},\ldots,x_{s})\) given by (1.15).
$$\begin{aligned}& \bigl(\mathrm{I}_{0,x}^{\alpha,\alpha',\beta,\beta',\eta }f \bigr) (x) \\& \quad =\frac{x^{-\alpha}}{\Gamma(\eta)}\int_{0}^{x}(x-t)^{\eta-1}t^{-\alpha'}F_{3} \biggl(\alpha,\alpha',\beta,\beta';\eta;1- \frac{t}{x},1-\frac{x}{t} \biggr) f(t)\, dt\quad (\operatorname{Re} \eta>0 ) \end{aligned}$$
(1.1)
$$\begin{aligned}& \bigl(\mathrm{I}_{x,\infty}^{\alpha,\alpha',\beta,\beta',\eta }f \bigr) (x) \\& \quad =\frac{x^{-\alpha'}}{\Gamma(\eta)}\int_{x}^{\infty}(t-x)^{\eta -1}t^{-\alpha}F_{3} \biggl(\alpha,\alpha',\beta,\beta';\eta;1- \frac{x}{t},1-\frac{t}{x} \biggr) f(t)\, dt \quad (\operatorname{Re} \eta>0 ), \end{aligned}$$
(1.2)
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A variant of such operators (integral transforms) was introduced by Marichev [13] as Mellin type convolution operators with a special function \(F_{3}(\cdot)\) in the kernel. These operators were rediscovered and studied by Saigo in [23] as a generalization of the so-called Saigo fractional integral operators, see [24]. The properties of these operators were studied by Saigo and Maeda [14]. In particular, relations of the operators with the Mellin transforms, hypergeometric operators (or Saigo fractional integral operators), their decompositions and acting properties in the McBride spaces \(F_{p;\mu}\) (see [25]) were found.
In (1.1), (1.2) the symbol \(F_{3}(\cdot)\) denotes the so-called 3rd Appell function (known also as Horn function) (see [26], p.413): The properties of this function are discussed in [26], pp.412-415. In particular, its relation to the Gauss hypergeometric function is presented: It is known that the 3rd Appell function cannot be expressed as a product of two \({}_{2} F_{1}\) functions and satisfy pairs of linear partial differential equations of the second order.
$$ F_{3}\bigl(\alpha,\alpha^{\prime}; \beta, \beta^{\prime}; \eta; x; y\bigr) = \sum_{m,n=0}^{\infty} \frac{(\alpha)_{m} (\alpha^{\prime})_{n} (\beta)_{m} (\beta^{\prime})_{n}}{(\eta )_{m+n} m! n!} x^{m} y^{n} \quad \bigl(\max \bigl\{ \vert x\vert , |y|\bigr\} < 1\bigr). $$
(1.3)
$$F_{3}(\alpha,\eta- \alpha; \beta, \eta- \beta; \eta; x; y) = {}_{2} F_{1}(\alpha, \beta; \eta; x + y - xy). $$
Operators (1.1), (1.2) can be reduced to the fractional integral operators introduced by Saigo [27] due to the following relations: and The multivariable H-function, introduced by Srivastava and Panda in a series of research papers (see [28], p.271, Eq. (4.1), [22], p.130, Eq. (1.3) and [29, 30]), can be defined and represented in the following manner (see, e.g., [31], p.251, Eq. (C.1)):1
where
In these formulas, for any \(i=1,\ldots,r\), \(L_{i}\) represents the contours which start at the point \(\tau_{i}-\omega\infty\) and go \(\tau_{i}+\omega\infty\) with \(\tau_{i}\in\mathbb{R}=(-\infty,\infty)\) such that all the poles of \(\Gamma (d_{j}^{(i)}-\delta_{j}^{(i)}\xi_{i} )\), \(j=1,\ldots,m_{i}\); \(i=1,\ldots,r\), are separated from those of \(\Gamma (1-c^{(i)}_{j}+\gamma^{(i)}_{j}\xi_{i} )\), \(j=1,\ldots,n_{i}\); \(i=1,\ldots,r\), and \(\Gamma (1-a_{j}+\sum^{r}_{i=1}\alpha^{(i)}_{j}\xi_{i} )\), \(j=1,\ldots,n\). Here, the integers n, p, q, \(m_{i}\), \(n_{i}\), \(p_{i}\), \(q_{i}\) satisfy the inequalities \(0\leq n\leq p\); \(q\geq0\), \(1\leq m_{i} \leq q_{i}\) and \(1\leq n_{i}\leq p_{i}\), \(i=1,\ldots,r\).
$$ \mathrm{I}_{0,x}^{\alpha,0,\beta,\beta',\gamma} f (x )= \mathrm{I}_{0,x}^{\gamma,\alpha-\gamma,-\beta} f (x )\quad (\gamma\in\mathbb{C} ) $$
(1.4)
$$ \mathrm{I}_{x,\infty}^{\alpha, 0,\beta, \beta',\gamma} f (x )= \mathrm{I}_{x,\infty}^{\gamma,\alpha-\gamma,-\beta} f (x )\quad (\gamma\in\mathbb{C} ). $$
(1.5)
$$\begin{aligned} H[x_{1},\ldots,x_{r}] =&H^{0,n:\{m_{r},n_{r}\}}_{p,q:\{ p_{r},q_{r}\}} \left [ \textstyle\begin{array}{@{}c@{}} x_{1} \\ \vdots \\ x_{r} \end{array}\displaystyle \Biggm| \textstyle\begin{array}{@{}c@{}} (a_{j};\alpha^{(1)}_{j},\ldots,\alpha^{(r)}_{j} )_{1,p}: \{ (c^{(r)}_{j},\gamma^{(r)}_{j} )_{1,p_{r}} \} \\ (b_{j};\beta^{(1)}_{j},\ldots,\beta^{(r)}_{j} )_{1,q}: \{ (d^{(r)}_{j},\delta^{(r)}_{j} )_{1,q_{r}} \} \end{array}\displaystyle \right ] \\ =&\frac{1}{(2\pi\omega)^{r}}\int_{L_{1}}\cdots\int _{L_{r}}\phi (\xi_{1},\ldots,\xi_{r}) \prod^{r}_{i=1} \bigl(\theta_{i}( \xi_{i})x^{\xi_{i}}_{i}\, d\xi_{i} \bigr), \end{aligned}$$
(1.6)
$$\begin{aligned}& \phi(\xi_{1},\ldots,\xi_{r})=\frac{ \prod^{n}_{j=1} \Gamma (1-a_{j}+\sum^{r}_{i=1}\alpha^{(i)}_{j}\xi_{i} )}{ \prod^{p}_{j=n+1} \Gamma (a_{j}-\sum^{r}_{i=1}\alpha^{(i)}_{j}\xi_{i} )\prod^{q}_{j=1}\Gamma (1-b_{j}+ \sum^{r}_{i=1}\beta^{(i)}_{j}\xi_{i} )}, \end{aligned}$$
(1.7)
$$\begin{aligned}& \theta_{i}(\xi_{i})=\frac{ \prod^{n_{i}}_{j=1} \Gamma (1-c^{(i)}_{j}+\gamma^{(i)}_{j}\xi_{i} )\prod^{m_{i}}_{j=1} \Gamma (d^{(i)}_{j}-\delta^{(i)}_{j}\xi_{i} )}{ \prod^{p_{i}}_{j=n_{i}+1} \Gamma (c^{(i)}_{j}-\gamma^{(i)}_{j}\xi_{i} )\prod^{q_{i}}_{j=m_{i}+1} \Gamma (1-d^{(i)}_{j}+\delta^{(i)}_{j}\xi_{i} )}, \quad \forall i\in \{1,2,\ldots,r\}. \end{aligned}$$
(1.8)
Further, suppose that the parameters \(a_{j}\), \(j=1,\ldots,p\); \(c_{j}^{(i)}\), \(j=1,\ldots,p_{i}\); \(i=1,\ldots,r\), \(b_{j}\), \(j=1,\ldots,q\); \(d_{j}^{(i)}\), \(j=1,\ldots,q_{i}\); \(i=1,\ldots,r\), are complex numbers, while the associated coefficients \(\alpha_{j}^{(i)}\), \(j=1,\ldots,p\); \(i=1,\ldots,r\); \(\gamma_{j}^{(i)}\), \(j=1,\ldots,p_{i}\); \(i=1,\ldots,r\), \(\beta_{j}^{(i)}\), \(j=1,\ldots,q\); \(i=1,\ldots,r\); \(\delta_{j}^{(i)}\), \(j=1,\ldots,q_{i}\); \(i=1,\ldots,r\), are positive real numbers such that
$$\begin{aligned}& \Lambda_{i}=\sum_{j=1}^{p} \alpha _{j}^{(i)}+\sum_{j=1}^{p_{i}} \gamma_{j}^{(i)}-\sum_{j=1}^{q} \beta_{j}^{(i)}-\sum_{j=1}^{q_{i}} \delta_{j}^{(i)}\leq0, \end{aligned}$$
(1.9)
$$\begin{aligned}& \Omega_{i}=\sum_{j=1}^{n} \alpha_{j}^{(i)}-\sum_{j=n+1}^{p} \alpha_{j}^{(i)}+ \sum_{j=1}^{q} \beta_{j}^{(i)}+\sum_{j=1}^{n_{i}} \gamma_{j}^{(i)} -\sum_{j=n_{i}+1}^{p_{i}} \gamma_{j}^{(i)}+\sum_{j=1}^{m_{i}} \delta_{j}^{(i)}-\sum_{j=m_{i}+1}^{q_{i}} \delta_{j}^{(i)}>0. \end{aligned}$$
(1.10)
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It is assumed that the poles of integrand of (1.6) are simple. Under the conditions (1.10) the integrals in (1.6) converge absolutely (see [31], p.251) in the domain and points \(z_{i}=0\), \(i=1,\ldots,r\), and various exceptional parameter values being tacitly excluded.
$$ \bigl\vert \arg(z_{i})\bigr\vert < \frac{\pi}{2} \Omega_{i}, \quad i=1,\ldots,r, $$
(1.11)
From Srivastava and Panda [32], we have where \(e_{i}=\min_{1\leq j\leq m_{i}} [\frac{\Re (d_{j}^{(i)} )}{\delta_{j}^{(i)}} ]\), \(i=1,\ldots,r\).
$$ H[z_{1},\ldots,z_{r}]=o\bigl(|z_{1}|^{e_{1}}, \ldots,|z_{r}|^{e_{r}}\bigr),\qquad \max_{1\leq j\leq r} \bigl[\vert z_{j}\vert \bigr]\rightarrow0, $$
(1.12)
If \(n=0\) in (1.6), then the following asymptotic expansion (Srivastava and Panda [32]) holds: where \(g_{i}=\max_{1\leq j\leq m_{i}} [\frac{\Re (c_{j}^{(i)} )}{\gamma_{j}^{(i)}} ]\), \(i=1,\ldots,r\), provided that (1.9), (1.10) and (1.11) hold true.
$$ H[z_{1},\ldots,z_{r}]=o\bigl(|z_{1}|^{g_{1}}, \ldots,|z_{r}|^{g_{r}}\bigr),\qquad \min_{1\leq j\leq r} \bigl[\vert z_{j}\vert \bigr]\rightarrow\infty, $$
(1.13)
Remark 1
Remark 2
It is interesting to observe that for \(n = p = q = 0\), the multivariable H-function breaks up into the product of r, H-functions; and consequently, there holds the following result (Saxena [21]):
$$ H^{0,0:\{m_{r},n_{r}\}}_{0,0:\{p_{r},q_{r}\}}\left [ \textstyle\begin{array}{@{}c@{}} x_{1} \\ \vdots \\ x_{r} \end{array}\displaystyle \Biggm| \textstyle\begin{array}{@{}c@{}} -: \{ (c^{(r)}_{j},\gamma^{(r)}_{j} )_{1,p_{r}} \} \\ -: \{ (d^{(r)}_{j},\delta^{(r)}_{j} )_{1,q_{r}} \} \end{array}\displaystyle \right ] =\prod_{i=1}^{r}H^{m_{i},n_{i}}_{p_{i},q_{i}} \left [ z_{i}\Bigm| \textstyle\begin{array}{@{}c@{}} \{ (c^{(i)}_{j},\gamma^{(i)}_{j} )_{1,p_{i}} \} \\ \{ (d^{(i)}_{j},\delta^{(i)}_{j} )_{1,q_{i}} \} \end{array}\displaystyle \right ]. $$
(1.14)
Remark 3
The function defined by (1.6) was introduced and studied by Srivastava and Panda [37]. When \(\alpha_{j}^{(1)}=\cdots=\alpha_{j}^{(r)}\), \(j=1,\ldots,p\); \(\beta _{j}^{(1)}=\cdots=\beta_{j}^{(r)}\), \(j=1,\ldots,q\), in (1.6) the multivariable H-function defined and studied by Saxena [21, 38] is obtained. In case all the Greek letters are assumed to be unity, the H-function of several complex variables (1.6) reduces to the G-function of several complex variables studied by Khadia and Goyal [39, 40].
Remark 4
In the sequel, Srivastava and Garg [46], p.686, Eq. (1.4), introduced the multivariable analogue of the polynomials \(S^{m}_{n} (x)\). This first class of multivariable polynomials \(S^{m_{1},\ldots,m_{s}}_{n} (x_{1},\ldots,x_{s})\) is defined by where \(m_{1},\ldots,m_{s}\) are arbitrary positive integers and the coefficients \(\Lambda (n;k_{1},\ldots,k_{s} )\) (\(n,k_{i}\geq 0\), \(i=1,\ldots,s\)) are arbitrary constants, real or complex.
$$ S^{m_{1},\ldots,m_{s}}_{n} (x_{1}, \ldots,x_{s})= \sum^{m_{1}k_{1}+\cdots+m_{s}k_{s}\leq n}_{k_{1},\ldots,k_{s}=0} (-n )_{m_{1}k_{1}+\cdots+m_{s}k_{s}} \Lambda (n;k_{1},\ldots,k_{s} ) \frac{x^{k_{1}}_{1}\cdots x^{k_{s}}_{s}}{k_{1}!\cdots k_{s}!}, $$
(1.15)
2 Main results
In this section, we prove two theorems on composition of the Marichev-Saigo-Maeda operators with the product of a multivariable H-function and the first class of multivariable polynomials. We start with presenting image formulas involving Marichev-Saigo-Maeda fractional integral operators.
Lemma 1
(see [14])
Let
\(\alpha,\alpha',\beta,\beta',\gamma \in\mathbb{C}\).
If
\(\Re (\gamma )>0\), \(\Re (p )> \max[0, \Re (\alpha+\alpha'+\beta-\gamma ),\Re (\alpha'-\beta' )]\), then
$$ \mathrm{I}_{0,x}^{\alpha,\alpha',\beta ,\beta',\gamma} x^{\rho-1} =x^{\rho-\alpha-\alpha'+\gamma-1}\frac {\Gamma (\rho )\Gamma (\rho+\gamma-\alpha-\alpha '-\beta )\Gamma (\rho+\beta'-\alpha' )}{\Gamma (\rho+\gamma-\alpha-\alpha' )\Gamma (\rho+\gamma-\alpha'-\beta )\Gamma (\rho+\beta' )}. $$
(2.1)
If
\(\Re (\gamma )>0\), \(\Re (p )<1+\min[\Re (-\beta ),\Re (\alpha+\alpha'-\gamma ),\Re (\alpha+\beta '-\gamma )]\), then
$$\begin{aligned} \begin{aligned}[b] \mathrm{I}_{x,\infty}^{\alpha,\alpha',\beta,\beta',\gamma} x^{\rho -1} ={}&x^{\rho-\alpha-\alpha' +\gamma-1} \\ &{}\times\frac {\Gamma (1+\alpha+\alpha'-\gamma-\rho )\Gamma (1+\alpha+\beta'-\gamma-\rho )\Gamma (1-\beta-\rho )}{\Gamma (1-\rho )\Gamma (1+\alpha+\alpha'+\beta '-\gamma-\rho )\Gamma (1+\alpha-\beta-\rho )}. \end{aligned} \end{aligned}$$
(2.2)
Theorem 1
Let
\(\Re (\gamma )>0\)
and
\(\alpha,\alpha ',\beta,\beta',\gamma,\mu,\eta,\delta_{j},v_{i},z_{i}, a,b,c_{j}\in\mathbb{C}\), \(\lambda_{j} ,\sigma_{i} >0\) (\(i\in\{ 1,\ldots,r\}\); \(j\in\{1,\ldots,s\}\)). Let the following conditions be satisfied, too:
(i)
\(|\arg z_{i}|<\frac{\pi}{2} \Omega_{i} \), \(\Omega_{i} >0\) (\(i = 1,\ldots,r\)), where
$$\begin{aligned} \Omega_{i} =&\sum_{j=1}^{n} \alpha_{j}^{ (i )}-\sum_{j=n+1}^{p} \alpha_{j}^{ (i )}-\sum_{j=1}^{q} \beta_{j}^{ (i )} +\sum_{j=1}^{n_{i} } \gamma_{j}^{ (i )} \\ &{} -\sum_{j=n_{i} +1}^{p_{i} } \gamma_{j}^{ (i )} +\sum_{j=1}^{m_{i} } \delta_{j}^{ (i )}-\sum_{j=m_{i}+1}^{q_{i} } \delta_{j}^{ (i )} >0 . \end{aligned}$$
(2.3)
(ii)
\(\min_{1\le j\le m_{i} ,1\le i\le r} [-\frac{\sigma _{i} \Re (d_{j}^{ (i )} )}{\delta_{j}^{ (i )} } ]<\Re (\mu )+\min[\Re (\alpha' ),\Re (\beta' ),\Re (\gamma-\alpha-\beta )]\).
(iii)
\(|\frac{a}{b}x|<1\), also we have
$$ \begin{aligned} &\Re (\mu )+\sum_{i=1}^{r} \sigma_{i} \min_{1\le j\le m_{i} } \biggl[\frac{\Re (d_{j}^{ (i )} )}{\delta _{j}^{ (i )} } \biggr] \\ &\quad >\max \bigl[0,\Re \bigl(\alpha+\alpha'+\beta -\gamma \bigr), \Re \bigl(\alpha'-\beta' \bigr)\bigr]\quad (i=1, \ldots,r ), \\ &\Re (\eta )+\sum_{i=1}^{r}v_{i} \min_{1\le j\le m_{i} } \biggl[\frac{\Re (d_{j}^{ (i )} )}{\delta _{j}^{ (i )} } \biggr] \\ &\quad > \max\bigl[0,\Re \bigl(\alpha+\alpha'+\beta -\gamma \bigr), \Re \bigl( \alpha'-\beta' \bigr)\bigr]\quad (i=1,\ldots,r ). \end{aligned} $$
(2.4)
Then the following result holds:
where
and
$$\begin{aligned}& \bigl\{ \mathrm{I}_{0+}^{\alpha,\alpha',\beta ,\beta',\gamma} \bigl(t^{\mu-1} (b-at )^{-\eta} S^{m_{1},\ldots,m_{s}}_{n} \bigl[c_{1} t^{\lambda_{1}} (b-at )^{-\delta _{1} }, \ldots,c_{s} t^{\lambda_{s}} (b-at )^{-\delta_{s} } \bigr] \\& \qquad {}\times H\bigl[z_{1} t^{\sigma_{1} } (b-at )^{-v_{1} } ,\ldots,z_{r} t^{\sigma_{r} } (b-at )^{-v_{r}} \bigr]\bigr) \bigr\} (x ) \\& \quad =b^{-\eta} x^{\mu-\alpha-\alpha'+\gamma-1} \sum^{m_{1}k_{1}+\cdots+m_{s}k_{s}\leq n}_{k_{1},\ldots,k_{s}=0} \frac{ (-n )_{m_{1}k_{1}+\cdots+m_{s}k_{s}}}{k_{1}!\cdots k_{s}!}\Lambda (n;k_{1},\ldots,k_{s} ) c_{1}^{k_{1} } \cdots c_{s}^{k_{s} } \\& \qquad {}\times b^{- \sum_{j=1}^{s}\delta_{j} k_{j}} x^{\sum_{j=1}^{s}\lambda _{j} k_{j}}H^{0,n+4:\{m_{r},n_{r}\};1,0}_{p+4,q+4:\{p_{r},q_{r}\};0,1} \\& \qquad \left [ \textstyle\begin{array}{@{}c@{}} z_{1}\frac{x^{\sigma_{1}}}{b^{v_{1}}} \\ \vdots \\ z_{r}\frac {x^{\sigma_{r}}}{b^{v_{r}}} \\ -\frac{a}{b}x \end{array}\displaystyle \left. \,\right|\, \textstyle\begin{array}{@{}c@{}} (a_{j};\alpha^{(1)}_{j},\ldots,\alpha^{(r)}_{j} )_{1,p},A: \{ (c^{(r)}_{j},\gamma^{(r)}_{j} )_{1,p_{r}} \} ;- \\ (b_{j};\beta^{(1)}_{j},\ldots,\beta^{(r)}_{j} )_{1,q},B: \{ (d^{(r)}_{j},\delta^{(r)}_{j} )_{1,q_{r}} \};(0,1) \end{array}\displaystyle \right ], \end{aligned}$$
(2.5)
$$\begin{aligned} A =& \Biggl(1-\eta-\sum_{j=1}^{s} \delta_{j} k_{j}; v_{1} ,\ldots,v_{r}, 1 \Biggr), \Biggl(1-\mu-\sum_{j=1}^{s} \lambda_{j} k_{j} ;\sigma _{1},\ldots, \sigma_{r},1 \Biggr) , \\ & \Biggl(1-\mu-\gamma+\alpha+\alpha'+\beta-\sum _{j=1}^{s}\lambda_{j} k_{j} ;\sigma_{1} ,\ldots,\sigma_{r}, 1 \Biggr) , \\ & \Biggl(1-\mu+\alpha'-\beta'-\sum _{j=1}^{s}\lambda_{j} k_{j} ;\sigma_{1} ,\ldots,\sigma_{r}, 1 \Biggr) \end{aligned}$$
$$\begin{aligned} B =& \Biggl(1-\eta-\sum_{j=1}^{s} \delta_{j} k_{j} ;v_{1},\ldots,v_{r} ,0 \Biggr), \Biggl(1-\mu-\gamma+\alpha+\alpha'-\sum _{j=1}^{s}\lambda_{j} k_{j} ;\sigma_{1} ,\ldots,\sigma_{r}, 1 \Biggr) \\ & \Biggl(1-\mu-\gamma+\alpha'+\beta-\sum _{j=1}^{s}\lambda_{j} k_{j}; \sigma_{1},\ldots,\sigma_{r}, 1 \Biggr), \\ &\Biggl(1-\mu- \beta'-\sum_{j=1}^{s} \lambda_{j} k_{j} ;\sigma_{1},\ldots, \sigma_{r},1 \Biggr) . \end{aligned}$$
Theorem 2
Let
\(\alpha,\alpha',\beta,\beta',\gamma,\mu,\eta,\delta _{j},v_{i},z_{i}, a,b,c_{j}\in\mathbb{C}\), \(\lambda_{j} ,\sigma_{i} >0\) (\(i\in\{ 1,\ldots,r\}\); \(j\in\{1,\ldots,s\}\)) and
\(\Re (\gamma )>0\). Let the following conditions be satisfied, too:
and the conditions (i)-(iii) in Theorem
1
are also satisfied.
$$\begin{aligned}& \Re (\mu )-\sum_{i=1}^{r} \sigma_{i} \min_{1\le j\le m_{i} } \biggl[\frac{\Re (d_{j}^{ (i )} )}{\delta _{j}^{ (i )} } \biggr] < 1+\max\bigl[\Re(-\beta),\Re \bigl(\alpha +\alpha'-\gamma \bigr), \Re \bigl(\alpha+\beta'-\gamma \bigr)\bigr] , \\& \Re (\eta )-\sum_{i=1}^{r}v_{i} \min_{1\le j\le m_{i} } \biggl[\frac{\Re (d_{j}^{ (i )} )}{\delta _{j}^{ (i )} } \biggr] < 1+\max\bigl[ \Re(-\beta),\Re \bigl(\alpha +\alpha'-\gamma \bigr), \Re \bigl( \alpha+\beta'-\gamma \bigr)\bigr] , \end{aligned}$$
Then the following result holds:
where
and
$$\begin{aligned}& \bigl\{ \mathrm{I}_{0-}^{\alpha,\alpha',\beta ,\beta',\gamma} \bigl(t^{\mu-1} (b-at )^{-\eta} S^{m_{1},\ldots,m_{s}}_{n} \bigl[c_{1} t^{\lambda_{1}} (b-at )^{-\delta _{1} }, \ldots,c_{s} t^{\lambda_{s}} (b-at )^{-\delta_{s} } \bigr] \\& \qquad {}\times H\bigl[z_{1} t^{\sigma_{1} } (b-at )^{-v_{1} } ,\ldots,z_{r} t^{\sigma_{r} } (b-at )^{-vr} \bigr]\bigr) \bigr\} (x ) \\& \quad =b^{-\eta} x^{\mu-\alpha-\alpha'+\gamma-1} \sum^{m_{1}k_{1}+\cdots+m_{s}k_{s}\leq n}_{k_{1},\ldots,k_{s}=0} \frac{ (-n )_{m_{1}k_{1}+\cdots+m_{s}k_{s}}}{k_{1}!\cdots k_{s}!}\Lambda (n;k_{1},\ldots,k_{s} ) c_{1}^{k_{1} } \cdots c_{s}^{k_{s} } \\& \qquad {}\times b^{- \sum_{j=1}^{s}\delta_{j} k_{j}} x^{\sum_{j=1}^{s}\lambda _{j} k_{j}}H^{0,n+4:\{m_{r},n_{r}\};1,0}_{p+4,q+4:\{p_{r},q_{r}\};0,1} \\& \qquad \left [ \textstyle\begin{array}{@{}c@{}} z_{1}\frac{x^{\sigma_{1}}}{b^{v_{1}}} \\ \vdots \\ z_{r}\frac {x^{\sigma_{r}}}{b^{v_{r}}} \\ -\frac{a}{b}x \end{array}\displaystyle \left. \,\right|\, \textstyle\begin{array}{@{}c@{}} (a_{j};\alpha^{(1)}_{j},\ldots,\alpha^{(r)}_{j} )_{1,p},C: \{ (c^{(r)}_{j},\gamma^{(r)}_{j} )_{1,p_{r}} \} ;- \\ (b_{j};\beta^{(1)}_{j},\ldots,\beta^{(r)}_{j} )_{1,q},D: \{ (d^{(r)}_{j},\delta^{(r)}_{j} )_{1,q_{r}} \};(0,1) \end{array}\displaystyle \right ], \end{aligned}$$
(2.6)
$$\begin{aligned} C =& \Biggl(1-\eta-\sum_{j=1}^{s} \delta_{j} k_{j}; v_{1} ,\ldots,v_{r}, 1 \Biggr), \Biggl(1-\mu-\sum_{j=1}^{s} \lambda_{j} k_{j} ;\sigma _{1},\ldots, \sigma_{r},1 \Biggr) , \\ & \Biggl(1+\alpha+\alpha'+\beta'-\mu-\gamma -\sum _{j=1}^{s}\lambda_{j} k_{j} ;\sigma_{1} ,\ldots,\sigma_{r}, 1 \Biggr) , \\ & \Biggl(1 +\alpha-\beta-\mu-\sum_{j=1}^{s} \lambda_{j} k_{j} ;\sigma _{1} ,\ldots, \sigma_{r}, 1 \Biggr) \end{aligned}$$
$$\begin{aligned} \begin{aligned} D ={}& \Biggl(1-\eta-\sum_{j=1}^{s} \delta_{j} k_{j} ;v_{1},\ldots,v_{r} ,0 \Biggr), \Biggl(1-\mu-\gamma+\alpha+\alpha'-\sum _{j=1}^{s}\lambda_{j} k_{j} ;\sigma_{1} ,\ldots,\sigma_{r}, 1 \Biggr) \\ & \Biggl(1-\mu-\gamma+\alpha+\beta' -\sum _{j=1}^{s}\lambda_{j} k_{j}; \sigma_{1},\ldots,\sigma_{r}, 1 \Biggr), \\ & \Biggl(1-\mu- \beta-\sum_{j=1}^{s}\lambda_{j} k_{j} ;\sigma_{1},\ldots,\sigma_{r},1 \Biggr) . \end{aligned} \end{aligned}$$
Proof
For convenience, let the left-hand side of (2.5) be denoted by \(\mathcal{I}\). Applying the Marichev-Saigo-Maeda generalized fractional operators (1.1) to the LHS of (2.5) and making use of (1.15) and (1.6), we obtain
$$\begin{aligned} \mathcal{I} =& \Biggl\{ \mathrm{I}_{0+}^{\alpha,\alpha',\beta,\beta ',\gamma} \Biggl(t^{\mu-1} (b-at )^{-\gamma} \sum^{m_{1}k_{1}+\cdots+m_{s}k_{s}\leq n}_{k_{1},\ldots,k_{s}=0} \frac{ (-n )_{m_{1}k_{1}+\cdots+m_{s}k_{s}}}{k_{1}!\cdots k_{s}!}\Lambda (n;k_{1},\ldots,k_{s} ) \\ & {}\times \bigl(c_{1}t^{\lambda _{1}}(b-at)^{\delta_{1}} \bigr)^{k_{1} } \cdots \bigl(c_{s}t^{\lambda _{s}}(b-at)^{\delta_{s}} \bigr)^{k_{s} } \times\frac{1}{ (2\pi i )^{r}} \int_{L_{1} } \cdots \int_{L_{r} } \phi (\xi_{1} ,\ldots, \xi_{r} ) \\ & {}\times \prod_{i=1}^{r} \theta_{i} (\xi_{i} ) \bigl(z_{i}t^{\sigma_{i}}(b-at)^{\nu_{i}} \bigr)^{\xi_{i}} \, d\xi_{1} \cdots \, d\xi _{r} \Biggr)\Biggr\} (x ). \end{aligned}$$
(2.7)
By changing the order of summation and integration, we have By using the binomial expansion for \((b-at)^{-\eta-\sum _{j=1}^{s}\lambda_{j} k_{j} -\sum_{i=1}^{r}\sigma_{i} \xi_{i}}\) and applying the Mellin-Barnes counter integral, we get
$$\begin{aligned} \mathcal{I} =&\sum^{m_{1}k_{1}+\cdots+m_{s}k_{s}\leq n}_{k_{1},\ldots,k_{s}=0} \frac{ (-n )_{m_{1}k_{1}+\cdots+m_{s}k_{s}}}{k_{1}!\cdots k_{s}!}\Lambda (n;k_{1},\ldots,k_{s} ) c_{1}^{k_{1} } \cdots c_{s}^{k_{s} } \\ &{}\times \Biggl[\frac{1}{ (2\pi i )^{r} } \int_{L_{1} } \cdots \int _{L_{r} } \phi (\xi_{1} ,\ldots,\xi_{r} ) \prod_{i=1}^{r} \bigl\{ \theta _{i} (\xi_{i} ) z_{i} \bigr\} ^{\xi_{i} } \bigl\{ \mathrm{I}_{0+}^{\alpha,\alpha',\beta,\beta' ,\gamma} \bigl(t^{\mu+\sum_{j=1}^{s}\lambda_{j} k_{j} +\sum _{i=1}^{r}\sigma_{i} \xi_{i} -1 } \\ &{}\times(b-at)^{-\eta-\sum_{j=1}^{s}\lambda _{j} k_{j} -\sum_{i=1}^{r}\sigma_{i} \xi_{i}} \bigr) \bigr\} (x ) \, d \xi_{1} \cdots\, d\xi_{r}\Biggr]. \end{aligned}$$
(2.8)
$$\begin{aligned} \mathcal{I} =&\sum^{m_{1}k_{1}+\cdots+m_{s}k_{s}\leq n}_{k_{1},\ldots,k_{s}=0} \frac{ (-n )_{m_{1}k_{1}+\cdots+m_{s}k_{s}}}{k_{1}!\cdots k_{s}!}\Lambda (n;k_{1},\ldots,k_{s} ) c_{1}^{k_{1} } \cdots c_{s}^{k_{s} }b^{-\sum _{j=1}^{s}\lambda_{j} k_{j}} \\ &{}\times\Biggl[\frac{1}{ (2\pi i )^{r} } \int_{L_{1} } \cdots \int _{L_{r} } \phi (\xi_{1} ,\ldots,\xi_{r} ) \times\prod_{i=1}^{r} \bigl\{ \theta_{i} (\xi_{i} ) z_{i}b^{-\sum_{i=1}^{r}\sigma_{i} \xi_{i}} \bigr\} ^{\xi_{i} } \\ &{} \times \int_{L_{r+1} }\frac{\Gamma(\eta+\sum_{j=1}^{s}\lambda_{j} k_{j}+\sum_{i=1}^{r}\sigma_{i} \xi_{i}+\xi_{r+1})}{\Gamma(\eta+\sum_{j=1}^{s}\lambda_{j} k_{j}+\sum_{i=1}^{r}\sigma_{i} \xi_{i})\Gamma (\xi_{r+1}+1)} \\ &{}\times \bigl\{ \mathrm{I}_{0+}^{\alpha,\alpha',\beta,\beta' ,\gamma} \bigl(t^{\mu+\sum_{j=1}^{s}\lambda_{j} k_{j} +\sum _{i=1}^{r}\sigma_{i} \xi_{i}+\xi_{r+1} -1 } \bigr) \bigr\} (x )\, d\xi_{1} \cdots\, d \xi_{r}\Biggr]. \end{aligned}$$
(2.9)
Here, we derive certain, presumably, new formulas involving Marichev-Saigo-Maeda type fractional integral operators. By setting \(n = p= q = 0\) in (2.5) and (2.6), respectively, we obtain two fractional integral formulas involving product of the r, H-functions stated in Corollaries 1 and 2 below.
Corollary 1
Let
\(\alpha,\alpha',\beta,\beta',\gamma,\mu,\eta ,\delta_{j}, v_{i}, z_{i}, a, b, c_{j} \in\mathbb{C}\), \(\lambda_{j},\sigma _{i} >0\) (\(i\in\{ 1,\ldots r\}\); \(j\in\{ 1,\ldots,s\}\)) and
\(\Re (\gamma )>0\). Then the following relation holds true:
where
A
and
B
are given in Theorem
1
and
\(H_{p,q}^{m,n} (\cdot )\)
is the familiar Fox
H-function.
$$\begin{aligned}& \Biggl\{ \mathrm{I}_{0+}^{\alpha,\alpha',\beta ,\beta',\gamma} \Biggl(t^{\mu-1} (b-at )^{-\eta} S^{m_{1},\ldots,m_{s}}_{n} \bigl[c_{1} t^{\lambda_{1}} (b-at )^{-\delta _{1} }, \ldots,c_{s} t^{\lambda_{s}} (b-at )^{-\delta_{s} } \bigr] \\& \qquad {}\times\prod^{r}_{i=1}H^{m_{i},n_{i}}_{p_{i},q_{i}} \left [z_{i} t^{\sigma_{i}} (b-at )^{-v_{i}}\Bigm| \textstyle\begin{array}{@{}c@{}} (c^{i}_{j},C^{i}_{j} )_{1,p_{i}} \\ (d^{i}_{j},D^{i}_{j} )_{1,q_{i}} \end{array}\displaystyle \right ]\Biggr)\Biggr\} (x ) \\& \quad =b^{-\eta} x^{\mu-\alpha-\alpha'+\gamma-1} \sum^{m_{1}k_{1}+\cdots+m_{s}k_{s}\leq n}_{k_{1},\ldots,k_{s}=0} \frac{ (-n )_{m_{1}k_{1}+\cdots+m_{s}k_{s}}}{k_{1}!\cdots k_{s}!}\Lambda (n;k_{1},\ldots,k_{s} ) c_{1}^{k_{1} } \cdots c_{s}^{k_{s} } \\& \qquad {}\times b^{- \sum_{j=1}^{s}\delta_{j} k_{j}} x^{\sum_{j=1}^{s}\lambda _{j} k_{j}}H^{0,4:m_{i},n_{i};1,0}_{4,4:p_{i},q_{i};0,1} \left [ \textstyle\begin{array}{@{}c@{}} z_{i}\frac{x^{\sigma_{i}}}{b^{v_{i}}} \\ \frac{-a}{b}x \end{array}\displaystyle \biggm| \textstyle\begin{array}{@{}c@{}} A: (c^{(r)}_{j},\gamma^{(r)}_{j} )_{1,p_{i}};- \\ B: (d^{(r)}_{j},\delta^{(r)}_{j} )_{1,q_{i}};(0,1) \end{array}\displaystyle \right ], \end{aligned}$$
(2.10)
Corollary 2
Let
\(\alpha,\alpha',\beta,\beta',\gamma,\mu,\eta ,\delta_{j}, v_{i}, z_{i}, a, b, c_{j} \in\mathbb{C}\), \(\lambda_{j},\sigma _{i} >0\) (\(i\in\{ 1,\ldots,r\}\); \(j\in\{ 1,\ldots,s\}\)) and
\(\Re (\gamma )>0\). Then the following result holds true:
where
and
where
\(H_{p,q}^{m,n} (\cdot )\)
is the familiar Fox
H-function.
$$\begin{aligned}& \Biggl\{ \mathrm{I}_{0-}^{\alpha,\alpha',\beta ,\beta',\gamma} \Biggl(t^{\mu-1} (b-at )^{-\eta} S^{m_{1},\ldots,m_{s}}_{n} \bigl[c_{1} t^{\lambda_{1}} (b-at )^{-\delta _{1} }, \ldots,c_{s} t^{\lambda_{s}} (b-at )^{-\delta_{s} } \bigr] \\& \qquad {}\times\prod^{r}_{i=1}H^{m_{i},n_{i}}_{p_{i},q_{i}} \left [z_{i} t^{\sigma_{i}} (b-at )^{-v_{i}}\Bigm| \textstyle\begin{array}{@{}c@{}} (c^{i}_{j},C^{i}_{j} )_{1,p_{i}} \\ (d^{i}_{j},D^{i}_{j} )_{1,q_{i}} \end{array}\displaystyle \right ]\Biggr)\Biggr\} (x ) \\& \quad =b^{-\eta} x^{\mu-\alpha-\alpha'+\gamma-1} \sum^{m_{1}k_{1}+\cdots+m_{s}k_{s}\leq n}_{k_{1},\ldots,k_{s}=0} \frac{ (-n )_{m_{1}k_{1}+\cdots+m_{s}k_{s}}}{k_{1}!\cdots k_{s}!}\Lambda (n;k_{1},\ldots,k_{s} ) c_{1}^{k_{1} } \cdots c_{s}^{k_{s}} \\& \qquad {}\times b^{- \sum_{j=1}^{s}\delta_{j} k_{j}} x^{\sum_{j=1}^{s}\lambda _{j} k_{j}}H^{0,4:m_{i},n_{i};1,0}_{4,4:p_{i},q_{i};0,1} \left [ \textstyle\begin{array}{@{}c@{}} z_{i}\frac{x^{\sigma_{i}}}{b^{v_{i}}} \\ \frac{-a}{b}x \end{array}\displaystyle \biggm| \textstyle\begin{array}{@{}c@{}} C': (c^{(r)}_{j},\gamma^{(r)}_{j} )_{1,p_{i}};- \\ D': (d^{(r)}_{j},\delta^{(r)}_{j} )_{1,q_{i}};(0,1) \end{array}\displaystyle \right ], \end{aligned}$$
(2.11)
$$\begin{aligned} C' ={}& \Biggl(1-\eta-\sum_{j=1}^{s} \delta_{j} k_{j}; v_{1} ,\ldots,v_{r}, 1 \Biggr), \\ &\Biggl(1-\mu-\sum_{j=1}^{s} \lambda_{j} k_{j} ;\sigma _{1},\ldots, \sigma_{r},1 \Biggr) , \\ & \Biggl(1-\mu-\gamma+\alpha+\alpha'+\beta'-\sum _{j=1}^{s}\lambda_{j} k_{j} ;\sigma_{1} ,\ldots,\sigma_{r}, 1 \Biggr) , \\ & \Biggl(1-\mu+\alpha-\beta-\sum_{j=1}^{s} \lambda_{j} k_{j} ;\sigma _{1} ,\ldots, \sigma_{r}, 1 \Biggr) \end{aligned}$$
$$\begin{aligned} D' =& \Biggl(1-\eta-\sum_{j=1}^{s} \delta_{j} k_{j} ;v_{1} ,\ldots,v_{r} ,0 \Biggr), \Biggl(1-\mu-\gamma+\alpha+\alpha'-\sum _{j=1}^{s}\lambda_{j} k_{j} ;\sigma_{1} ,\ldots,\sigma_{r}, 1 \Biggr) , \\ & \Biggl(1-\mu-\gamma+\alpha+\beta' -\sum _{j=1}^{s}\lambda_{j} k_{j}; \sigma_{1},\ldots,\sigma_{r}, 1 \Biggr), \\ &\Biggl(1-\mu- \beta-\sum_{j=1}^{s}\lambda_{j} k_{j} ;\sigma_{1},\ldots,\sigma_{r},1 \Biggr) , \end{aligned}$$
Interestingly, in view of relation (1.4), we arrive at the following corollaries concerning Saigo fractional integral operator. It is similar to the results due to Agarwal [47], p.587 and p.590, Eqs. (18) and (25).
Corollary 3
Let
\(\alpha,\beta,\gamma,\mu,\eta,\delta _{j},v_{i},z_{i}, a,b,c_{j}\in\mathbb{C}\), \(\lambda_{j} ,\sigma_{i} >0\) (\(i\in\{ 1,\ldots,r\}\); \(j\in\{1,\ldots,s\}\)), and
\(\Re(\alpha)>0\). Let the following conditions be satisfied, too:
(i)
\(|\arg z_{i}|<\frac{\pi}{2} \Omega_{i} \), \(\Omega_{i} >0\) (\(i = 1,\ldots,r\)), where
$$\begin{aligned} \Omega_{i} =&-\sum_{j=n+1}^{p} \alpha_{j}^{ (i )}-\sum_{j=1}^{q} \beta_{j}^{ (i )} +\sum_{j=1}^{n_{i} } \gamma _{j}^{ (i )} -\sum_{j=n_{i} +1}^{p_{i} } \gamma_{j}^{ (i )} \\ &{}+\sum_{j=1}^{m_{i} } \delta_{j}^{ (i )}-\sum_{j=m_{i}+1}^{q_{i} } \delta_{j}^{ (i )} >0 \quad \bigl(i\in\{1,\ldots,r\}\bigr). \end{aligned}$$
(ii)
$$\begin{aligned}& \Re (\mu )+\sum_{i=1}^{r} \sigma_{i} \min_{1\le j\le m_{i} } \biggl[\frac{\Re (d_{j}^{ (i )} )}{\delta _{j}^{ (i )} } \biggr] >\max\bigl[0,\Re (\beta-\gamma )\bigr] , \\& \Re (\eta )+\sum_{i=1}^{r}v_{i} \min_{1\le j\le m_{i} } \biggl[\frac{\Re (d_{j}^{ (i )} )}{\delta _{j}^{ (i )} } \biggr] > \max\bigl[0,\Re (\beta-\gamma )\bigr] \quad \textit{and}\quad \biggl\vert \frac{a}{b}x \biggr\vert < 1 . \end{aligned}$$
Then the following result holds:
where
and
and the conditions of existence of the above corollary follow easily from Theorem
1.
$$\begin{aligned}& \bigl\{ \mathrm{I}_{0+}^{\alpha,\beta,\gamma} \bigl(t^{\mu-1} (b-at )^{-\eta} S^{m_{1},\ldots,m_{s}}_{n} \bigl[c_{1} t^{\lambda_{1}} (b-at )^{-\delta_{1} }, \ldots,c_{s} t^{\lambda_{s}} (b-at )^{-\delta_{s} } \bigr] \\& \qquad{}\times H\bigl[z_{1} t^{\sigma_{1} } (b-at )^{-v_{1} } ,\ldots,z_{r} t^{\sigma_{r} } (b-at )^{-v_{r}} \bigr]\bigr) \bigr\} (x ) \\& \quad =b^{-\eta} x^{\mu-\beta-1} \sum^{m_{1}k_{1}+\cdots+m_{s}k_{s}\leq n}_{k_{1},\ldots,k_{s}=0} \frac{ (-n )_{m_{1}k_{1}+\cdots+m_{s}k_{s}}}{k_{1}!\cdots k_{s}!}\Lambda (n;k_{1},\ldots,k_{s} ) c_{1}^{k_{1} } \cdots c_{s}^{k_{s} } \\& \qquad {}\times b^{- \sum_{j=1}^{s}\delta_{j} k_{j}} x^{\sum_{j=1}^{s}\lambda _{j} k_{j}}H^{0,n+3:\{m_{r},n_{r}\};1,0}_{p+3,q+3:\{p_{r},q_{r}\};0,1} \\& \qquad \left [ \textstyle\begin{array}{@{}c@{}} z_{1}\frac{x^{\sigma_{1}}}{b^{v_{1}}} \\ \vdots \\ z_{r}\frac {x^{\sigma_{r}}}{b^{v_{r}}} \\ -\frac{a}{b}x \end{array}\displaystyle \left. \,\right|\, \textstyle\begin{array}{@{}c@{}} (a_{j};\alpha^{(1)}_{j},\ldots,\alpha^{(r)}_{j} )_{1,p},A': \{ (c^{(r)}_{j},\gamma^{(r)}_{j} )_{1,p_{r}} \} ;- \\ (b_{j};\beta^{(1)}_{j},\ldots,\beta^{(r)}_{j} )_{1,q},B': \{ (d^{(r)}_{j},\delta^{(r)}_{j} )_{1,q_{r}} \};(0,1) \end{array}\displaystyle \right ], \end{aligned}$$
(2.12)
$$\begin{aligned} A' =& \Biggl(1-\eta-\sum_{j=1}^{s} \delta_{j} k_{j}; v_{1} ,\ldots,v_{r}, 1 \Biggr), \Biggl(1-\mu-\sum_{j=1}^{s} \lambda_{j} k_{j} ;\sigma _{1},\ldots, \sigma_{r},1 \Biggr) , \\ & \Biggl(1-\mu-\gamma+\beta-\sum_{j=1}^{s} \lambda_{j} k_{j} ;\sigma _{1} ,\ldots, \sigma_{r}, 1 \Biggr) \end{aligned}$$
$$\begin{aligned} B' =& \Biggl(1-\eta-\sum_{j=1}^{s} \delta_{j} k_{j} ;v_{1},\ldots,v_{r} ,0 \Biggr), \Biggl(1-\mu+\beta-\sum_{j=1}^{s} \lambda_{j} k_{j} ;\sigma_{1} ,\ldots, \sigma_{r}, 1 \Biggr) \\ & \Biggl(1-\mu-\gamma-\alpha-\sum_{j=1}^{s} \lambda_{j} k_{j}; \sigma _{1},\ldots, \sigma_{r}, 1 \Biggr) , \end{aligned}$$
Corollary 4
Let
\(\alpha,\beta,\gamma,\mu,\eta,\delta _{j},v_{i},z_{i}, a,b,c_{j}\in\mathbb{C}\), \(\lambda_{j} ,\sigma_{i} >0\) (\(i\in\{ 1,\ldots,r\}\); \(j\in\{1,\ldots,s\}\)), and
\(\Re(\alpha)>0\). Let the following conditions be satisfied, too:
(i)
\(|\arg z_{i}|<\frac{\pi}{2} \Omega_{i} \), \(\Omega_{i}>0\) (\(i = 1,\ldots,r\)), where
$$\Omega_{i} =-\sum_{j=n+1}^{p} \alpha_{j}^{ (i )}-\sum_{j=1}^{q} \beta_{j}^{ (i )} +\sum_{j=1}^{n_{i} } \gamma _{j}^{ (i )} -\sum_{j=n_{i} +1}^{p_{i} } \gamma_{j}^{ (i )} +\sum_{j=1}^{m_{i} } \delta_{j}^{ (i )}-\sum_{j=m_{i}+1}^{q_{i} } \delta_{j}^{ (i )} >0. $$
(ii)
$$\begin{aligned}& \Re (\mu )-\sum_{i=1}^{r} \sigma_{i} \min_{1\le j\le m_{i} } \biggl[\frac{\Re (d_{j}^{ (i )} )}{\delta _{j}^{ (i )} } \biggr] < 1+\max\bigl[\Re (\beta ),\Re (\gamma )\bigr]\quad (i=1,\ldots,r ) , \\& \Re (\eta )-\sum_{i=1}^{r}v_{i} \min_{1\le j\le m_{i} } \biggl[\frac{\Re (d_{j}^{ (i )} )}{\delta _{j}^{ (i )} } \biggr]< 1+\max\bigl[\Re ( \beta ),\Re (\gamma )\bigr]\quad (i=1,\ldots,r ) . \end{aligned}$$
Then the following result holds:
where
and
and the conditions of existence of the above corollary follow easily from Theorem
2.
$$\begin{aligned}& \bigl\{ \mathrm{I}_{0-}^{\alpha,\beta,\gamma} \bigl(t^{\mu-1} (b-at )^{-\eta} S^{m_{1},\ldots,m_{s}}_{n} \bigl[c_{1} t^{\lambda_{1}} (b-at )^{-\delta_{1} }, \ldots,c_{s} t^{\lambda_{s}} (b-at )^{-\delta_{s} } \bigr] \\& \qquad {}\times H\bigl[z_{1} t^{\sigma_{1} } (b-at )^{-v_{1} } ,\ldots,z_{r} t^{\sigma_{r} } (b-at )^{-v_{r}} \bigr]\bigr) \bigr\} (x ) \\& \quad =b^{-\eta} x^{\mu-\beta-1} \sum^{m_{1}k_{1}+\cdots+m_{s}k_{s}\leq n}_{k_{1},\ldots,k_{s}=0} \frac{ (-n )_{m_{1}k_{1}+\cdots+m_{s}k_{s}}}{k_{1}!\cdots k_{s}!}\Lambda (n;k_{1},\ldots,k_{s} ) c_{1}^{k_{1} } \cdots c_{s}^{k_{s} } \\& \qquad {}\times b^{- \sum_{j=1}^{s}\delta_{j} k_{j}} x^{\sum_{j=1}^{s}\lambda _{j} k_{j}}H^{0,n+3:\{m_{r},n_{r}\};1,0}_{p+3,q+3:\{p_{r},q_{r}\};0,1} \\& \qquad \left [ \textstyle\begin{array}{@{}c@{}} z_{1}\frac{x^{\sigma_{1}}}{b^{v_{1}}} \\ \vdots \\ z_{r}\frac {x^{\sigma_{r}}}{b^{v_{r}}} \\ -\frac{a}{b}x \end{array}\displaystyle \left. \,\right|\, \textstyle\begin{array}{@{}c@{}} (a_{j};\alpha^{(1)}_{j},\ldots,\alpha^{(r)}_{j} )_{1,p},C'': \{ (c^{(r)}_{j},\gamma^{(r)}_{j} )_{1,p_{r}} \} ;- \\ (b_{j};\beta^{(1)}_{j},\ldots,\beta^{(r)}_{j} )_{1,q},D'': \{ (d^{(r)}_{j},\delta^{(r)}_{j} )_{1,q_{r}} \};(0,1) \end{array}\displaystyle \right ], \end{aligned}$$
(2.13)
$$\begin{aligned} C'' =& \Biggl(1-\eta-\sum _{j=1}^{s}\delta_{j} k_{j}; v_{1} ,\ldots,v_{r}, 1 \Biggr), \Biggl(1-\mu-\sum _{j=1}^{s}\lambda_{j} k_{j} ;\sigma_{1},\ldots,\sigma_{r},1 \Biggr) , \\ & \Biggl(1+\alpha+\beta-\mu-\gamma-\sum_{j=1}^{s} \lambda_{j} k_{j} ;\sigma_{1} ,\ldots, \sigma_{r}, 1 \Biggr) \end{aligned}$$
$$\begin{aligned} D'' =& \Biggl(1-\eta-\sum _{j=1}^{s}\delta_{j} k_{j} ;v_{1},\ldots,v_{r} ,0 \Biggr), \Biggl(1-\gamma-\mu-\sum _{j=1}^{s}\lambda_{j} k_{j} ;\sigma _{1} ,\ldots,\sigma_{r}, 1 \Biggr) \\ & \Biggl(1+\beta-\mu-\sum_{j=1}^{s} \lambda_{j} k_{j}; \sigma _{1},\ldots, \sigma_{r}, 1 \Biggr) , \end{aligned}$$
3 Special cases and concluding remarks
Here we consider another variation of the results derived in the preceding sections. The multivariable H-function occurring in these results can be suitably specialized to yield a wide variety of special functions (or product of such functions) of one or more variables (see [31], pp.18-19, 88-93, 253-254 and [48]). Again, by suitably specializing the coefficient of the first class of multivariable polynomials, it can be reduced to other multivariable hypergeometric polynomials and classical orthogonal polynomials of one or more variables.
(i) If we reduce the multivariable H-functions into the product of two Fox H-functions in Theorem 1 and then reduce one H-function to the exponential function by taking \(\sigma_{1} =1\), \(v_{1} \to0\), we obtain the following result after a little simplification: where
$$\begin{aligned}& \Biggl\{ \mathrm{I}_{0+}^{\alpha,\alpha',\beta ,\beta',\gamma} \Biggl(t^{\mu-1} (b-at )^{-\eta} S^{m_{1},\ldots,m_{s}}_{n} \bigl[c_{1} t^{\lambda_{1}} (b-at )^{-\delta _{1} }, \ldots,c_{s} t^{\lambda_{s}} (b-at )^{-\delta_{s} } \bigr] \\& \qquad {}\times e^{-z_{1}t}H^{m_{2},n_{2}}_{p_{2},q_{2}}\left [z_{2} t^{\sigma_{2} } (b-at )^{-v_{2} }\Bigm| \textstyle\begin{array}{@{}c@{}} (c_{j},C_{j} )_{1,p_{2}} \\ (d_{j},D_{j} )_{1,q_{2}} \end{array}\displaystyle \right ]\Biggr)\Biggr\} (x ) \\& \quad =b^{-\eta} x^{\mu -\alpha-\alpha'+\gamma-1} \sum^{m_{1}k_{1}+\cdots+m_{s}k_{s}\leq n}_{k_{1},\ldots,k_{s}=0} \frac{ (-n )_{m_{1}k_{1}+\cdots+m_{s}k_{s}}}{k_{1}!\cdots k_{s}!}\Lambda (n;k_{1},\ldots,k_{s} ) c_{1}^{k_{1} } \cdots c_{s}^{k_{s} } \\& \qquad {}\times b^{- \sum_{j=1}^{s}\delta_{j} k_{j}} x^{\sum_{j=1}^{s}\lambda _{j} k_{j}}H^{0,4:1,0;m_{2},n_{2};1,0}_{4,4:0,1;p_{2},q_{2};0,1} \\& \qquad \left [ \textstyle\begin{array}{@{}c@{}} z_{1}x \\ z_{2}\frac{x^{\sigma_{2}}}{b^{v_{2}}} \\ -\frac{a}{b}x \end{array}\displaystyle \Biggm| \textstyle\begin{array}{@{}c@{}} E: (c_{j},\gamma_{j} )_{1,p_{2}};- \\ F:(0,1); \{ (d_{j},\delta_{j} )_{1,q_{2}} \};(0,1) \end{array}\displaystyle \right ], \end{aligned}$$
(3.1)
$$\begin{aligned}& E=\Biggl(1-\eta-\sum^{s}_{j=1} \delta_{j}k_{j};1,v_{2},1\Biggr),\Biggl(1-\mu- \sum^{s}_{j=1}\lambda_{j}k_{j};1, \sigma_{2},1\Biggr), \\& \hphantom{E={}}\Biggl(1-\mu-\gamma+\alpha+\alpha'+\beta -\sum ^{s}_{j=1}\lambda_{j}k_{j};1, \sigma_{2},1\Biggr), \\& \hphantom{E={}}\Biggl(1-\mu+\alpha'- \beta'-\sum^{s}_{j=1} \lambda_{j}k_{j};1,\sigma_{2},1\Biggr) , \\& F=\Biggl(1-\eta-\sum^{s}_{j=1} \delta_{j}k_{j};1,v_{2},0\Biggr), \Biggl(1- \mu-\gamma+\alpha+\alpha-\sum^{s}_{j=1} \lambda_{j}k_{j};1,\sigma_{2},1\Biggr), \\& \hphantom{F={}}\Biggl(1-\mu-\gamma+\alpha'+\beta-\sum ^{s}_{j=1}\lambda_{j}k_{j};1, \sigma _{2},1\Biggr),\Biggl(1-\mu-\beta'-\sum ^{s}_{j=1}\lambda_{j}k_{j};1, \sigma_{2},1\Biggr) . \end{aligned}$$
The conditions of validity of the above result easily follow from Theorem 1.
Remark 6
If we reduce the Marichev-Saigo-Maeda type fractional integral operator into the Riemann-Liouville fractional integral operator and put \(S_{n}^{m_{1},\ldots,m_{s} } =1\), \(\eta,v_{2} =0\), and make a suitable adjustment in the parameters in the above equation, then we obtain the known result by Kilbas and Saigo [49], p.52, Eq. (2.7.9); in addition, if we put \(S_{n_{j} }^{m_{j} } =1\), \(\eta ,v_{2} =0\), \(z_{2} =1/4\), \(\sigma_{2} =2\) and reduce the Fox H-function to the Bessel function of the first kind, then we obtain the known result given by Kilbas and Sebastian [50], p.873, Eqs. (25)-(29).
(ii) Next if we take \(z_{2} ,\sigma_{2} =1\) and \(v_{2} =0\) in the result (3.1) and reduce the H-function of one variable to a generalized Mittag-Leffler function [51] (see also [52]), we can easily obtain the following result after a few simplifications: where
$$\begin{aligned}& \bigl\{ \mathrm{I}_{0+}^{\alpha,\alpha',\beta ,\beta',\gamma} \bigl(t^{\mu-1} (b-at )^{-\eta} S^{m_{1},\ldots,m_{s}}_{n} \bigl[c_{1} t^{\lambda_{1}} (b-at )^{-\delta _{1} }, \ldots,c_{s} t^{\lambda_{s}} (b-at )^{-\delta_{s} } \bigr] \\& \qquad {}\times e^{-z_{1}t}E^{v}_{\omega,\xi} [t ]\bigr) \bigr\} (x ) \\& \quad =\frac{b^{-\eta} x^{\mu-\alpha-\alpha'+\gamma-1}}{\Gamma(v)} \sum^{m_{1}k_{1}+\cdots+m_{s}k_{s}\leq n}_{k_{1},\ldots,k_{s}=0} \frac{ (-n )_{m_{1}k_{1}+\cdots+m_{s}k_{s}}}{k_{1}!\cdots k_{s}!}\Lambda (n;k_{1},\ldots,k_{s} ) c_{1}^{k_{1} } \cdots c_{s}^{k_{s} } \\& \qquad {}\times b^{- \sum_{j=1}^{s}\delta_{j} k_{j}} (-x)^{\sum _{j=1}^{s}\lambda_{j} k_{j}}H^{0,4:1,0;1,1;1,0}_{4,3:0,1;1,3;0,1} \\& \qquad \left [ \textstyle\begin{array}{@{}c@{}} z_{1}x \\ x \\ -\frac{a}{b}x \end{array}\displaystyle \Biggm| \textstyle\begin{array}{@{}c@{}} E:-;(1-v,1);- \\ F:(0,1);(0,1),(1-\eta;0);(1-\xi,\omega);(0,1) \end{array}\displaystyle \right ], \end{aligned}$$
(3.2)
$$\begin{aligned}& E=\Biggl(1-\eta-\sum^{s}_{j=1} \delta_{j}k_{j};1,0,1\Biggr),\Biggl(1-\mu-\sum ^{s}_{j=1}\lambda_{j}k_{j};1,1,1 \Biggr), \\& \hphantom{E={}}\Biggl(1-\mu-\gamma+\alpha+\alpha'+\beta-\sum ^{s}_{j=1}\lambda_{j}k_{j};1,1,1 \Biggr),\Biggl(1-\mu+\alpha'-\beta'-\sum ^{s}_{j=1}\lambda _{j}k_{j};1,1,1 \Biggr), \\& F=\Biggl(1-\eta-\sum^{s}_{j=1} \delta_{j}k_{j};1,0,0\Biggr),\Biggl(1-\mu-\gamma+\alpha+ \alpha '-\sum^{s}_{j=1} \lambda_{j}k_{j};1,1,1\Biggr), \\& \hphantom{F={}}\Biggl(1-\mu-\gamma+\alpha'+\beta-\sum ^{s}_{j=1}\lambda_{j}k_{j};1,1,1 \Biggr),\Biggl(1-\mu-\beta'-\sum^{s}_{j=1} \lambda_{j}k_{j};1,1,1\Biggr) . \end{aligned}$$
Remark 7
If we reduce the Marichev-Saigo-Maeda type fractional integral operator to the Riemann-Liouville fractional integral operator and set \(S_{n}^{m_{1},\ldots,m_{s} } =1\), \(\eta,v_{2} =0\), and make a suitable adjustment in the parameters in equation (3.2), then we arrive at the known result given by Saxena et al. [53], p.168, Eq. (2.1).
(iii) Again, if we reduce the H-function of one variable to the generalized Wright hypergeometric function [31], p.19, Eq. (2.6.11), in the result (3.1), we arrive at the following new and interesting result: where
$$\begin{aligned}& \Biggl\{ \mathrm{I}_{0+}^{\alpha,\alpha',\beta ,\beta',\gamma} \Biggl(t^{\mu-1} (b-at )^{-\eta} S^{m_{1},\ldots,m_{s}}_{n} \bigl[c_{1} t^{\lambda_{1}} (b-at )^{-\delta _{1} }, \ldots,c_{s} t^{\lambda_{s}} (b-at )^{-\delta_{s} } \bigr] \\& \qquad {}\times e^{-z_{1}t}{}_{p_{2}}\psi_{q_{2}}\left [-z_{2} t^{\sigma_{2} } (b-at )^{-v_{2} }\Bigm| \textstyle\begin{array}{@{}c@{}} (1-c_{j},C_{j} )_{1,p_{2}} \\ (0,1), (1-d_{j},D_{j} )_{1,q_{2}} \end{array}\displaystyle \right ]\Biggr)\Biggr\} (x ) \\& \quad =b^{-\eta} x^{\mu -\alpha-\alpha'+\gamma-1} \sum^{m_{1}k_{1}+\cdots+m_{s}k_{s}\leq n}_{k_{1},\ldots,k_{s}=0} \frac{ (-n )_{m_{1}k_{1}+\cdots+m_{s}k_{s}}}{k_{1}!\cdots k_{s}!}\Lambda (n;k_{1},\ldots,k_{s} ) c_{1}^{k_{1} } \cdots c_{s}^{k_{s} } \\& \qquad {}\times b^{- \sum_{j=1}^{s}\delta_{j} k_{j}} x^{\sum_{j=1}^{s}\lambda _{j} k_{j}}H^{0,4:1,0;1,p_{2};1,0}_{4,4:0,1;p_{2},q_{2};0,1} \\& \qquad \left [ \textstyle\begin{array}{@{}c@{}} z_{1}x \\ -z_{2}\frac{x^{\sigma_{2}}}{b^{v_{2}}} \\ -\frac{a}{b}x \end{array}\displaystyle \Biggm| \textstyle\begin{array}{@{}c@{}}E: (c_{j},C_{j} )_{1,p_{2}};- \\ F:(0,1); \{ (d_{j},D_{j} )_{1,q_{2}} \};(0,1) \end{array}\displaystyle \right ], \end{aligned}$$
(3.3)
$$\begin{aligned}& E=\Biggl(1-\eta-\sum^{s}_{j=1} \delta_{j}k_{j};1,v_{2},1\Biggr),\Biggl(1-\mu- \sum^{s}_{j=1}\lambda_{j}k_{j};1, \sigma_{2},1\Biggr), \\& \hphantom{E={}}\Biggl(1-\mu-\gamma+\alpha+\alpha'+\beta -\sum ^{s}_{j=1}\lambda_{j}k_{j};1, \sigma_{2},1\Biggr), \Biggl(1-\mu+\alpha'- \beta'-\sum^{s}_{j=1} \lambda_{j}k_{j};1,\sigma_{2},1\Biggr), \\& F=\Biggl(1-\eta-\sum^{s}_{j=1} \delta_{j}k_{j};1,v_{2},0\Biggr),\Biggl(1-\mu- \gamma+\alpha+\alpha '-\sum^{s}_{j=1} \lambda_{j}k_{j};1,\sigma_{2},1\Biggr), \\& \hphantom{F={}}\Biggl(1-\mu-\gamma+\alpha'+\beta -\sum ^{s}_{j=1}\lambda_{j}k_{j};1, \sigma_{2},1\Biggr),\Biggl(1-\mu-\beta'-\sum ^{s}_{j=1}\lambda_{j}k_{j};1, \sigma_{2},1\Biggr) . \end{aligned}$$
(iv) Further Theorem 1, if we reduce the multivariable H-function to the product of r-different Whittaker functions [31], p.18, Eq. (2.6.7), and take \(v_{i} \to0\), \(\sigma_{i} =1\), then we arrive at the following result: where
$$\begin{aligned}& \Biggl\{ \mathrm{I}_{0+}^{\alpha,\alpha',\beta ,\beta',\gamma} \Biggl(t^{\mu-1} (b-at )^{-\eta} S^{m_{1},\ldots,m_{s}}_{n} \bigl[c_{1} t^{\lambda_{1}} (b-at )^{-\delta _{1} }, \ldots,c_{s} t^{\lambda_{s}} (b-at )^{-\delta_{s} } \bigr] \\& \qquad {}\times\prod^{r}_{i=1}e^{\frac {-z_{i}t}{2}}W_{\lambda_{i},\mu_{i}}(z_{i}t) \Biggr)\Biggr\} (x) \\& \quad =b^{-\eta} x^{\mu-\alpha-\alpha'+\gamma-1} \sum^{m_{1}k_{1}+\cdots+m_{s}k_{s}\leq n}_{k_{1},\ldots,k_{s}=0} \frac{ (-n )_{m_{1}k_{1}+\cdots+m_{s}k_{s}}}{k_{1}!\cdots k_{s}!}\Lambda (n;k_{1},\ldots,k_{s} ) c_{1}^{k_{1} } \cdots c_{s}^{k_{s} } \\& \qquad {}\times b^{- \sum_{j=1}^{s}\delta_{j} k_{j}} x^{\sum_{j=1}^{s}\lambda _{j} k_{j}}H^{0,4:2,0;\ldots;2,0;1,0}_{4,4:1,2;\ldots;1,2;0,1} \\& \qquad \left [ \textstyle\begin{array}{@{}c@{}} z_{1}x \\ \vdots \\ z_{r}x \\ -\frac{a}{b}x \end{array}\displaystyle \left. \,\right|\, \textstyle\begin{array}{@{}c@{}} G:-;(1-\lambda_{1},1);\ldots;(1-\lambda_{r},1);- \\ H:(1/2\pm\mu_{1},1);\ldots;(1/2\pm\mu_{r},1)(0,1) \end{array}\displaystyle \right ], \end{aligned}$$
(3.4)
$$\begin{aligned}& G=\Biggl(1-\eta-\sum^{s}_{j=1} \delta_{j}k_{j};\underbrace{1,\ldots,1}_{r \text{ times}},1 \Biggr),\Biggl(1-\mu-\sum^{s}_{j=1} \lambda_{j}k_{j};\underbrace{1,\ldots,1}_{r \text{ times}},1 \Biggr), \\& \hphantom{G={}}\Biggl(1-\mu-\gamma+\alpha+\alpha'+\beta-\sum ^{s}_{j=1}\lambda _{j}k_{j}; \underbrace{1,\ldots,1}_{r \text{ times}},1\Biggr), \\& \hphantom{G={}}\Biggl(1-\mu+ \alpha'-\beta '-\sum^{s}_{j=1} \lambda_{j}k_{j};\underbrace{1,\ldots,1}_{r \text{ times}},1 \Biggr) , \\& H=\Biggl(1-\eta-\sum^{s}_{j=1} \delta_{j}k_{j};\underbrace{1,\ldots,1}_{r \text{ times}},0 \Biggr),\Biggl(1-\mu-\gamma+\alpha+\alpha'-\sum ^{s}_{j=1}\lambda _{j}k_{j}; \underbrace{1,\ldots,1}_{r \text{ times}},1\Biggr), \\& \hphantom{H={}}\Biggl(1-\mu-\gamma+\alpha '+\beta-\sum ^{s}_{j=1}\lambda_{j}k_{j}; \underbrace{1,\ldots,1}_{r \text{ times}},1\Biggr),\Biggl(1-\mu- \beta'-\sum^{s}_{j=1} \lambda_{j}k_{j};\underbrace {1,\ldots,1}_{r \text{ times}},1 \Biggr) . \end{aligned}$$
The conditions of validity of the above result easily follow from Theorem 1.
Remark 8
If we reduce the Marichev-Saigo-Maeda type fractional integral operator to the Riemann-Liouville fractional integral operator and set \(S_{n}^{m_{1},\ldots,m_{s} } =1\), \(\eta,v_{2} =0\), and make a suitable adjustment in the parameters in equation (3.4), we arrive at the known result given by Kilbas [54], p.117, Eq. (11).
(v) Finally, if we reduce the multivariable H-function into the product of two Fox H-functions in Theorem 1 and then reduce one H-function to the exponential function by setting \(\sigma_{1}\), \(\lambda_{j} \), \(v_{1} \), \(\delta_{j} =0\), \(c_{j}\), \(s=1\) and \(S_{n}^{m}\) to the Hermite polynomial [6, 55], and we set \(S_{n}^{2} [x]=x^{n/2}H_{n} [\frac{1}{2\sqrt{x}} ]\), in this case \(m=2\), \(A_{n,k}=(-1)^{k}\), we obtain the following interesting result: where
$$\begin{aligned}& \Biggl\{ \mathrm{I}_{0+}^{\alpha,\alpha',\beta,\beta ',\gamma}\Biggl(t^{\mu-1} (b-at )^{-\eta} t^{n/2}H_{n} \biggl[\frac{1}{2\sqrt{t}} \biggr] \\& \qquad {}\times e^{-z_{1}t}H^{m_{2},n_{2}}_{p_{2},q_{2}}\left [z_{1} t^{\sigma_{1} } (b-at )^{-v_{1} }\Bigm| \textstyle\begin{array}{c} (c_{j},C_{j} )_{1,p_{2}} \\ (d_{j},D_{j} )_{1,q_{2}} \end{array}\displaystyle \right ]\Biggr)\Biggr\} (x ) \\& \quad =b^{-\eta} x^{\mu -\alpha-\alpha'+\gamma-1} \sum^{m_{1}k_{1}+\cdots+m_{s}k_{s}\leq n}_{k_{1},\ldots,k_{s}=0} \frac{ (-n )_{m_{1}k_{1}+\cdots+m_{s}k_{s}}}{k_{1}!\cdots k_{s}!}\Lambda (n;k_{1},\ldots,k_{s} ) c_{1}^{k_{1} } \cdots c_{s}^{k_{s} } \\& \qquad {}\times b^{- \sum_{j=1}^{s}\delta_{j} k_{j}} x^{\sum_{j=1}^{s}\lambda _{j} k_{j}}H^{0,4:1,0;\{m_{2},n_{2}\};1,0}_{4,4:0,1;\{p_{2},q_{2}\} ;0,1} \\& \qquad \left [ \textstyle\begin{array}{@{}c@{}} z_{1}x \\ z_{2}\frac{x^{\sigma_{2}}}{b^{v_{2}}} \\ -\frac{a}{b}x \end{array}\displaystyle \Biggm| \textstyle\begin{array}{@{}c@{}} G: (c^{(r)}_{j},\gamma^{(r)}_{j} )_{1,p_{r}};- \\ H:(0,1); \{ (d_{j},\delta_{j} )_{1,q_{2}} \};(0,1) \end{array}\displaystyle \right ], \end{aligned}$$
(3.5)
$$\begin{aligned}& G=\Biggl(1-\eta-\sum^{s}_{j=1} \delta_{j}k_{j};1,v_{2},1\Biggr),\Biggl(1-\mu- \sum^{s}_{j=1}\lambda_{j}k_{j};1, \sigma_{2},1\Biggr), \\& \hphantom{G={}}\Biggl(1-\mu-\gamma+\alpha+\alpha'+\beta -\sum ^{s}_{j=1}\lambda_{j}k_{j};1, \sigma_{2},1\Biggr),\Biggl(1-\mu+\alpha'- \beta'-\sum^{s}_{j=1} \lambda_{j}k_{j};1,\sigma_{2},1\Biggr), \\& H=\Biggl(1-\eta-\sum^{s}_{j=1} \delta_{j}k_{j};1,v_{2},0\Biggr),\Biggl(1-\mu- \gamma+\alpha+\alpha -\sum^{s}_{j=1} \lambda_{j}k_{j};1,\sigma_{2},1\Biggr), \\& \hphantom{H={}}\Biggl(1-\mu-\gamma+\alpha'+\beta-\sum ^{s}_{j=1}\lambda_{j}k_{j};1, \sigma_{2},1\Biggr),\Biggl(1-\mu-\beta'-\sum ^{s}_{j=1}\lambda _{j}k_{j};1, \sigma_{2},1\Biggr) . \end{aligned}$$
The conditions of validity of the above result can be easily derived from Theorem 1.
A large number of other special cases of our main result can also be obtained, but we do not mention them here on account of the lack of space.
Now, we conclude our present investigation by remarking that the fractional integration (of Marichev-Saigo-Maeda type) of the products of multivariable H-functions and the first class of multivariable polynomials established in this paper will be useful for investigators in various disciplines of applied sciences and engineering physics. We are also trying to find certain possible applications of those results presented here to some other research areas, for example, obtaining a closed form solution of a fractional generalization of a free electron equation.
Acknowledgements
We are grateful to Dr. S Jain for her useful suggestions during discussions and to anonymous reviewers for their valuable remarks.
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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors have participated in the obtained results. The collaboration of each one cannot be separated in different parts of the paper. All of them have made substantial contributions to the theoretical results. All authors have been involved in drafting the manuscript and revising it critically for important intellectual content. All authors read and approved the final manuscript.