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Modified Saigo fractional integral operators involving multivariable H-function and general class of multivariable polynomials

Abstract

In this paper, we establish modified Saigo fractional integral operators involving the product of a general class of multivariable polynomials and the multivariable H-function. The results established here are of general nature and provide extension of some results obtained recently by Saxena et al.

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 ‘rH-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)

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). □

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

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 \(r=1\).

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Suthar, D.L., Debalkie, B. & Andualem, M. Modified Saigo fractional integral operators involving multivariable H-function and general class of multivariable polynomials. Adv Differ Equ 2019, 213 (2019). https://doi.org/10.1186/s13662-019-2150-0

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MSC

  • 26A33
  • 26C05

Keywords

  • Multivariable H-function
  • General class of multivariable polynomial
  • Modified Saigo operators