1 Introduction and preliminaries

The multivariable H-function is defined and studied by Srivastava and Panda ([1], p. 271, Eq. (4.1)) in terms of Mellin–Barnes type contour integral as follows:

$$\begin{aligned} H [z_{1} ,\ldots,z_{r} ] =&H_{p,q:p_{1} ,q_{1} ;\ldots;p_{r} ,q _{r} }^{0,n:m_{1} ,n_{1} ;\ldots;m_{r} ,n_{r} } \left [\left.\textstyle\begin{array}{c} {z_{{1} } } \\ {\vdots } \\ {z_{{r} } } \end{array}\displaystyle \right| \textstyle\begin{array}{c} { (a_{j} ;\alpha _{j}^{\prime } ,\ldots,\alpha _{j}^{ (r )} ) _{1,p} : } \\ { (b_{j} ;\beta _{j}^{\prime } ,\ldots,\beta _{j}^{ (r )} ) _{1,q} : } \end{array}\displaystyle \right . \\ &{} \left . \vphantom{\left [\textstyle\begin{array}{c} {z_{{1} } } \\ {\vdots } \\ {z_{{r} } } \end{array}\displaystyle \right .} \textstyle\begin{array}{c} { (c_{j}^{\prime } ,\gamma _{j}^{\prime } )_{1,p_{{1} } } ;\ldots; (c _{j}^{ (r )} ,\gamma _{j}^{ (r )} )_{1,p _{r} } } \\ { (d_{j}^{\prime } ,\delta _{j}^{\prime } )_{1,q_{{1} } } ;\ldots; (d _{j}^{ (r )} ,\delta _{j}^{ (r )} )_{1,q _{r} } } \end{array}\displaystyle \right ] \end{aligned}$$
(1)
$$\begin{aligned} =&\frac{1}{ (2\pi \omega )^{r} } \int _{L_{1} }\dots \int _{L _{r} }\phi (\xi _{1} ,\ldots,\xi _{r} ) \Biggl\{ \prod_{i=1} ^{r} \theta _{i} (\xi _{i} )z_{i} ^{\xi _{i} } \Biggr\} \,d\xi _{1} \cdots d\xi _{r} , \end{aligned}$$
(2)

where \(\omega =\sqrt{-1} \); and

$$\begin{aligned}& \phi (\xi _{1} ,\ldots,\xi _{r} )= \frac{\prod_{j=1}^{n}\varGamma (1-a_{j} +\sum_{i=1}^{r}\alpha _{j}^{ (i )} \xi _{i} ) }{\prod_{j=n+1}^{p}\varGamma (a_{j} -\sum_{i=1}^{r}\alpha _{j}^{ (i )} \xi _{i} )\prod_{j=1}^{n}\varGamma (1-b _{j} +\sum_{i=1}^{r}\beta _{j}^{ (i )} \xi _{i} ) } , \end{aligned}$$
(3)
$$\begin{aligned}& \theta _{i} (\xi _{i} )=\frac{\prod_{j=1}^{n_{i} }\varGamma (1-c_{j}^{ (i )} +\gamma _{j}^{ (i )} \xi _{i} +\sum_{i=1}^{r}\alpha _{j}^{ (i )} \xi _{i} ) \prod_{j=1}^{m_{i} }\varGamma (d_{j}^{ (i )} -\delta _{j}^{ (i )} \xi _{i} ) }{\prod_{j=n_{i} +1}^{p_{i} } \varGamma (c_{j}^{ (i )} -\gamma _{j}^{ (i )} \xi _{i} )\prod_{j=m_{i} +1}^{q_{i} }\varGamma (1-d_{j}^{ (i )} +\delta _{j}^{ (i )} \xi _{i} )}, \end{aligned}$$
(4)

and \(L_{j} =L_{\omega \tau _{j} \infty } \) represents the contours which start at the point \(\tau _{j} -\omega \infty \) and terminate at the points \(\tau _{j} +\omega \infty \) with \(\tau _{j} \in \Re = (- \infty ,\infty )\) (\(j=1,\ldots,r \)).

In the case \(r=2\), (1) reduces to the H-function of two variables. For a detailed definition and convergence conditions of the multivariable H-function, the reader is referred to the original papers [2,3,4,5,6,7,8,9]. From Srivastava and Panda ([10], p. 131), we have

$$ H[z_{1} ,\ldots,z_{r} ] = \mathrm{O} \bigl( \vert z_{1} \vert ^{e _{1} }\cdots \vert z_{r} \vert ^{e_{r} } \bigr) \bigl(\mathop{\max } _{1\le j \le r} \Vert z_{j} \Vert \to 0 \bigr) , $$
(5)

where

$$ e_{i} = \mathop{\min } _{1\le j \le r} \biggl[ \frac{\operatorname{Re} (d _{j}^{(i)} )}{\delta _{j}^{(i)} } \biggr]\quad (i = 1,\ldots,r). $$
(6)

For \(n=p=q=0\), the multivariable H-function breaks up into product of ‘r’ H-function; consequently, there holds the following results:

$$\begin{aligned}& H_{0, 0 : p_{ 1} , q_{ 1} ; \ldots ; p_{ r} , q _{r} }^{0, 0 : m_{ 1} , n _{1} ; \ldots ; m_{ r} , n _{r} } \left [\left. \textstyle\begin{array}{c} { \textstyle\begin{array}{l} {z_{1} } \\ { \vdots } \end{array}\displaystyle } \\ {z_{r} } \end{array}\displaystyle \right| \textstyle\begin{array}{c} { (c_{j}^{\prime}, \gamma _{j}^{\prime} )_{1, p_{ 1} } ; \ldots ; (c _{j}^{(r)}, \gamma _{j}^{(r)} )_{1, p_{ r}} } \\ { (d_{j}^{\prime} , \delta _{j}^{\prime} )_{1, q_{ 1} } ; \ldots ; (d_{j}^{(r)}, \delta _{j}^{(r)} )_{1, q_{ r} } } \end{array}\displaystyle \right ] \\& \quad = \prod_{i=1}^{r}H_{p_{i} , q _{i} }^{ m_{ i} , n _{i}} \left [z\left| \textstyle\begin{array}{c} { (c_{j}^{(i)}, \gamma _{j}^{(i)} )_{1, p_{i}}} \\ { (d_{j}^{(i)}, \delta _{j}^{(i)} )_{1, q_{i}}} \end{array}\displaystyle \right. \right ] , \end{aligned}$$
(7)

where \(H_{p, q }^{m, n} (\cdot)\) is the familiar H-function.

In the sequel, Srivastava and Garg ([11], p. 686, Eq. (1.4)) gave the definition of multivariable generalization of the polynomials \(S_{n}^{m} (x )\) as follows:

$$\begin{aligned} S_{L}^{h_{1} ,\ldots,h_{s} } (x_{1} , \dots , x_{s} ) =& \sum_{k_{1} , \dots ,k_{s} =0}^{h_{1} k_{1} + \cdots +h_{s} k_{s} \le L } (-L )_{h_{1} k_{1} + \cdots +h_{s} k_{s} } \\ &{}\times A (L; k_{1} , \dots , k_{s} ) \frac{x_{1}^{k_{1} } }{k_{1} !} \dots \frac{x_{s}^{k_{s} } }{k_{s}!}, \end{aligned}$$
(8)

where the coefficients \(A (L; h _{1} , \ldots,h_{s} )\) (\(L,h_{i} \in N_{0}\), \(i=1,\ldots,s\)) are arbitrary. Choosing constants to be real or complex, as Srivastava [12] defined by \(s = 1\) on the above polynomial, we obtain a polynomial of the form \(S_{n}^{m} (x )\).

Let \(\alpha '\), \(\beta '\), \(\eta '\) be complex numbers and \(\theta > 0\). The modified Saigo integral operators are denoted by \(I_{0, x , \theta } ^{\alpha ' , \beta ' , \eta '} \) and \(J_{x , \infty , \theta }^{ \alpha ' , \beta ' , \eta '} \) respectively for \(\Re (\alpha ') >\) 0:

$$\begin{aligned}& I_{0, x, \theta }^{\alpha ', \beta ', \eta '} f = \frac{\theta x^{- \theta (\alpha ' + \beta ' )} }{\varGamma (\alpha ')} \int _{0}^{x} \bigl(x ^{\theta } -t^{\theta } \bigr)^{\alpha ' - 1} \\& \hphantom{I_{0, x, \theta }^{\alpha ', \beta ', \eta '} f =}{}\times {}_{2} F_{1} \bigl( \alpha ' + \beta ' , -\eta ' ; \alpha '; 1-{t ^{\theta } / x^{\theta } } \bigr) t^{\theta - 1} f(t)\,dt \end{aligned}$$
(9)
$$\begin{aligned}& \hphantom{I_{0, x, \theta }^{\alpha ', \beta ', \eta '} f}{}=\frac{d^{n} }{d(x^{\theta } )^{n} } I_{0, x, \theta }^{\alpha ' + n, \beta ' - n, \eta ' - n} f,\quad 0 < \Re \bigl(\alpha '\bigr) + n \mathbin{\underline{\le }}1, \end{aligned}$$
(10)
$$\begin{aligned}& J_{x , \infty , \theta }^{\alpha ', \beta ', \eta '} f= \frac{\theta }{\varGamma (\alpha ')} \int _{x}^{\infty }\bigl(t^{\theta } -x^{\theta } \bigr)^{ \alpha ' - 1} t^{-\theta (\alpha ' + \beta ')} \\& \hphantom{J_{x , \infty , \theta }^{\alpha ', \beta ', \eta '} f=}{}\times {}_{2} F_{1} \bigl(\alpha ' + \beta ' , -\eta ' ; \alpha '; 1-{x ^{\theta } / t^{\theta } } \bigr) t^{\theta - 1} f(t)\,dt \end{aligned}$$
(11)
$$\begin{aligned}& \hphantom{J_{x , \infty , \theta }^{\alpha ', \beta ', \eta '} f}{}= (-1)^{n} \frac{d^{n} }{d(x^{\theta } )^{n} } J_{x, \infty , \theta }^{\alpha ' + n, \beta ' - n, \eta ' - n} f,\quad 0 < \Re \bigl(\alpha ' \bigr) + n\mathbin{\underline{\le }} 1. \end{aligned}$$
(12)

Sufficient conditions for the existence of (9) and (11) are

$$ \theta > 0,\qquad \Re \bigl(\alpha '\bigr)>1 - {1 / 2\theta };\qquad f(x)\in L_{2} ( \Re {}_{+}) $$
(13)

and max\([0, \Re (\beta ' - \eta ' ) ] >1 - {1 / 2\theta }\); min\([\Re (\beta ' ), \Re (\eta ' )]> - {1 / 2\theta } \). If these conditions are satisfied, then \(I_{0,x,\theta }^{\alpha ', \beta ', \eta '} f(x)\), \(J_{x ,\infty , \theta }^{\alpha ', \beta ', \eta '} f (x)\) both exist and both \(\in L_{2}(\Re {}_{+})\).

The operators \(I_{0, x , \theta }^{\alpha ', \beta ', \eta '} \) and \(J_{x , \infty , \theta }^{\alpha ', \beta ', \eta '}\) include as their special case, \(\beta ' = - \alpha '\), the fractional calculus operators of Riemann–Liouville and Weyl types:

$$ I_{0, x ,\theta }^{\alpha ', -\alpha ', \eta '} f = R_{0 , x , \theta }^{\alpha '} f ,\qquad J_{x , \infty , \theta }^{\alpha ', -\alpha ', \eta '} f = W_{x , \infty , \theta }^{\alpha '} f. $$
(14)

Also, we obtain the following identities and inverses:

$$\begin{aligned}& I_{0 , x , \theta }^{0, 0, \eta '} f = f(x) ;\qquad J_{x , \infty , \theta }^{0, 0, \eta '} f = f(x). \end{aligned}$$
(15)
$$\begin{aligned}& \bigl[I_{0, x ,\theta }^{\alpha ', \beta ', \eta '} \bigr]^{-1} = I_{0, x , \theta }^{-\alpha ', -\beta ', \alpha ' + \eta '} ;\qquad \bigl[ J_{x , \infty , \theta }^{\alpha ', \beta ', \eta '} \bigr]^{-1} = J_{x , \infty , \theta }^{-\alpha ', -\beta ', \alpha ' + \eta '}. \end{aligned}$$
(16)

For the operators \(I_{0, x , \theta }^{\alpha ', \beta ', \eta '} \) and \(J_{x , \infty , \theta }^{\alpha ', \beta ', \eta '} \) there holds interesting results similar to the ones derived in a series of earlier papers [13,14,15,16,17,18,19].

In this paper, we shall study another generalization of (9) and (11) which is given in the following manner:

$$\begin{aligned}& I_{0, x, \theta ; r, \varepsilon , q; m, k, l}^{\rho ; \alpha , \beta , \tau ; C, D, \alpha ', \beta ', \eta ' } \bigl\{ f(x) \bigr\} \\& \quad = \frac{\theta x^{-\theta (\alpha ' + \beta ' )} }{\varGamma (\alpha ')} \int _{0}^{x} \bigl(x^{\theta } -t^{\theta } \bigr)^{\alpha ' - 1} {}_{2} F_{1} \bigl( \alpha ' + \beta ' , -\eta ' ; \alpha '; 1-{t^{\theta } / x^{\theta } } \bigr) t^{\theta - 1} \\& \qquad {}\times \Im _{n}^{\alpha , \beta , \tau } \bigl[zt^{\rho } ; r, \varepsilon , q, C, D, m, k, l \bigr] f (t)\,dt , \end{aligned}$$
(17)

and

$$\begin{aligned}& J_{x, \infty , \theta ; r, \varepsilon , q; m, k, l}^{\rho ; \alpha , \beta , \tau ; C, D, \alpha ', \beta ', \eta ' } \bigl\{ f(x) \bigr\} \\& \quad = \frac{\theta }{\varGamma (\alpha ')} \int _{x}^{\infty } \bigl(t^{\theta } -x ^{\theta } \bigr)^{\alpha ' - 1} t^{-\theta (\alpha ' + \beta ')} {}_{2} F _{1} \bigl(\alpha ' + \beta ' , -\eta ' ; \alpha '; 1-{x^{\theta } / t^{\theta } } \bigr) t^{\theta - 1} \\& \qquad {}\times \Im _{n}^{\alpha , \beta , \tau } \bigl[zt^{\rho } ; r, \varepsilon , q, C, D, m, k, l \bigr] f (t)\,dt, \end{aligned}$$
(18)

where \(\Re (\alpha ') > 0\), and \(\Im _{n}^{\alpha , \beta , \tau } [z] \) stands for the generalized polynomial set defined by the following Rodrigues type formula ([20], p. 64, Eq. (2.18)):

$$\begin{aligned}& \Im _{n}^{\alpha , \beta , \tau } [x ; r, \varepsilon , q, C, D, m, k, l ] \\& \quad = (Cx + D)^{- \alpha } \bigl( 1-\tau x^{r} \bigr)^{\frac{-\beta }{\tau } } T _{k, l}^{m + n} \bigl[(Cx + D)^{ \alpha + q n} \bigl( 1 - \tau x^{r} \bigr)^{\frac{ \beta }{\tau +\varepsilon n} } \bigr], \end{aligned}$$
(19)

with the differential operator \(T_{k, l} \) being defined as

$$ T_{k, l} \equiv x^{l} \biggl(k + x \frac{d}{dx} \biggr). $$
(20)

An explicit form of this generalized polynomial set ([20], p. 71, Eq. (2.34)) is given by

$$\begin{aligned}& \Im _{n}^{\alpha , \beta , \tau } [x ; r, \varepsilon , q, C, D, m, k, l ] \\& \quad = D^{q n} x^{l ( m + n)} \bigl(1 - \tau x^{r} \bigr)^{\varepsilon n} l^{m + n } \\& \qquad {}\times \sum_{\upsilon = 0}^{m + n} \sum _{u = 0}^{\upsilon } \sum _{j = 0}^{m + n} \sum_{i = 0}^{j} \frac{(-1)^{j} (-j_{i} ) (\alpha )_{j} (- \upsilon )_{u} (-\alpha - q n)_{i} }{i ! j ! u ! v ! (1 - \alpha - j)_{i} } \\& \qquad {}\times \biggl(-\frac{\beta }{\tau } -\varepsilon n \biggr)_{\upsilon } \biggl(\frac{i +k+ru}{l} \biggr)_{m + n} \biggl( \frac{-\tau x^{r} }{1 - \tau x^{r} } \biggr)^{\upsilon } \biggl(\frac{C x}{D} \biggr)^{j}. \end{aligned}$$
(21)

It may be noted that the polynomial set defined by (19) is of general character and unifies and extends a number of classical polynomials introduced and studied by various authors (see [21,22,23,24,25,26]). Two special cases of (17) are given below ([20], p. 65).

  1. 1.

    If we set \(C = 1 \), \(D = 0\) in (19), it gives

    $$\begin{aligned}& \Im _{n}^{\alpha , \beta , \tau } [x ; r, \varepsilon , q, 1, 0, m, k, l ] \\& \quad = x^{q n + l ( m + n)} \bigl(1 - \tau x^{r} \bigr)^{\varepsilon n} l^{m + n } \\& \qquad {}\times \sum_{\upsilon = 0}^{m + n} \sum_{u = 0}^{\upsilon } \frac{(- \upsilon )_{u} }{u ! v!} \biggl(- \frac{\beta }{\tau } - \varepsilon n \biggr) _{\upsilon } \biggl( \frac{\alpha + q n +k+ru}{l} \biggr)_{m + n} \biggl(\frac{-\tau x^{r} }{1 - \tau x^{r} } \biggr)^{\upsilon }. \end{aligned}$$
    (22)
  2. 2.

    As \(\tau \to 0\) in (21), by virtue of the well-known confluence principle \(\mathop{\lim }_{ \vert b \vert \to \infty } (b_{n} ) (\frac{z}{b} )^{n} = z ^{n} \), it yields the following polynomial set:

$$\begin{aligned}& \Im _{n}^{\alpha , \beta , 0} [x ; r, \varepsilon , q, 1, 0, m, k, l ] \\& \quad = x^{q n + l ( m + n)} l^{m + n } \sum _{\upsilon = 0}^{m + n} \sum_{u = 0}^{\upsilon } \frac{(-\upsilon )_{u} }{u ! v!} \biggl(\frac{\alpha + q n +k+ru}{l} \biggr)_{m + n} \bigl( \beta x^{r} \bigr)^{\upsilon }. \end{aligned}$$
(23)

2 Main results

It will be shown here that

(I)

$$\begin{aligned}& I_{0, x, \theta ; r, \varepsilon , q; m, k, l}^{\rho ; \alpha , \beta , \tau ; C, D, \alpha ', \beta ', \eta ' } \bigl\{ t ^{ \lambda } S_{L}^{h_{1} ,\ldots,h_{s} } \bigl( y_{1} t^{\eta _{1}} , \dots , y_{s} t^{\eta _{s}} \bigr) H \bigl[a_{1} t^{\zeta _{1} } ,\ldots,a _{r} t^{\zeta _{r} } \bigr] \bigr\} \\& \quad = \sum_{\upsilon = 0}^{m + n} \sum _{u = 0}^{\upsilon } \sum_{j = 0} ^{m + n} \sum_{i = 0}^{j} \varOmega ( i , j , u , v ) z^{ l ( m + n ) + r \upsilon + j } \\& \qquad {}\times \sum_{k_{1} , \dots ,k_{s} =0}^{h_{1} k_{1} + \cdots +h_{s} k _{s} \le L } \sum _{w=0}^{\infty }\frac{ (-L )_{h_{1} k_{1} + \cdots +h_{s} k_{s} } ( \upsilon -\varepsilon n )_{w} (\tau )^{w} z ^{r w }}{k_{1} !\cdots k_{s}! w!}A (L; k_{1} , \dots , k_{s} ) \\& \qquad {}\times {y_{1}^{k_{1}}} \cdots {y_{s}^{k_{s} }} x^{ \lambda +G+ \rho [ l ( m + n ) + r\upsilon + r w + j ] - \theta \beta '} H_{p+2,q+2:p_{1} ,q_{1} ;\dots ,p _{r} ,q_{r} }^{0,n+2:m_{1} ,n_{1} ;\dots ,m_{r} ,n_{r} } \left [\left. \textstyle\begin{array}{c} {a_{1} x^{\zeta _{1} } } \\ {\vdots } \\ {a_{r} x^{\zeta _{r} } } \end{array}\displaystyle \right| \textstyle\begin{array}{c} {} \\ {} \end{array}\displaystyle \right . \\& \qquad {}\textstyle\begin{array}{c} { (1 - \frac{\ell }{\theta }, \frac{\zeta _{1} }{\theta } ,\ldots,\frac{ \zeta _{r} }{\theta } ) , (1 - \frac{\ell }{\theta } - \eta ' + \beta ' , \frac{\zeta _{1} }{\theta } ,\ldots,\frac{\zeta _{r} }{ \theta } ) , } \\ { (1 - \frac{\ell }{\theta } + \beta ' , \frac{\zeta _{1} }{ \theta } ,\ldots,\frac{\zeta _{r} }{\theta } ), (1 - \frac{ \ell }{\theta } - \eta ' - \alpha ' , \frac{\zeta _{1} }{\theta } ,\ldots,\frac{ \zeta _{r} }{\theta } ),} \end{array}\displaystyle \\& \qquad {}\left . \textstyle\begin{array}{c} { (a_{j} ;\alpha _{j}^{\prime } ,\ldots,\alpha _{j}^{ (r )} ) _{1,p} : (c_{j}^{\prime } ,\gamma _{j}^{\prime } )_{1,p_{{1} } } ;\ldots; (c_{j}^{ (r )} ,\gamma _{j}^{ (r )} ) _{1,p_{r} } } \\ { (b_{j} ;\beta _{j}^{\prime } ,\ldots,\beta _{j}^{ (r )} ) _{1,q} : (d_{j}^{\prime } ,\delta _{j}^{\prime } )_{1,q_{{1} } } ;\ldots; (d_{j}^{ (r )} ,\delta _{j}^{ (r )} ) _{1,q_{r} } } \end{array}\displaystyle \vphantom{\textstyle\begin{array}{c} {a_{1} x^{\zeta _{1} } } \\ {\vdots } \\ {a_{r} x^{\zeta _{r} } } \end{array}\displaystyle } \right ], \end{aligned}$$
(24)

where

$$\begin{aligned}& \varOmega ( i , j , u , v )= D^{q n} l^{m + n } \frac{(-1)^{j} (-j_{i} ) (\alpha )_{j} (-\upsilon )_{u} (-\alpha - q n)_{i} }{i ! j ! u ! v ! (1 - \alpha - j)_{i} } \\& \hphantom{\varOmega ( i , j , u , v )=}{}\times \biggl(-\frac{\beta }{\tau } -\varepsilon n \biggr)_{\upsilon } \biggl(\frac{i +k +ru}{l} \biggr)_{m + n} \biggl( \frac{C}{D} \biggr) ^{j} (-\tau )^{\upsilon } , \end{aligned}$$
(25)
$$\begin{aligned}& \ell =\lambda + \theta + G + \rho l ( m + n ) + \rho r \upsilon + \rho r w + \rho j. \end{aligned}$$
(26)

Proof

In view of definition (17) and by using the general binomial theorem, we expand the term

$$ (\alpha -\beta x )^{-\omega } =\alpha ^{-\omega } \sum _{w=0} ^{\infty }\frac{ (\omega )_{w} }{w !} \biggl( \frac{\beta x}{\alpha } \biggr)^{w} $$

for (\(\vert \frac{\beta x}{\alpha } \vert <1 \)) and the L.H.S. of (24)

$$\begin{aligned}& = \frac{\theta x^{-\theta (\alpha ' + \beta ' )} }{\varGamma (\alpha ')} \int _{0}^{x} \bigl(x^{\theta } -t^{\theta } \bigr)^{\alpha ' - 1} t^{ \lambda + \theta - 1} {}_{2} F_{1} \bigl(\alpha ' + \beta ' , -\eta ' ; \alpha '; 1- {t^{\theta } / x^{\theta } } \bigr) \\& \hphantom{=}{}\times \Im _{n}^{\alpha , \beta , \tau } \bigl[z t^{\rho } ; r, \varepsilon , q, C, D, m, k, l \bigr] S_{L}^{h_{1} ,\ldots,h_{s} } \bigl( y_{1} t^{\eta _{1}} , \dots , y_{s} t^{\eta _{s}} \bigr) H \bigl[a_{1} t^{\zeta _{1} } , \ldots,a_{r} t^{\zeta _{r} } \bigr]\,dt, \end{aligned}$$
(27)

using (21), (8), and (2), it is found that the L.H.S. of (24)

$$\begin{aligned}& = \sum_{\upsilon = 0}^{m + n} \sum _{u = 0}^{\upsilon } \sum_{j = 0} ^{m + n} \sum_{i = 0}^{j} \varOmega ( i, j, u, v) z^{ l ( m + n ) + r \upsilon + j } \\& \hphantom{=}{}\times \sum_{k_{1} , \dots ,k_{s} =0}^{h_{1} k_{1} + \cdots +h_{s} k _{s} \le L } \sum_{w=0}^{\infty }\frac{ (-L )_{h_{1} k_{1} + \cdots +h_{s} k_{s} } ( \upsilon -\varepsilon n )_{w} (\tau )^{w} z ^{r w }}{k_{1} !\cdots k_{s}! w!} A (L; k_{1} , \dots , k _{s} ) \\& \hphantom{=}{}\times {y_{1}^{k_{1}}} \cdots {y_{s}^{k_{s} }} \frac{1}{ (2\pi \omega )^{r} } \int _{L_{1} }\cdots \int _{L_{r} }\phi (\xi _{1} ,\ldots,\xi _{r} ) \Biggl\{ \prod_{i=1}^{r} \theta _{i} (\xi _{i} ) a_{i} ^{\xi _{i} } \Biggr\} \,d\xi _{1} \cdots d\xi _{r} \\& \hphantom{=}{}\times \frac{\theta x^{-\theta (\alpha ' + \beta ' )} }{\varGamma ( \alpha ')} \int _{0}^{x} \bigl(x^{\theta } -t^{\theta } \bigr)^{\alpha ' - 1} t ^{ \lambda + \theta + G+\rho l ( m + n ) + \rho r v + \rho r w + \rho j+L - 1} \\& \hphantom{=}{}\times {}_{2} F_{1} \bigl(\alpha ' + \beta ' , -\eta ' ; \alpha '; 1-{t ^{\theta } / x^{\theta } } \bigr)\,dt, \end{aligned}$$
(28)

where \(G=\sum_{i=1}^{s}\eta _{i} k_{i} \),\(L=\sum_{i=1}^{r}\zeta _{i} \xi _{i} \), \(\varOmega ( i , j , u , v )\) and \(\phi (\xi _{1} ,\ldots, \xi _{r} )\) are defined by (25) and (3), respectively. □

Applying the following result given by Saigo and Saxena ([27], p. 57, Eq. (4.16))

$$\begin{aligned}& A \int _{0}^{x} u^{ \rho - 1} \bigl( x^{ A} - u^{A} \bigr)^{ \alpha - 1} {}_{2 } F_{ 1} \biggl(\alpha + \beta , - \eta ; \alpha ; 1 - \frac{u^{A} }{x ^{A} } \biggr)\,du \\& \quad = \frac{\varGamma ( \alpha ) \varGamma (\frac{\rho }{A} ) \varGamma (\frac{\rho }{A} + \eta - \beta )}{\varGamma (\frac{ \rho }{A} - \beta ) \varGamma (\frac{\rho }{A} + \eta + \alpha )} x^{\alpha A + \rho -A} , \end{aligned}$$
(29)

where \(\Re (\alpha )> 0\), \(\Re (\rho )> 0\), \(\Re ({( \rho / A ) + \eta -\beta )}> 0\), \(A > 0\) in (28) and interchanging the order of integration and summation, we obtain (24).

Next, we prove that

(II)

$$\begin{aligned}& J_{x, \infty , \theta ; r, \varepsilon , q; m, k, l} ^{\rho ; \alpha , \beta , \tau ; C, D, \alpha ', \beta ', \eta ' } \bigl\{ t^{ \lambda } S_{L}^{h_{1} ,\ldots,h_{s} } \bigl( y_{1} t^{\eta _{1}} , \dots , y_{s} t^{\eta _{s}} \bigr) H \bigl[a_{1} t^{\zeta _{1} } , \ldots,a_{r} t^{\zeta _{r} } \bigr] \bigr\} \\& \quad = \sum_{\upsilon = 0}^{m + n} \sum _{u = 0}^{\upsilon } \sum_{j = 0} ^{m + n} \sum_{i = 0}^{j} \varOmega ( i, j, u, v ) z^{ l ( m + n ) + r \upsilon + j } \\& \qquad {}\times \sum_{k_{1} , \dots ,k_{s} =0}^{h_{1} k_{1} + \cdots +h_{s} k _{s} \le L } \sum _{w=0}^{\infty }\frac{ (-L )_{h_{1} k_{1} + \cdots +h_{s} k_{s} } ( \upsilon -\varepsilon n )_{w} (\tau )^{w} z ^{r w }}{k_{1} !\cdots k_{s}! w!} A (L; k_{1} , \dots , k_{s} ) \\& \qquad {}\times {y_{1}^{k_{1}}} \cdots {y_{s}^{k_{s} }} x^{ \lambda +G+ \rho [ l ( m + n ) + r\upsilon + r w + j ] - \theta \beta '}H_{p+2,q+2:p_{1} ,q_{1} ;\dots ,p_{r} ,q_{r} }^{0,n+2: m_{1} ,n _{1} ;\dots ,m_{r} ,n_{r} } \left [\left. \textstyle\begin{array}{c} {a_{1}t^{\xi _{1} } } \\ {\vdots } \\ {a_{r}t^{\xi _{r} } } \end{array}\displaystyle \right| \right . \\& \qquad \textstyle\begin{array}{c} { (1 - \alpha ' - \beta '- \frac{\ell '}{\theta } , \frac{\zeta _{1} }{\theta } ,\ldots,\frac{\zeta _{r} }{\theta } ), (1 + \eta ' - \frac{\ell '}{\theta } , \frac{\zeta _{1} }{\theta } ,\ldots,\frac{ \zeta _{r} }{\theta } ) , } \\ { (1 - \alpha ' - \frac{\ell '}{\theta }, \frac{\zeta _{1} }{ \theta } ,\ldots,\frac{\zeta _{r} }{\theta } ), (1 - \alpha ' - \beta ' + \eta '- \frac{\ell '}{\theta } , \frac{\zeta _{1} }{\theta } ,\ldots,\frac{\zeta _{r} }{\theta } ) ,} \end{array}\displaystyle \\& \qquad \left . \textstyle\begin{array}{c} { (a_{j} ;\alpha _{j}^{\prime } ,\ldots,\alpha _{j}^{ (r )} ) _{1,p} : (c_{j}^{\prime } ,\gamma _{j}^{\prime } )_{1,p_{{1} } } ;\ldots; (c_{j}^{ (r )} ,\gamma _{j}^{ (r )} ) _{1,p_{r} } } \\ { (b_{j} ;\beta _{j}^{\prime } ,\ldots,\beta _{j}^{ (r )} ) _{1,q} : (d_{j}^{\prime } ,\delta _{j}^{\prime } )_{1,q_{{1} } };\ldots; (d_{j}^{ (r )} ,\delta _{j}^{ (r )} ) _{1,q_{r} } } \end{array}\displaystyle \vphantom{\textstyle\begin{array}{c} {a_{1}t^{\xi _{1} } } \\ {\vdots } \\ {a_{r}t^{\xi _{r} } } \end{array}\displaystyle } \right ], \end{aligned}$$
(30)

where

$$ \ell ' = \lambda + \theta +G+ \rho l ( m + n ) + \rho r v + \rho r w + \rho j - \theta \alpha ' - \theta \beta '; $$
(31)

\(\varOmega ( i, j, u, v )\) and \(S_{L}^{h_{1} ,\ldots,h_{s} } (x ) \) are defined in (25) and (8).

Proof

In view of definition (18), the L.H.S. of (30)

$$\begin{aligned}& = \frac{\theta }{\varGamma (\alpha ')} \int _{x}^{\infty } \bigl(t^{\theta } -x ^{\theta } \bigr)^{\alpha ' - 1} t^{\lambda - \theta (\alpha ' + \beta ' ) + \theta - 1} {}_{2} F_{1} \bigl(\alpha ' + \beta ' , -\eta ' ; \alpha '; 1- {x^{\theta } / t^{\theta } } \bigr) \\& \quad {}\times \Im _{n}^{\alpha , \beta , \tau } \bigl[zt^{\rho } ; r, \varepsilon , q, C, D, m, k, l \bigr] S_{L}^{h_{1} ,\ldots,h_{s} } \bigl( y_{1} t^{\eta _{1}} , \dots , y_{s} t ^{\eta _{s}} \bigr)H \bigl[a_{1} t^{\zeta _{1} } ,\ldots,a_{r} t^{\zeta _{r} } \bigr]\,dt. \end{aligned}$$
(32)

If we apply (21), (8), and (2) in the above term, we get

$$\begin{aligned}& = \sum_{\upsilon = 0}^{m + n} \sum _{u = 0}^{\upsilon } \sum_{j = 0} ^{m + n} \sum_{i = 0}^{j} \varOmega ( i , j , u , v ) z^{ l ( m + n ) + r \upsilon + j } \\& \quad {}\times \sum_{k_{1} , \dots ,k_{s} =0}^{h_{1} k_{1} + \cdots +h_{s} k _{s} \le L } \sum _{w=0}^{\infty }\frac{ (-L )_{h_{1} k_{1} + \cdots +h_{s} k_{s} } ( \upsilon -\varepsilon n )_{w} (\tau )^{w} z ^{r w }}{k_{1} !\cdots k_{s}! w!} A (L; k_{1} , \dots , k _{s} ) {y_{1}^{k_{1}}} \cdots {y_{s}^{k_{s} }} \\& \quad {}\times \frac{1}{ (2\pi \omega )^{r} } \int _{L_{1} }\cdots \int _{L_{r} }\phi (\xi _{1} ,\ldots,\xi _{r} ) \Biggl\{ \prod_{i=1}^{r} \theta _{i} (\xi _{i} )a_{i} ^{\xi _{i} } \Biggr\} \,d\xi _{1} \dots d\xi _{r} \\& \quad {}\times \frac{\theta }{\varGamma (\alpha ')} \int _{x}^{\infty } \bigl(t^{ \theta } -x^{\theta } \bigr)^{\alpha ' - 1} t^{ \lambda + \theta + G+ \rho l ( m + n ) + \rho r v + \rho j - \theta ( \alpha ' + \beta ' ) +H - 1} \\& \quad {}\times {}_{2} F_{1} \bigl(\alpha ' + \beta ' , -\eta ' ; \alpha '; 1-{x ^{\theta } / t^{\theta } } \bigr)\,dt. \end{aligned}$$
(33)

Now, by applying the integral given by Saigo and Saxena ([27], p. 57, Eq. (4.17))

$$\begin{aligned}& A \int _{x}^{\infty } u^{ \rho - 1} \bigl( u^{ A} - x^{A} \bigr)^{ \alpha - 1} {}_{2 } F_{ 1} \biggl(\alpha + \beta , - \eta ; \alpha ; 1 - \frac{x ^{A} }{u^{A} } \biggr)\,du \\& \quad = \frac{\varGamma ( \alpha ) \varGamma (1 - \alpha - \frac{\rho }{A} ) \varGamma (1 - \alpha - \beta + \eta - \frac{\rho }{A} )}{ \varGamma (1 - \alpha - \beta - \frac{\rho }{A} ) \varGamma (1 + \eta - \frac{\rho }{A} )} x ^{\alpha A + \rho - A} , \end{aligned}$$
(34)

where \(\Re (\alpha ) > 0\), \(\Re (1 - \alpha -{\rho / A} ) > 0\), \(\Re (1 - \alpha - \beta + \eta - {\rho / A} ) > 0\), \(A > 0\) in (33) and interchanging the order of integration and summation, we arrive at the result (30). □

3 Special cases

(i) If we use the identity \(I_{0 , x , \theta }^{\alpha ', - \alpha ', \eta '} f = R_{0 , x , \theta }^{\alpha '} f \) with \(\theta = 1\) in (24), we find that

$$\begin{aligned}& R _{0, x, 1; r, \varepsilon , q; m, k, l}^{\rho ; \alpha , \beta , \tau ; C, D, \alpha ' } \bigl\{ t^{ \lambda } S_{L}^{h_{1} ,\ldots,h _{s} } \bigl( y_{1} t^{\eta _{1}} , \dots , y_{s} t^{\eta _{s}} \bigr) H \bigl[a_{1} t^{\zeta _{1} } , \ldots,a_{r} t^{\zeta _{r} } \bigr] \bigr\} \\& \quad = \sum_{\upsilon = 0}^{m + n} \sum _{u = 0}^{\upsilon } \sum_{j = 0} ^{m + n} \sum_{i = 0}^{j} \varOmega ( i , j , u , v ) z^{ l ( m + n ) + r \upsilon + j } \\& \qquad {}\times \sum_{k_{1} , \dots ,k_{s} =0}^{h_{1} k_{1} + \cdots +h_{s} k _{s} \le L } \sum _{w=0}^{\infty }\frac{ (-L )_{h_{1} k_{1} + \cdots +h_{s} k_{s} } ( \upsilon -\varepsilon n )_{w} (\tau )^{w} z ^{r w }}{k_{1} !\cdots k_{s}! w!} \\& \qquad {}\times A (L; k_{1} , \dots , k_{s} ) {y_{1}^{k_{1}}} \cdots {y_{s}^{k_{s} }} x^{ \lambda + G+ \rho [ l ( m + n ) + r\upsilon + r w + j ] - \alpha '} \\& \qquad {}\times H_{p+1,q+1:p_{1} ,q_{1} ;\dots ,p_{r} ,q_{r} }^{0,n+1:m_{1} ,n _{1} ;\dots ,m_{r} ,n_{r} } \left [ \left.\textstyle\begin{array}{c} {a_{1} x^{\zeta _{1} } } \\ {\vdots } \\ {a_{r} x^{\zeta _{r} } } \end{array}\displaystyle \right| \textstyle\begin{array}{c} { (1 - \varLambda , \zeta _{1} ,\ldots,\zeta _{r} ) , } \\ { (1 - \varLambda - \alpha ' , \zeta _{1} ,\ldots,\zeta _{r} ),} \end{array}\displaystyle \right . \\& \qquad {} \left . \textstyle\begin{array}{c} { (a_{j} ;\alpha _{j}^{\prime } ,\ldots,\alpha _{j}^{ (r )} ) _{1,p} : (c_{j}^{\prime } ,\gamma _{j}^{\prime } )_{1,q_{{1} } } ;\ldots; (c_{j}^{ (r )} ,\gamma _{j}^{ (r )} ) _{1,p_{r} } } \\ { (b_{j} ;\beta _{j}^{\prime } ,\ldots,\beta _{j}^{ (r )} ) _{1,p} : (d_{j}^{\prime } ,\delta _{j}^{\prime } )_{1,q_{{1} } } ;\ldots; (d_{j}^{ (r )} ,\delta _{j}^{ (r )} ) _{1,q_{r} } } \end{array}\displaystyle \vphantom{\textstyle\begin{array}{c} {a_{1} x^{\zeta _{1} } } \\ {\vdots } \\ {a_{r} x^{\zeta _{r} } } \end{array}\displaystyle } \right ], \end{aligned}$$
(35)

where \(\varLambda =\lambda + G + \rho l ( m + n ) + \rho r \upsilon + \rho r w + \rho j+1\).

(ii) The formula \(J_{x, \infty , \theta }^{\alpha ', -\alpha ', \eta '} f = W_{x , \infty , \theta }^{\alpha '} f \) with \(\theta = 1\), when used in (30), gives

$$\begin{aligned}& W _{x, \infty , 1; r, \varepsilon , q; m, k, l}^{\rho ; \alpha , \beta , \tau ; C, D, \alpha ' } \bigl\{ t^{ \lambda } S_{L}^{h_{1} ,\ldots,h _{s} } \bigl( y_{1} t^{\eta _{1}} , \dots , y_{s} t^{\eta _{s}} \bigr) H \bigl[a_{1} t^{\zeta _{1} } , \ldots,a_{r} t^{\zeta _{r} } \bigr] \bigr\} \\& \quad = \sum_{\upsilon = 0}^{m + n} \sum _{u = 0}^{\upsilon } \sum_{j = 0} ^{m + n} \sum_{i = 0}^{j} \varOmega ( i , j , u , v ) z^{ l ( m + n ) + r \upsilon + j } \\& \qquad {}\times \sum_{k_{1} , \dots ,k_{s} =0}^{h_{1} k_{1} + \cdots +h_{s} k _{s} \le L } \sum _{w=0}^{\infty }\frac{ (-L )_{h_{1} k_{1} + \cdots +h_{s} k_{s} } ( \upsilon -\varepsilon n )_{w} (\tau )^{w} z ^{r w }}{k_{1} !\cdots k_{s}! w!} \\& \qquad {}\times A (L; k_{1} , \dots , k_{s} ) {y_{1}^{k_{1}}} \cdots {y_{s}^{k_{s} }} x^{ \lambda + G+ \rho [ l ( m + n ) + r\upsilon + r w + j ] +\alpha '} \\& \qquad {}\times H_{p+1,q+1:p_{1} ,q_{1} ;\dots ,p_{r} ,q_{r} }^{0,n+1:m_{1} ,n _{1} ;\dots ,m_{r} ,n_{r} } \left [ \left.\textstyle\begin{array}{c} {a_{1}t^{\xi _{1} } } \\ {\vdots } \\ {a_{r}t^{\xi _{r} } } \end{array}\displaystyle \right| \textstyle\begin{array}{c} { (1 - \varLambda ' , \zeta _{1} ,\ldots,\zeta _{r} ), } \\ { (1 - \alpha ' - \varLambda ' , \zeta _{1} ,\ldots,\zeta _{r} ), } \end{array}\displaystyle \right . \\& \qquad {} \left . \textstyle\begin{array}{c} { (a_{j} ;\alpha _{j}^{\prime } ,\ldots,\alpha _{j}^{ (r )} ) _{1,p} : (c_{j}^{\prime } ,\gamma _{j}^{\prime } )_{1,p_{{1} } } ;\ldots; (c_{j}^{ (r )} ,\gamma _{j}^{ (r )} ) _{1,p_{r} } } \\ { (b_{j} ;\beta _{j}^{\prime } ,\ldots,\beta _{j}^{ (r )} ) _{1,q} : (d_{j}^{\prime } ,\delta _{j}^{\prime } )_{1,q_{{1} } } ;\ldots; (d_{j}^{ (r )} ,\delta _{j}^{ (r )} ) _{1,q_{r} } } \end{array}\displaystyle \vphantom{\textstyle\begin{array}{c} {a_{1}t^{\xi _{1} } } \\ {\vdots } \\ {a_{r}t^{\xi _{r} } } \end{array}\displaystyle } \right ], \end{aligned}$$
(36)

where \(\varLambda '=\lambda + G + \rho l ( m + n ) + \rho r \upsilon + \rho r w + \rho j + 1 \).

(iii) If we take \(n=p=q=0\) in (24) and (30) with respect to H-function respectively, we obtain two fractional integral formulas involving product of the r, H-functions stated as follows:

$$\begin{aligned}& I_{0, x, \theta ; r, \varepsilon , q; m, k, l}^{\rho ; \alpha , \beta , \tau ; C, D, \alpha ', \beta ', \eta ' } \left \{t^{ \lambda } S_{L}^{h_{1} ,\ldots,h_{s} } \bigl( y_{1} t^{\eta _{1}} , \dots , y_{s} t^{\eta _{s}} \bigr) \prod _{i=1}^{r}H_{p_{i} , q _{i} }^{ m_{ i} , n _{i}} \left [ a_{i} t^{\zeta _{i}} \left| \textstyle\begin{array}{c} { (c_{j}^{(i)}, \gamma _{j}^{(i)} )_{1, p_{i}}} \\ { (d_{j}^{(i)}, \delta _{j}^{(i)} )_{1, q_{i}}} \end{array}\displaystyle \right. \right ] \right \} \\& \quad = \sum_{\upsilon = 0}^{m + n} \sum _{u = 0}^{\upsilon } \sum_{j = 0} ^{m + n} \sum_{i = 0}^{j} \varOmega ( i , j , u , v ) z^{ l ( m + n ) + r \upsilon + j } \\& \qquad {}\times \sum_{k_{1} , \dots ,k_{s} =0}^{h_{1} k_{1} + \cdots +h_{s} k _{s} \le L } \sum _{w=0}^{\infty }\frac{ (-L )_{h_{1} k_{1} + \cdots +h_{s} k_{s} } ( \upsilon -\varepsilon n )_{w} (\tau )^{w} z ^{r w }}{k_{1} !\cdots k_{s}! w!} \\& \qquad {}\times A (L; k_{1} , \dots , k_{s} ) {y_{1}^{k_{1}}} \cdots {y_{s}^{k_{s} }} x^{ \lambda +G+ \rho [ l ( m + n ) + r\upsilon + r w + j ] - \theta \beta '} \\& \qquad {}\times H_{2,2:p_{1} ,q_{1} ;\dots ,p_{r} ,q_{r} }^{0,2:m_{1} ,n_{1} ;\dots ,m_{r} ,n _{r} } \left [ \textstyle\begin{array}{c} a_{i} x^{\zeta _{i}} \end{array}\displaystyle \left| \textstyle\begin{array}{c} { (1 - \frac{\ell }{\theta }, \frac{\zeta _{1} }{\theta } ,\ldots,\frac{ \zeta _{r} }{\theta } ) , } \\ { (1 - \frac{\ell }{\theta } + \beta ' , \frac{\zeta _{1} }{ \theta } ,\ldots,\frac{\zeta _{r} }{\theta } ),} \end{array}\displaystyle \right. \right . \\& \qquad \left . \textstyle\begin{array}{c} { (1 - \frac{\ell }{\theta } - \eta ' + \beta ' , \frac{\zeta _{1} }{\theta } ,\ldots,\frac{\zeta _{r} }{\theta } ) , (c _{j}^{ (1 )} ,\gamma _{j}^{ (1 )} )_{1,p _{i} } } \\ { (1 - \frac{\ell }{\theta } - \eta ' - \alpha ' , \frac{\zeta _{1} }{\theta } ,\ldots,\frac{\zeta _{r} }{\theta } ), (d _{j}^{ (1 )} ,\delta _{j}^{ (1 )} )_{1,q _{i} } } \end{array}\displaystyle \right ]; \end{aligned}$$
(37)

and

$$\begin{aligned}& J_{x, \infty , \theta ; r, \varepsilon , q; m, k, l}^{\rho ; \alpha , \beta , \tau ; C, D, \alpha ', \beta ', \eta ' } \left \{t^{ \lambda } S_{L}^{h_{1} ,\ldots,h_{s} } \bigl( y_{1} t^{\eta _{1}} , \dots , y_{s} t^{\eta _{s}} \bigr) \prod _{i=1}^{r}H_{p_{i} , q _{i} }^{ m_{ i} , n _{i}} \left [ a_{i} t^{\zeta _{i}} \left| \textstyle\begin{array}{c} { (c_{j}^{(i)}, \gamma _{j}^{(i)} )_{1, p_{i}}} \\ { (d_{j}^{(i)}, \delta _{j}^{(i)} )_{1, q_{i}}} \end{array}\displaystyle \right. \right ] \right \} \\& \quad = \sum_{\upsilon = 0}^{m + n} \sum _{u = 0}^{\upsilon } \sum_{j = 0} ^{m + n} \sum_{i = 0}^{j} \varOmega ( i , j , u , v ) z^{ l ( m + n ) + r \upsilon + j } \\& \qquad {}\times \sum_{k_{1} , \dots ,k_{s} =0}^{h_{1} k_{1} + \cdots +h_{s} k _{s} \le L } \sum _{w=0}^{\infty }\frac{ (-L )_{h_{1} k_{1} + \cdots +h_{s} k_{s} } ( \upsilon -\varepsilon n )_{w} (\tau )^{w} z ^{r w }}{k_{1} !\cdots k_{s}! w!} \\& \qquad {}\times A (L; k_{1} , \dots , k_{s} ) {y_{1}^{k_{1}}} \cdots {y_{s}^{k_{s} }} x^{ \lambda +G+ \rho [ l ( m + n ) + r\upsilon + r w + j ] - \theta \beta '} \\& \qquad {}\times H_{2, 2:p_{1} ,q_{1} ;\dots ,p_{r} ,q_{r} }^{0,2: m_{1} ,n_{1} ;\dots ,m _{r} ,n_{r} } \left [ \textstyle\begin{array}{c} a_{i} x^{\zeta _{i}} \end{array}\displaystyle \left| \textstyle\begin{array}{c} { (1 - \alpha ' - \beta '- \frac{\ell '}{\theta } , \frac{\zeta _{1} }{\theta } ,\ldots,\frac{\zeta _{r} }{\theta } ),} \\ { (1 - \alpha ' - \frac{\ell '}{\theta }, \frac{\zeta _{1} }{ \theta } ,\ldots,\frac{\zeta _{r} }{\theta } ),} \end{array}\displaystyle \right. \right . \\& \qquad {} \left . \textstyle\begin{array}{c} { (1 + \eta ' - \frac{\ell '}{\theta } , \frac{\zeta _{1} }{ \theta } ,\ldots,\frac{\zeta _{r} }{\theta } ) , (c_{j}^{ (1 )} ,\gamma _{j}^{ (1 )} )_{1,p_{r} } } \\ { (1 - \alpha ' - \beta ' + \eta '- \frac{\ell '}{\theta } , \frac{ \zeta _{1} }{\theta } ,\ldots,\frac{\zeta _{r} }{\theta } ) , (d _{j}^{ (1 )} ,\delta _{j}^{ (1 )} )_{1,q _{r} } } \end{array}\displaystyle \right ]. \end{aligned}$$
(38)

(iv) If we set \(S_{L}^{h_{j} } (x)\) to reduce to unity, i.e., \(S_{0}^{ h_{j} }(x) \to 1\), in (24) and (30) respectively, then we arrive at the interesting results.

$$\begin{aligned}& I_{0, x, \theta ; r, \varepsilon , q; m, k, l}^{\rho ; \alpha , \beta , \tau ; C, D, \alpha ', \beta ', \eta ' } \bigl\{ t^{ \lambda } H \bigl[a_{1} t^{\zeta _{1} } ,\ldots,a_{r} t^{\zeta _{r} } \bigr] \bigr\} \\& \quad = \sum_{\upsilon = 0}^{m + n} \sum _{u = 0}^{\upsilon } \sum_{j = 0} ^{m + n} \sum_{i = 0}^{j} \varOmega ( i , j , u , v ) z^{ l ( m + n ) + r \upsilon + j } \sum_{w=0}^{\infty } \frac{( \upsilon -\varepsilon n )_{w} (\tau )^{w} z^{r w } }{w!} \\& \qquad {}\times x^{ \lambda + \rho [ l ( m + n ) + r\upsilon + r w + j ] - \theta \beta '}H_{p+2,q+2:p_{1} ,q_{1} ;\dots ,p_{r} ,q_{r} }^{0,n+2:m _{1} ,n_{1} ;\dots ,m_{r} ,n_{r} } \left [\left. \textstyle\begin{array}{c} {a_{1} x^{\zeta _{1} } } \\ {\vdots } \\ {a_{r} x^{\zeta _{r} } } \end{array}\displaystyle \right| \right . \\& \qquad {}\textstyle\begin{array}{c} { (1 - \frac{\ell ''}{\theta } , \frac{\zeta _{1} }{\theta } ,\ldots,\frac{ \zeta _{r} }{\theta } ) , (1 - \frac{\ell ''}{\theta } - \eta ' + \beta ' , \frac{\zeta _{1} }{\theta } ,\ldots,\frac{\zeta _{r} }{ \theta } ) , } \\ { (1 - \frac{\ell ''}{\theta } + \beta ' , \frac{\zeta _{1} }{ \theta } ,\ldots,\frac{\zeta _{r} }{\theta } ), (1 - \frac{ \ell ''}{\theta } - \eta ' - \alpha ' , \frac{\zeta _{1} }{\theta } ,\ldots,\frac{ \zeta _{r} }{\theta } ), } \end{array}\displaystyle \\& \qquad {} \left . \textstyle\begin{array}{c} { (a_{j} ;\alpha _{j}^{\prime } ,\ldots,\alpha _{j}^{ (r )} ) _{1,p} : (c_{j}^{\prime } ,\gamma _{j}^{\prime } )_{1,p_{{1} } } ;\ldots; (c_{j}^{ (r )} ,\gamma _{j}^{ (r )} ) _{1,p_{r} } } \\ { (b_{j} ;\beta _{j}^{\prime } ,\ldots,\beta _{j}^{ (r )} ) _{1,q} : (d_{j}^{\prime } ,\delta _{j}^{\prime } )_{1,q_{{1} } } ;\ldots; (d_{j}^{ (r )} ,\delta _{j}^{ (r )} ) _{1,q_{r} } } \end{array}\displaystyle \vphantom{\textstyle\begin{array}{c} {a_{1} x^{\zeta _{1} } } \\ {\vdots } \\ {a_{r} x^{\zeta _{r} } } \end{array}\displaystyle } \right ], \end{aligned}$$
(39)

where \(\ell ''=\lambda + \theta + \rho l ( m + n ) + \rho r \upsilon + \rho r w + \rho j\); and

$$\begin{aligned}& J_{x, \infty , \theta ; r, \varepsilon , q; m, k, l}^{\rho ; \alpha , \beta , \tau ; C, D, \alpha ', \beta ', \eta ' } \bigl\{ t^{ \lambda } H \bigl[a_{1} t^{\zeta _{1} } ,\ldots,a_{r} t^{\zeta _{r} } \bigr] \bigr\} \\& \quad = \sum_{\upsilon = 0}^{m + n} \sum _{u = 0}^{\upsilon } \sum_{j = 0} ^{m + n} \sum_{i = 0}^{j} \varOmega ( i , j , u , v ) z^{ l ( m + n ) + r \upsilon + j } \sum_{w=0}^{\infty } \frac{( \upsilon -\varepsilon n )_{w} (\tau )^{w} z^{r w } }{w!} \\& \qquad {}\times x^{ \lambda + \rho [ l ( m + n ) + r\upsilon + r w + j ] - \theta \beta '} H_{p+2,q+2:p_{1} ,q_{1} ;\dots ,p_{r} ,q_{r} }^{0,n+2:m _{1} ,n_{1} ;\dots ,m_{r} ,n_{r} } \left [\left. \textstyle\begin{array}{c} {a_{1}t^{\xi _{1} } } \\ {\vdots } \\ {a_{r}t^{\xi _{r} } } \end{array}\displaystyle \right| \right . \\& \qquad {}\textstyle\begin{array}{c} { (1 - \alpha ' - \beta '- \frac{\ell '''}{\theta } , \frac{\zeta _{1} }{\theta } ,\ldots,\frac{\zeta _{r} }{\theta } ), (1 + \eta ' - \frac{\ell '''}{\theta } , \frac{\zeta _{1} }{\theta } ,\ldots,\frac{ \zeta _{r} }{\theta } ) , } \\ { (1 - \alpha ' - \frac{\ell '''}{\theta } , \frac{\zeta _{1} }{ \theta } ,\ldots,\frac{\zeta _{r} }{\theta } ), (1 - \alpha ' - \beta ' + \eta '- \frac{\ell '''}{\theta } , \frac{\zeta _{1} }{ \theta } ,\ldots,\frac{\zeta _{r} }{\theta } ) , } \end{array}\displaystyle \\& \qquad {} \left . \textstyle\begin{array}{c} { (a_{j} ;\alpha _{j}^{\prime } ,\ldots,\alpha _{j}^{ (r )} ) _{1,p} : (c_{j}^{\prime } ,\gamma _{j}^{\prime } )_{1,p_{{1} } } ;\ldots; (c_{j}^{ (r )} ,\gamma _{j}^{ (r )} ) _{1,p_{r} } } \\ { (b_{j} ;\beta _{j}^{\prime } ,\ldots,\beta _{j}^{ (r )} ) _{1,q} : (d_{j}^{\prime } ,\delta _{j}^{\prime } )_{1,q_{{1} } } ;\ldots; (d_{j}^{ (r )} ,\delta _{j}^{ (r )} ) _{1,q_{r} } } \end{array}\displaystyle \vphantom{\textstyle\begin{array}{c} {a_{1}t^{\xi _{1} } } \\ {\vdots } \\ {a_{r}t^{\xi _{r} } } \end{array}\displaystyle } \right ], \end{aligned}$$
(40)

where \(\ell ''' = \lambda + \theta + \rho l ( m + n ) + \rho r v + \rho r w +\rho j - \theta \alpha ' - \theta \beta '\).

4 Concluding remarks

The modified Saigo fractional integral operators have advantage that they generalize the Saigo, Erdélyi–Kober, Riemann–Liouville, and Weyl fractional integral operators. Therefore, several authors called them general operators. We also derived analogous results in the form of Riemann–Liouville and Weyl fractional integral operators, which have been depicted in corollaries. Now, we conclude this paper by interesting results that can be derived as the specific cases of our leading results I and II in the form of I-function and H-function. On the other hand, by putting the appropriate values to the arbitrary constant, the family of polynomials (defined by (8)) provide several well-known classical orthogonal polynomials as its special cases, which includes the Hermite, the Laguerre, the Jacobi, the Konhauser polynomials, and so on. Finally, it is interesting to observe that the results given earlier by Saxena et al. ([28], Eqs. (2.1), (2.11))) can be derived from the results (24) and (30) of this paper by virtue of the identity \(r=1\).