2 Preliminaries
The well-known lower and upper gamma functions of incomplete type [16 ] γ (𝔳,𝔜) and Γ (𝔳,𝔜) respectively, are presented as:
(1)
γ ( v , Y ) = ∫ 0 Y u v − 1 e − u d u , ( ℜ ( v ) > 0 ; Y ≧ 0 ) ,
\gamma (\mathfrak{v},\mathfrak{Y}) = \int_0^\mathfrak{Y} {\mathfrak{u}^{\mathfrak{v} - 1}}\;{e^{ - \mathfrak{u}}}\;d\mathfrak{u},\quad \quad (\Re (\mathfrak{v}) > 0;{\kern 1pt} {\kern 1pt} \mathfrak{Y} \mathbin{\lower.3ex\hbox{$\buildrel>\over{\smash{\scriptstyle=}\vphantom{_x}}$}} 0),
and
(2)
Γ ( v , Y ) = ∫ Y ∞ u v − 1 e − u d u , ( Y ≧ 0 ; ℜ ( v ) > 0 when Y = 0 ) .
\Gamma (\mathfrak{v},\mathfrak{Y}) = \int_\mathfrak{Y}^\infty {\mathfrak{u}^{\mathfrak{v} - 1}}\;{e^{ - \mathfrak{u}}}\;d\mathfrak{u},\quad \quad (\mathfrak{Y} \mathbin{\lower.3ex\hbox{$\buildrel>\over{\smash{\scriptstyle=}\vphantom{_x}}$}} 0;{\kern 1pt} {\kern 1pt} \Re (\mathfrak{v}) > 0\quad when\quad \mathfrak{Y} = 0).
The following connection (sometimes referred to as the decomposition formula) is satisfied by these incomplete gamma functions.
(3)
γ ( v , Y ) + Γ ( v , Y ) = Γ ( v ) , ( ℜ ( v ) > 0 ) .
\gamma (\mathfrak{v},\mathfrak{Y}) + \Gamma (\mathfrak{v},\mathfrak{Y}) = \Gamma (\mathfrak{v}),\quad \quad (\Re (\mathfrak{v}) > 0).
The Srivastava investigated a broad category of polynomials [17 ], which is described as follows (see [18 ] also):
(4)
S Q P [ t ] = ∑ O = 0 [ Q P ] ( − Q ) P O O ! A Q , O t O ,
S_\mathfrak{Q}^\mathfrak{P}[t] = \sum\limits_{\mathfrak{O} = 0}^{[\mathfrak{Q}\mathfrak{P}]} \frac{{{{( - \mathfrak{Q})}_{\mathfrak{P}\mathfrak{O}}}}}{{\mathfrak{O}!}}{A_{\mathfrak{Q},\mathfrak{O}}}{\kern 1pt} {t^\mathfrak{O}}{\kern 1pt} ,
where 𝔓 ∈ ℤ+ and A 𝔔,𝔒 are real or complex numbers arbitrary constants.
The notations [k ] indicates the floor function and (κ )μ denote the Pochhammer symbol described by:
( κ ) 0 = 1 and ( κ ) μ = Γ ( κ + μ ) Γ ( κ ) , ( μ ∈ ℂ ) ,
{(\kappa )_0} = 1\quad and\quad {(\kappa )_\mu } = \frac{{\Gamma (\kappa + \mu )}}{{\Gamma (\kappa )}},\quad (\mu \in \mathbb{C}),
in the form of the Gamma function. Numerous FC results relating to the incomplete ℵ-functions are presented in this paper. For ϛ , ϛ ', ϰ , ϰ ', ϖ ∈ ℂ and x > 0 with ℜ (ϖ ) > 0, the MSM FIO [19 ] with the left-and right-hand sides are explained as:
(5)
( ℐ 0 + ς , ς ' , ϰ , ϰ ' , ϖ f ) ( x ) = x − ς Γ ( ϖ ) ∫ 0 x ( x − y ) ϖ − 1 y − ς ' × F 3 ( ς , ς ' , ϰ , ϰ ' ; ϖ ; 1 − y x , 1 − x y ) f ( y ) d y ,
\left( {\mathcal{I}_{0 + }^{\varsigma ,{\kern 1pt} {\varsigma ^\prime },\varkappa ,{\kern 1pt} {\varkappa ^\prime },\varpi }f} \right)(x) = \frac{{{x^{ - \varsigma }}}}{{\Gamma (\varpi )}}\int_0^x {(x - y)^{\varpi - 1}}{y^{ - {\varsigma ^\prime }}} \times {F_3}\left( {\varsigma ,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime };{\kern 1pt} \varpi ;{\kern 1pt} 1 - \frac{y}{x},{\kern 1pt} 1 - \frac{x}{y}} \right)f(y)dy,
and
(6)
( ℐ − ς , ς ' , ϰ , ϰ ' , ϖ f ) ( x ) = x − ς ' Γ ( ϖ ) ∫ x ∞ ( y − x ) ϖ − 1 y − ς × F 3 ( ς , ς ' , ϰ , ϰ ' ; ϖ ; 1 − x y , 1 − y x ) f ( y ) d y ,
\left( {\mathcal{I}_ - ^{\varsigma ,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },{\kern 1pt} \varpi }f} \right)(x) = \frac{{{x^{ - {\varsigma ^\prime }}}}}{{\Gamma (\varpi )}}\int_x^\infty {(y - x)^{\varpi - 1}}{y^{ - \varsigma }} \times {F_3}\left( {\varsigma ,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime };{\kern 1pt} \varpi ;{\kern 1pt} 1 - \frac{x}{y},{\kern 1pt} 1 - \frac{y}{x}} \right)f(y)dy,
respectively.
According to a description, the left-and right-hand handed MSM fractional differential operators are (see [20 ]):
(7)
( D 0 + ς , ς ' , ϰ , ϰ ' , ϖ f ) ( x ) = ( d d x ) α ( ℐ 0 + − ς ' , − ς , − ϰ ' + α , − ϰ , − ϖ + α f ) ( x ) ,
\left( {\mathcal{D}_{0 + }^{\varsigma ,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },{\kern 1pt} \varpi }f} \right)(x) = {\left( {\frac{d}{{dx}}} \right)^\alpha }\left( {\mathcal{I}_{0 + }^{ - {\varsigma ^\prime },{\kern 1pt} - \varsigma ,{\kern 1pt} - {\varkappa ^\prime } + \alpha ,{\kern 1pt} - \varkappa ,{\kern 1pt} - \varpi + \alpha }f} \right)(x),
and
(8)
( D − ς , ς ' , ϰ , ϰ ' , ϖ f ) ( x ) = ( − d d x ) [ α ] ( ℐ − − ς ' , − ς , − ϰ ' , − ϰ + α , − ϖ + α f ) ( x ) ,
\left( {\mathcal{D}_ - ^{\varsigma ,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },{\kern 1pt} \varpi }f} \right)(x) = {\left( { - \frac{d}{{dx}}} \right)^{[\alpha ]}}\left( {\mathcal{I}_ - ^{ - {\varsigma ^\prime },{\kern 1pt} - \varsigma ,{\kern 1pt} - {\varkappa ^\prime },{\kern 1pt} - \varkappa + \alpha ,{\kern 1pt} - \varpi + \alpha }f} \right)(x),
where, α = [ℜ (ϖ )] + 1 and [[ℜ(ϖ )] represent the integer component in [ℜ(ϖ ). For max {|x |, |y |} < 1, the third Appell function F 3 has the following definition:
(9)
F 3 ( ς , ς ' , ϰ , ϰ ' ; ϖ ; x ; y ) = ∑ i , j = 0 ∞ ( ς ) i ( ς ' ) j ( ϰ ) i ( ϰ ' ) j ( ϖ ) i + j x i y j i ! j ! ,
{F_3}(\varsigma ,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime };{\kern 1pt} \varpi ;{\kern 1pt} x;{\kern 1pt} y) = \sum\limits_{i,{\kern 1pt} j = 0}^\infty \frac{{{{(\varsigma )}_i}{{({\varsigma ^\prime })}_j}{{(\varkappa )}_i}{{({\varkappa ^\prime })}_j}}}{{{{(\varpi )}_{i + j}}}}\frac{{{x^i}{\kern 1pt} {y^j}}}{{i!{\kern 1pt} j!}},
here, (ϛ )n is the Pochhammer symbol. Current articles [21 , 22 ] include a comprehensive demonstration associated with the MSM operators along with the uses and characteristics. Saigo [10 ] instigate the fractional operators related with the Gauss hypergeometric function 2 F 1 ( ). The left-and right-handed Saigo FIO are given the following descriptions for ϛ , ϰ , ϖ ∈ ℂ, x > 0 and ℜ(ϛ ) > 0.
(10)
( ℐ 0 + ς , ϰ , ϖ f ) ( x ) = x − ς − ϰ Γ ( ς ) ∫ 0 x ( x − y ) ς − 1 2 F 1 ( ς + ϰ , − ϖ ; ς ; 1 − y x ) f ( y ) d y ,
\left( {\mathcal{I}_{0 + }^{\varsigma ,{\kern 1pt} \varkappa ,{\kern 1pt} \varpi }f} \right)(x) = \frac{{{x^{ - \varsigma - \varkappa }}}}{{\Gamma (\varsigma )}}\int_0^x {(x - y)^{\varsigma - 1}}{{\kern 1pt} _2}{F_1}\left( {\varsigma + \varkappa ,{\kern 1pt} - \varpi ;{\kern 1pt} \varsigma ;{\kern 1pt} 1 - \frac{y}{x}} \right)f(y)dy,
and
(11)
( ℐ − ς , ϰ , ϖ f ) ( x ) = 1 Γ ( ς ) ∫ x ∞ ( y − x ) ς − 1 y − ς − ϰ 2 F 1 ( ς + ϰ , − ϖ ; ς ; 1 − x y ) f ( y ) d y ,
\left( {\mathcal{I}_ - ^{\varsigma ,{\kern 1pt} \varkappa ,{\kern 1pt} \varpi }f} \right)(x) = \frac{1}{{\Gamma (\varsigma )}}\int_x^\infty {(y - x)^{\varsigma - 1}}{y^{ - \varsigma - \varkappa }}{{\kern 1pt} _2}{F_1}\left( {\varsigma + \varkappa ,{\kern 1pt} - \varpi ;{\kern 1pt} \varsigma ;{\kern 1pt} 1 - \frac{x}{y}} \right)f(y)dy,
respectively.
The following definitions are given for the left-and right-sided Saigo differential operators:
(12)
( D 0 + ς , ϰ , ϖ f ) ( x ) = ( d d x ) [ ℜ ( ς ) ] + 1 ( ℐ 0 + − ς + [ ℜ ( ς ) ] + 1 , − ϰ − [ ℜ ( ς ) ] − 1 , ς + ϖ − [ ℜ ( ς ) ] − 1 f ) ( x ) ,
\left( {\mathcal{D}_{0 + }^{\varsigma ,{\kern 1pt} \varkappa ,{\kern 1pt} \varpi }f} \right)(x) = {\left( {\frac{d}{{dx}}} \right)^{[\Re (\varsigma )] + 1}}\left( {\mathcal{I}_{0 + }^{ - \varsigma + [\Re (\varsigma )] + 1,{\kern 1pt} - \varkappa - [\Re (\varsigma )] - 1,{\kern 1pt} \varsigma + \varpi - [\Re (\varsigma )] - 1}f} \right)(x),
and
(13)
( D − ς , ϰ , ϖ f ) ( x ) = ( − d d x ) [ ℜ ( ς ) ] + 1 ( ℐ − − ς + [ ℜ ( ς ) ] + 1 , − ϰ − [ ℜ ( ς ) ] − 1 , ς + ϖ f ) ( x ) .
\left( {\mathcal{D}_ - ^{\varsigma ,{\kern 1pt} \varkappa ,{\kern 1pt} \varpi }f} \right)(x) = {\left( { - \frac{d}{{dx}}} \right)^{[\Re (\varsigma )] + 1}}\left( {\mathcal{I}_ - ^{ - \varsigma + [\Re (\varsigma )] + 1,{\kern 1pt} - \varkappa - [\Re (\varsigma )] - 1,{\kern 1pt} \varsigma + \varpi }f} \right)(x).
For ϰ = −ϛ and ϰ = 0 in (10 )–(13 ), the Riemann-Liouville (R-L) and Erdélyi-Kober (E-K) fractional operators are attained respectively (for further explanation see [23 ]). 2 F 1 is associated with F 3 as
F 3 ( ς , γ − ς , ϰ , γ − ϰ ; γ ; x ; y ) = 2 F 1 ( ς , ϰ ; γ ; x + y − x y ) .
{F_3}(\varsigma ,{\kern 1pt} \gamma - \varsigma ,{\kern 1pt} \varkappa ,{\kern 1pt} \gamma - \varkappa ;{\kern 1pt} \gamma ;{\kern 1pt} x;{\kern 1pt} y) = {{\kern 1pt} _2}{F_1}(\varsigma ,{\kern 1pt} \varkappa ;{\kern 1pt} \gamma ;{\kern 1pt} x + y - xy).
The MSM fractional operators (5 )–(8 ) are associated to Saigo operators (10 )–(13 ) by
(14)
( ℐ 0 + ς , 0 , ϰ , ϰ ' , ϖ f ) ( x ) = ( ℐ 0 + ϖ , ς − ϖ , − ϰ f ) ( x ) ,
\left( {\mathcal{I}_{0 + }^{\varsigma ,{\kern 1pt} 0,{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },\varpi }f} \right)(x) = \left( {\mathcal{I}_{0 + }^{\varpi ,{\kern 1pt} \varsigma - \varpi ,{\kern 1pt} - \varkappa }f} \right)(x),
(15)
( ℐ − ς , 0 , ϰ , ϰ ' , ϖ f ) ( x ) = ( ℐ − ϖ , ς − ϖ , − ϰ f ) ( x ) ,
\left( {\mathcal{I}_ - ^{\varsigma ,{\kern 1pt} 0,{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },\varpi }f} \right)(x) = \left( {\mathcal{I}_ - ^{\varpi ,{\kern 1pt} \varsigma - \varpi ,{\kern 1pt} - \varkappa }f} \right)(x),
and
(16)
( D 0 + 0 , ς ' , ϰ , ϰ ' , ϖ f ) ( x ) = ( D 0 + ϖ , ς ' − ϖ , ϰ ' − ϖ f ) ( x ) ,
\left( {\mathcal{D}_{0 + }^{0,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },{\kern 1pt} \varpi }f} \right)(x) = \left( {\mathcal{D}_{0 + }^{\varpi ,{\kern 1pt} {\varsigma ^\prime } - \varpi ,{\kern 1pt} {\varkappa ^\prime } - \varpi }f} \right)(x),
(17)
( D − 0 , ς ' , ϰ , ϰ ' , ϖ f ) ( x ) = ( D − ϖ , ς ' − ϖ , ϰ ' − ϖ f ) ( x ) .
\left( {\mathcal{D}_ - ^{0,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },{\kern 1pt} \varpi }f} \right)(x) = \left( {\mathcal{D}_ - ^{\varpi ,{\kern 1pt} {\varsigma ^\prime } - \varpi ,{\kern 1pt} {\varkappa ^\prime } - \varpi }f} \right)(x).
Lemma 1
Let ϛ, ϛ ', ϰ , ϰ ', ϖ ,λ ∈ ℂ and ℜ(ϖ ) > 0.
(a) If ℜ(λ ) > max {0, ℜ(ϛ ' − ϰ '), ℜ(ϛ +ϛ ' + ϰ −ϖ )}, then
(18)
( ℐ 0 + ς , ς ' , ϰ , ϰ ' , ϖ t λ − 1 ) ( x ) = x − ς − ς ' + ϖ + λ − 1 Γ ( λ ) Γ ( − ς ' + ϰ ' + λ ) Γ ( − ς − ς ' − ϰ + ϖ + λ ) Γ ( ϰ ' + λ ) Γ ( − ς − ς ' + ϖ + λ ) Γ ( − ς ' − ϰ + ϖ + λ ) .
\left( {\mathcal{I}_{0 + }^{\varsigma ,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },\varpi }{\kern 1pt} {t^{\lambda - 1}}} \right)(x) = {x^{ - \varsigma - {\varsigma ^\prime } + \varpi + \lambda - 1}}\frac{{\Gamma (\lambda )\Gamma ( - {\varsigma ^\prime } + {\varkappa ^\prime } + \lambda )\Gamma ( - \varsigma - {\varsigma ^\prime } - \varkappa + \varpi + \lambda )}}{{\Gamma ({\varkappa ^\prime } + \lambda )\Gamma ( - \varsigma - {\varsigma ^\prime } + \varpi + \lambda )\Gamma ( - {\varsigma ^\prime } - \varkappa + \varpi + \lambda )}}.
(b) If ℜ(λ ) > max {ℜ(ϰ ), ℜ(−ϛ −ϛ ' +ϖ ), ℜ(−ϛ − ϰ ' +ϖ )}, then
(19)
( ℐ − ς , ς ' , ϰ , ϰ ' , ϖ t − λ ) ( x ) = x − ς − ς ' + ϖ − λ Γ ( − ϰ + λ ) Γ ( ς + ς ' − ϖ + λ ) Γ ( ς + ϰ ' − ϖ + λ ) Γ ( λ ) Γ ( ς − ϰ + λ ) Γ ( ς + ς ' + ϰ ' − ϖ + λ ) .
\left( {\mathcal{I}_ - ^{\varsigma ,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },\varpi }{\kern 1pt} {t^{ - \lambda }}} \right)(x) = {x^{ - \varsigma - {\varsigma ^\prime } + \varpi - \lambda }}{\kern 1pt} \frac{{\Gamma ( - \varkappa + \lambda )\Gamma (\varsigma + {\varsigma ^\prime } - \varpi + \lambda )\Gamma (\varsigma + {\varkappa ^\prime } - \varpi + \lambda )}}{{\Gamma (\lambda )\Gamma (\varsigma - \varkappa + \lambda )\Gamma (\varsigma + {\varsigma ^\prime } + {\varkappa ^\prime } - \varpi + \lambda )}}.
Lemma 2
Let ϛ , ϛ ', ϰ , ϰ ', ϖ , λ ∈ ℂ.
(a) If ℜ(λ ) > max {0, ℜ(−ϛ + ϰ ), ℜ(−ϛ −ϛ ' − ϰ ' +ϖ )}, then
(20)
( D 0 + ς , ς ' , ϰ , ϰ ' , ϖ t λ − 1 ) ( x ) = x ς + ς ' − ϖ + λ − 1 Γ ( λ ) Γ ( ς − ϰ + λ ) Γ ( ς + ς ' + ϰ ' − ϖ + λ ) Γ ( − ϰ + λ ) Γ ( ς + ς ' − ϖ + λ ) Γ ( ς + ϰ ' − ϖ + λ ) .
\left( {\mathcal{D}_{0 + }^{\varsigma ,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },\varpi }{\kern 1pt} {t^{\lambda - 1}}} \right)(x) = {x^{\varsigma + {\varsigma ^\prime } - \varpi + \lambda - 1}}\frac{{\Gamma (\lambda )\Gamma (\varsigma - \varkappa + \lambda )\Gamma (\varsigma + {\varsigma ^\prime } + {\varkappa ^\prime } - \varpi + \lambda )}}{{\Gamma ( - \varkappa + \lambda )\Gamma (\varsigma + {\varsigma ^\prime } - \varpi + \lambda )\Gamma (\varsigma + {\varkappa ^\prime } - \varpi + \lambda )}}.
(b) If ℜ(λ ) > max {ℜ(−ϰ '), ℜ(ϛ ' + ϰ −ϖ ), ℜ(ϛ +ϛ ' −ϖ ) + [ℜ(ϖ )] + 1}, then
(21)
( D − ς , ς ' , ϰ , ϰ ' , ϖ t − λ ) ( x ) = x ς + ς ' − ϖ − λ Γ ( ϰ ' + λ ) Γ ( − ς − ς ' + ϖ + λ ) Γ ( − ς ' − ϰ + ϖ + λ ) Γ ( λ ) Γ ( − ς ' + ϰ ' + λ ) Γ ( − ς − ς ' − ϰ + ϖ + λ ) .
\left( {\mathcal{D}_ - ^{\varsigma ,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },\varpi }{\kern 1pt} {t^{ - \lambda }}} \right)(x) = {x^{\varsigma + {\varsigma ^\prime } - \varpi - \lambda }}\frac{{\Gamma ({\varkappa ^\prime } + \lambda )\Gamma ( - \varsigma - {\varsigma ^\prime } + \varpi + \lambda )\Gamma ( - {\varsigma ^\prime } - \varkappa + \varpi + \lambda )}}{{\Gamma (\lambda )\Gamma ( - {\varsigma ^\prime } + {\varkappa ^\prime } + \lambda )\Gamma ( - \varsigma - {\varsigma ^\prime } - \varkappa + \varpi + \lambda )}}.
2.1 Incomplete ℵ-function
In this paper, we introduced the incomplete ℵ-function
Γ ℵ r j , s j , ρ j ; m U , V ( Z )
^\Gamma \aleph _{{r_j},{s_j},{\rho _j};m}^{U,V}(\mathcal{Z})
and
γ ℵ r j , s j , ρ j ; m U , V ( Z )
^\gamma \aleph _{{r_j},{s_j},{\rho _j};m}^{U,V}(\mathcal{Z})
[24 , 25 ] as follows:
(22)
γ ℵ r j , s j , ρ j ; m U , V ( Z ) = γ ℵ r j , s j , ρ j ; m U , V [ Z | ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( ε n , E n ) 1 , U , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] = 1 2 π ι ∫ $ Φ ( q , Y ) Z − q d q ,
\begin{array}{*{20}{l}}{{{\kern 1pt} ^\gamma }\aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V}{\kern 1pt} (\mathcal{Z}){ = ^\gamma }\aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V}{\kern 1pt} \left[ {\mathcal{Z}\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\end{array}} \right]}\\{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\kern 1pt} = \frac{1}{{2\pi \iota }}\int_\$ \;\Phi (q,\mathcal{Y})\;{\mathcal{Z}^{ - q}}\;dq,}\end{array}
where
(23)
Φ ( q , Y ) = γ ( 1 − Λ 1 − D 1 q ; Y ) ∏ n = 1 U Γ ( ε n + E n q ) ∏ n = 2 V Γ ( 1 − Λ n − D n q ) ∑ j = 1 m ρ j [ ∏ n = U + 1 s j Γ ( 1 − ε n j − E n j q ) ∏ n = V + 1 r j Γ ( Λ n j + D n j ) ] ,
\Phi (q,\mathcal{Y}) = \frac{{\gamma (1 - {\Lambda _1} - {\mathfrak{D}_1}q;\mathcal{Y}){\kern 1pt} \prod\limits_{n = 1}^U \Gamma ({\varepsilon _n} + {\mathfrak{E}_n}q){\kern 1pt} \prod\limits_{n = 2}^V \Gamma (1 - {\Lambda _n} - {\mathfrak{D}_n}q)}}{{\sum\limits_{j = 1}^m {\rho _j}{\kern 1pt} [\prod\limits_{n = U + 1}^{{s_j}} \Gamma (1 - {\varepsilon _{nj}} - {\mathfrak{E}_{nj}}q){\kern 1pt} \prod\limits_{n = V + 1}^{{r_j}} \Gamma ({\Lambda _{nj}} + {\mathfrak{D}_{nj}})]}},
and
(24)
Γ ℵ r j , s j , ρ j ; m U , V ( Z ) = Γ ℵ r j , s j , ρ j ; m U , V [ Z | ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( ε n , E n ) 1 , U , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] = 1 2 π ι ∫ $ Ψ ( q , Y ) Z − q d q ,
\begin{array}{*{20}{l}}{{{\kern 1pt} ^\Gamma }\aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V}{\kern 1pt} (\mathcal{Z}){ = ^\Gamma }\aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V}{\kern 1pt} \left[ {\mathcal{Z}\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\end{array}} \right]}\\{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\kern 1pt} = \frac{1}{{2\pi \iota }}\int_\$ \;\Psi (q,\mathcal{Y})\;{\mathcal{Z}^{ - q}}\;dq,}\end{array}
where
(25)
Ψ ( q , Y ) = Γ ( 1 − Λ 1 − D 1 q ; Y ) ∏ n = 1 U Γ ( ε n + E n q ) ∏ n = 2 V Γ ( 1 − Λ n − D n q ) ∑ j = 1 m ρ j [ ∏ n = U + 1 s j Γ ( 1 − ε n j − E n j q ) ∏ n = V + 1 r j Γ ( Λ n j + D n j ) ] ,
\Psi (q,\mathcal{Y}) = \frac{{\Gamma (1 - {\Lambda _1} - {\mathfrak{D}_1}q;\mathcal{Y}){\kern 1pt} \prod\limits_{n = 1}^U \Gamma ({\varepsilon _n} + {\mathfrak{E}_n}q){\kern 1pt} \prod\limits_{n = 2}^V \Gamma (1 - {\Lambda _n} - {\mathfrak{D}_n}q)}}{{\sum\limits_{j = 1}^m {\rho _j}{\kern 1pt} [\prod\limits_{n = U + 1}^{{s_j}} \Gamma (1 - {\varepsilon _{nj}} - {\mathfrak{E}_{nj}}q){\kern 1pt} \prod\limits_{n = V + 1}^{{r_j}} \Gamma ({\Lambda _{nj}} + {\mathfrak{D}_{nj}})]}},
for 𝒵 ≠ 0,𝒴 ≧ 0, the incomplete ℵ-functions
γ ℵ r j , s j , ρ j ; m U , V ( Z )
^\gamma \aleph _{{r_j},{s_j},{\rho _j};m}^{U,V}(\mathcal{Z})
and
Γ ℵ r j , s j , ρ j ; m U , V ( Z )
^\Gamma \aleph _{{r_j},{s_j},{\rho _j};m}^{U,V}(\mathcal{Z})
in (22 ) and (24 ) exist in the circumstances listed as follows:
The complex- plane contour $ extended from γ − i ∞ to γ + i ∞, γ ∈ ℝ, and the poles of the gamma functions Γ (1 − Λ n − 𝔇nq ) for n = 1, 2, ...,V are not perfectly matched with the gamma function poles Γ (ϵ n + 𝔈nq ) for n = 1,2,..,U . The parameters rj and sj ∈ ℤ+ satisfying 0 ≤ V ≤ rj ,0 ≤ U ≤ sj for 1 ≤ j ≤ m . The parameters 𝔇n , 𝔈n , 𝔈nj , 𝔇nj are positive numbers, and Λ n , ε n , Λ nj , ε nj are complex. The void product is considered to represent unity and all of the poles Φ (q ,𝒴) and Ψ (q ,𝒴) should be simple.
A number of unique remarks are made about incomplete ℵ-functions and are as follows:
Remark 1
When 𝒴 = 0, Equation (24) changes to the suggested ℵ-function of Sudland [26 , 27 ]:
(26)
Γ ℵ r j , s j , ρ j ; m U , V [ Z | ( Λ 1 , D 1 : 0 ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( ε n , E n ) 1 , U , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] = ℵ r j , s j , ρ j ; m U , V [ Z | ( Λ n , D n ) 1 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( ε n , E n ) 1 , U , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] .
\begin{array}{*{20}{l}}{{{\kern 1pt} ^\Gamma }\aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V}{\kern 1pt} \left[ {\mathcal{Z}\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:0),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\end{array}} \right]}\\{\;\;\;\;\;\;\;\;\;\;\;\; = \aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V}{\kern 1pt} \left[ {\mathcal{Z}\Biggl|\begin{array}{*{20}{c}}{{{({\Lambda _n},{\mathfrak{D}_n})}_{1,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\end{array}} \right].}\end{array}
Remark 2
Again, when ρj = 1 in (22 ) and (24 ), then it changes to the incomplete I -function of Bansal and Kumar [28 ]:
(27)
γ ℵ r j , s j , ρ j ; m U , V [ Z | ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ 1 ( Λ n j , D n j ) ] V + 1 , r j ( ε n , E n ) 1 , U , [ 1 ( ε n j , E n j ) ] U + 1 , s j ] = γ I r j , s j ; m U , V [ Z | ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , ( Λ n j , D n j ) V + 1 , r j ( ε n , E n ) 1 , U , ( ε n j , E n j ) U + 1 , s j ] ,
\begin{array}{l}{{\kern 1pt} ^\gamma }\aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V}{\kern 1pt} \left[ {\mathcal{Z}\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[1({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{[1({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\\\end{array}} \right]\\\;\;\;\;\;\;\;\;\;\;\;\; = {{\kern 1pt} ^\gamma }I_{{r_j},{\kern 1pt} {s_j};{\kern 1pt} m}^{U,V}{\kern 1pt} \left[ {\mathcal{Z}\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{({\Lambda _{nj}},{\mathfrak{D}_{nj}})}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{({\varepsilon _{nj}},{\mathfrak{E}_{nj}})}_{U + 1,{s_j}}}}\\\end{array}} \right],\end{array}
and
(28)
Γ ℵ r j , s j , ρ j ; m U , V [ Z | ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ 1 ( Λ n j , D n j ) ] V + 1 , r j ( ε n , E n ) 1 , U , [ 1 ( ε n j , E n j ) ] U + 1 , s j ] = Γ I r j , s j ; m U , V [ Z | ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , ( Λ n j , D n j ) V + 1 , r j ( ε n , E n ) 1 , U , ( ε n j , E n j ) U + 1 , s j
]
.
\begin{array}{l}{{\kern 1pt} ^\Gamma }\aleph _{{r_j},{\kern 1pt} {s_j},{\rho _j};{\kern 1pt} m}^{U,V}{\kern 1pt} \left[ {\mathcal{Z}\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[1({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{[1({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\\ \end{array}} \right]\\\;\;\;\;\;\;\;\;\;\;\;\; = {{\kern 1pt} ^\Gamma }I_{{r_j},{\kern 1pt} {s_j};{\kern 1pt} m}^{U,V}{\kern 1pt} \left[ {\mathcal{Z}\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{({\Lambda _{nj}},{\mathfrak{D}_{nj}})}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{({\varepsilon _{nj}},{\mathfrak{E}_{nj}})}_{U + 1,{s_j}}}}\\ \end{array}} \right].\end{array}
Remark 3
Next, taking 𝒴 = 0 and ρj = 1 in (24 ), then it turns into the Saxena I -function [29 ]:
(29)
Γ ℵ r j , s j , 1 ; m U , V [ Z | ( Λ 1 , D 1 : 0 ) , ( Λ n , D n ) 2 , V , [ 1 ( Λ n j , D n j ) ] V + 1 , r j ( ε n , E n ) 1 , U , [ 1 ( ε n j , E n j ) ] U + 1 , s j ] = I r j , s j ; m U , V [ Z | ( Λ n , D n ) 1 , V , ( Λ n j , D n j ) V + 1 , r j ( ε n , E n ) 1 , U , ( ε n j , E n j ) U + 1 , s j ] .
\begin{array}{l}{{\kern 1pt} ^\Gamma }\aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} 1;{\kern 1pt} m}^{U,V}{\kern 1pt} \left[ {\mathcal{Z}\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:0),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[1({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{[1({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\\ \end{array}} \right]\\\;\;\;\;\;\;\;\;\;\;\;\; = {\kern 1pt} I_{{r_j},{\kern 1pt} {s_j};{\kern 1pt} m}^{U,V}{\kern 1pt} \left[ {\mathcal{Z}\Biggl|\begin{array}{*{20}{c}}{{{({\Lambda _n},{\mathfrak{D}_n})}_{1,V}},{\kern 1pt} {{({\Lambda _{nj}},{\mathfrak{D}_{nj}})}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{({\varepsilon _{nj}},{\mathfrak{E}_{nj}})}_{U + 1,{s_j}}}}\\ \end{array}} \right].\end{array}
Remark 4
Further taking ρj = 1 and m = 1 in (22 ) and (24 ), then it turns into the incomplete H -function(see [30 , 31 ] also) of Srivastava [32 ]:
(30)
γ ℵ r j , s j , 1 ; 1 U , V [ Z | ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ 1 ( Λ n j , D n j ) ] V + 1 , r j ( ε n , E n ) 1 , U , [ 1 ( ε n j , E n j ) ] U + 1 , s j ] = γ r , s U , V [ Z | ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , r ( ε n , E n ) 1 , s ] ,
\begin{array}{l}{{\kern 1pt} ^\gamma }\aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} 1;{\kern 1pt} 1}^{U,V}{\kern 1pt} \left[ {\mathcal{Z}\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[1({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{[1({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\\ \end{array}} \right]\\\;\;\;\;\;\;\;\;\;\;\;\; = {\kern 1pt} \gamma _{r,{\kern 1pt} s}^{U,V}{\kern 1pt} \left[ {\mathcal{Z}\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,r}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,s}}}\\ \end{array}} \right],\end{array}
and
(31)
Γ ℵ r j , s j , 1 ; 1 U , V [ Z | ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ 1 ( Λ n j , D n j ) ] V + 1 , r j ( ε n , E n ) 1 , U , [ 1 ( ε n j , E n j ) ] U + 1 , s j ] = Γ r , s U , V [ Z | ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , r ( ε n , E n ) 1 , s ] .
\begin{array}{l}{{\kern 1pt} ^\Gamma }\aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} 1;{\kern 1pt} 1}^{U,V}{\kern 1pt} \left[ {\mathcal{Z}\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[1({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{[1({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\\ \end{array}} \right]\\\;\;\;\;\;\;\;\;\;\;\;\; = {\kern 1pt} \Gamma _{r,{\kern 1pt} s}^{U,V}{\kern 1pt} \left[ {\mathcal{Z}\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,r}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,s}}}\\ \end{array}} \right].\end{array}
Remark 5
Next, we take 𝒴 = 0, ρj = 1, and m = 1 in (24 ), then it turns into the H -function of Srivastava [33 ]:
(32)
Γ ℵ r j , s j , 1 ; 1 U , V [ Z | ( Λ 1 , D 1 : 0 ) , ( Λ n , D n ) 2 , V , [ 1 ( Λ n j , D n j ) ] V + 1 , r j ( ε n , E n ) 1 , U , [ 1 ( ε n j , E n j ) ] U + 1 , s j ] = H r , s U , V [ Z | ( Λ n , D n ) 1 , r ( ε n , E n ) 1 , s ] .
\begin{array}{l}{{\kern 1pt} ^\Gamma }\aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} 1;{\kern 1pt} 1}^{U,V}{\kern 1pt} \left[ {\mathcal{Z}\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:0),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[1({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{[1({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\\ \end{array}} \right]\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = {\kern 1pt} H_{r,{\kern 1pt} s}^{U,V}{\kern 1pt} \left[ {\mathcal{Z}\Biggl|\begin{array}{*{20}{c}}{{{({\Lambda _n},{\mathfrak{D}_n})}_{1,r}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,s}}}\\ \end{array}} \right].\end{array}
We developed the FC findings linked to the incomplete ℵ-functions, which were influenced by the work of Srivastava et al. [34 ].
3 Fractional integral formulas
In this part, we create two formulas for fractional integrals that multiply incomplete ℵ-functions and the generic class of polynomials specified in equation (24) and (4) , respectively.
Theorem 3
Let ϛ , ϛ ', ϰ , ϰ ', ϖ ,z ,α ∈ ℂ and ℜ(ϖ ) > 0,μ > 0,λk > 0 (k = 1,2,3,··· ,s ),
ℜ ( α ) + μ min 1 ≤ j ≤ U ℜ ( ε j ϰ j ) > max [ 0 , ℜ ( ς + ς ' + ϰ − ϖ ) , ℜ ( ς ' − ϰ ' ) ] .
\Re (\alpha ) + \mu \mathop {\min }\limits_{1 \le j \le U} \Re \left( {\frac{{{\varepsilon _j}}}{{{\varkappa _j}}}} \right) > \max [0,\Re (\varsigma + {\varsigma ^\prime } + \varkappa - \varpi ),{\kern 1pt} \Re ({\varsigma ^\prime } - {\varkappa ^\prime })].
Further the parameters rj , s j , 𝔓 ∈ ℤ+ satisfying 0 ≤ V ≤ rj ,0 ≤ U ≤ sj for j = 1,2,···m , 𝔇n , 𝔈n , 𝔈nj , 𝔇nj ∈ ℝ+ , Λn ,εn ,Λnj ,εnj ∈ ℂ (j = 1,2,··· , rj ; n = 1,2,··· , sj ),
A Q k , P k k
A_{{\mathfrak{Q}_k},{\kern 1pt} {\mathfrak{P}_k}}^k
are real or complex numbers arbitrary constant for k = 1,2,··· ,s and ρi > 0 for i = 1,2,··· ,m, then the outcome shown below is accurate:
(33)
ℐ 0 + ς , ς ' , ϰ , ϰ ' , ϖ ( t α − 1 ∏ j = 1 s S Q j P j [ c j t λ j ] × Γ ℵ r j , s j , ρ j ; m U , V [ z t μ | ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( ε n , E n ) 1 , U , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] ) ( x ) = x α − ς − ς ' + ϖ − 1 ∑ k 1 = 0 [ Q 1 / P 1 ] ⋯ ∑ k s = 0 [ Q s / P s ] ( − Q 1 ) P 1 k 1 ⋯ ( − Q s ) P s k s k 1 ! ⋯ k s ! A Q 1 , P 1 ( 1 ) ⋯ A Q s , P s ( s ) c 1 k 1 ⋯ c s k s ( x ) ∑ j = 1 s λ j k j × Γ ℵ r j + 3 , s j + 3 , ρ j ; m U , V + 3 [ z x μ | ( 1 − α − ∑ j = 1 s λ j k j , μ ) , ( 1 − α + ς + ς ' + ϰ − ϖ − ∑ j = 1 s λ j k j , μ ) , ( ε n , E n ) 1 , U , ( 1 − α + ς + ς ' − ϖ − ∑ j = 1 s λ j k j , μ ) , ( 1 − α + ς ' − ϰ ' − ∑ j = 1 s λ j k j , μ ) , ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( 1 − α − ϰ ' − ∑ j = 1 s λ j k j , μ ) , ( 1 + ς ' + ϰ − ϖ − α − ∑ j = 1 s λ j k j , μ ) , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] .
\begin{array}{l}\mathcal{I}_{0 + }^{\varsigma ,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },\varpi }{\kern 1pt} \left( {{t^{\alpha - 1}}\prod\limits_{j = 1}^s S_{{\mathfrak{Q}_j}}^{{\mathfrak{P}_j}}[{c_j}{\kern 1pt} {t^{{\lambda _j}}}]} \right.\\\left. {{ \times ^\Gamma }\aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V}{\kern 1pt} \left[ {z{\kern 1pt} {t^\mu }\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\\ \end{array}} \right]} \right)(x)\\ = {x^{\alpha - \varsigma - {\varsigma ^\prime } + \varpi - 1}}\sum\limits_{{\mathfrak{k}_1} = 0}^{[{\mathfrak{Q}_1}/{\mathfrak{P}_1}]} \cdots \sum\limits_{{\mathfrak{k}_s} = 0}^{[{\mathfrak{Q}_s}/{\mathfrak{P}_s}]} \frac{{{{( - {\mathfrak{Q}_1})}_{{\mathfrak{P}_1}{\kern 1pt} {\mathfrak{k}_1}}} \cdots {{( - {\mathfrak{Q}_s})}_{{\mathfrak{P}_s}{\kern 1pt} {\mathfrak{k}_s}}}}}{{{\mathfrak{k}_1}! \cdots {\mathfrak{k}_s}!}}A_{{\mathfrak{Q}_1},{\kern 1pt} {\mathfrak{P}_1}}^{(1)} \cdots A_{{\mathfrak{Q}_s},{\kern 1pt} {\mathfrak{P}_s}}^{(s)}{\kern 1pt} {\kern 1pt} c_1^{{\mathfrak{k}_1}} \cdots c_s^{{\mathfrak{k}_s}}{\kern 1pt} {(x)^{\sum\limits_{j = 1}^s {\lambda _j}{\kern 1pt} {\mathfrak{k}_j}}}\\{ \times ^\Gamma }\aleph _{{r_j} + 3,{\kern 1pt} {s_j} + 3,{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V + 3}{\kern 1pt} \left[ {z{\kern 1pt} {x^\mu }\Biggl|\begin{array}{*{20}{c}}{(1 - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),(1 - \alpha + \varsigma + {\varsigma ^\prime } + \varkappa - \varpi - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} (1 - \alpha + \varsigma + {\varsigma ^\prime } - \varpi - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),}\\ \end{array}} \right.\\\;\;\left. {\;\;\begin{array}{*{20}{c}}{(1 - \alpha + {\varsigma ^\prime } - {\varkappa ^\prime } - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} ({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{(1 - \alpha - {\varkappa ^\prime } - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} (1 + {\varsigma ^\prime } + \varkappa - \varpi - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\end{array}} \right].\end{array}
Proof
The LHS of equation (33) is:
(34)
T 1 = ℐ 0 + ς , ς ' , ϰ , ϰ ' , ϖ ( t α − 1 ∏ j = 1 s S Q j P j [ c j t λ j ] × Γ ℵ r j , s j , ρ j ; m U , V [ z t μ | ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( ε n , E n ) 1 , U , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] ) .
\begin{array}{l}{T_1} = \mathcal{I}_{0 + }^{\varsigma ,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },\varpi }{\kern 1pt} \left( {{t^{\alpha - 1}}\prod\limits_{j = 1}^s S_{{\mathfrak{Q}_j}}^{{\mathfrak{P}_j}}[{c_j}{\kern 1pt} {t^{{\lambda _j}}}]} \right.\\\left. {{ \times ^\Gamma }\aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V}{\kern 1pt} \left[ {z{\kern 1pt} {t^\mu }\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\\ \end{array}} \right]} \right).\end{array}
Replace the incomplete ℵ-function and Srivastava polynomial by equation (24) and (4) respectively and by reversing the summation order, we discover the subsequent form:
(35)
T 1 = ∑ k 1 = 0 [ Q 1 / P 1 ] ⋯ ∑ k s = 0 [ Q s / P s ] ( − Q 1 ) P 1 k 1 ⋯ ( − Q s ) P s k s k 1 ! ⋯ k s ! × A Q 1 , P 1 ( 1 ) ⋯ A Q s , P s ( s ) c 1 k 1 ⋯ c s k s × 1 2 π ι ∫ $ Ψ ( q , Y ) z − q ( ℐ 0 + ς , ς ' , ϰ , ϰ ' , ϖ t α + ∑ j = 1 s λ j k j − μ q − 1 ) ( x ) d q ,
\begin{array}{l}{T_1} = \sum\limits_{{\mathfrak{k}_1} = 0}^{[{\mathfrak{Q}_1}/{\mathfrak{P}_1}]} \cdots \sum\limits_{{\mathfrak{k}_s} = 0}^{[{\mathfrak{Q}_s}/{\mathfrak{P}_s}]} \frac{{{{( - {\mathfrak{Q}_1})}_{{\mathfrak{P}_1}{\kern 1pt} {\mathfrak{k}_1}}} \cdots {{( - {\mathfrak{Q}_s})}_{{\mathfrak{P}_s}{\kern 1pt} {\mathfrak{k}_s}}}}}{{{\mathfrak{k}_1}! \cdots {\mathfrak{k}_s}!}}{\kern 1pt} \times A_{{\mathfrak{Q}_1},{\kern 1pt} {\mathfrak{P}_1}}^{(1)} \cdots A_{{\mathfrak{Q}_s},{\kern 1pt} {\mathfrak{P}_s}}^{(s)}{\kern 1pt} {\kern 1pt} c_1^{{\mathfrak{k}_1}} \cdots c_s^{{\mathfrak{k}_s}}\\ \times \frac{1}{{2\pi \iota }}\int_\$ \;\Psi (q,\mathcal{Y})\;{z^{ - q}}\left( {\mathcal{I}_{0 + }^{\varsigma ,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },\varpi }{\kern 1pt} {t^{\alpha + \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} - \mu q - 1}}} \right)(x)dq,\end{array}
where Ψ (q ,𝒴) is defined in equation (25) .
Using equation (18) of Lemma 1 , we discover the subsequent form:
(36)
T 1 = ∑ k 1 = 0 [ Q 1 / P 1 ] ⋯ ∑ k s = 0 [ Q s / P s ] ( − Q 1 ) P 1 k 1 ⋯ ( − Q s ) P s k s k 1 ! ⋯ k s ! × A Q 1 , P 1 ( 1 ) ⋯ A Q s , P s ( s ) c 1 k 1 ⋯ c s k s × 1 2 π ι ∫ $ x α − ς − ς ' + ϖ + ∑ j = 1 s λ j k j − 1 Ψ ( q , Y ) ( z x μ )
− q Γ ( α + ∑ j = 1 s λ j k j − μ q ) Γ ( ϰ ' + α + ∑ j = 1 s λ j k j − μ q ) × Γ ( − ς ' + ϰ ' + α + ∑ j = 1 s λ j k j − μ q ) Γ ( − ς − ς ' − ϰ + ϖ + α + ∑ j = 1 s λ j k j − μ q ) Γ ( − ς − ς ' + ϖ + α + ∑ j = 1 s λ j k j − μ q ) Γ ( − ς ' − ϰ + ϖ + α + ∑ j = 1 s λ j k j − μ q ) d q .
\begin{array}{l}{T_1} = \sum\limits_{{\mathfrak{k}_1} = 0}^{[{\mathfrak{Q}_1}/{\mathfrak{P}_1}]} \cdots \sum\limits_{{\mathfrak{k}_s} = 0}^{[{\mathfrak{Q}_s}/{\mathfrak{P}_s}]} \frac{{{{( - {\mathfrak{Q}_1})}_{{\mathfrak{P}_1}{\kern 1pt} {\mathfrak{k}_1}}} \cdots {{( - {\mathfrak{Q}_s})}_{{\mathfrak{P}_s}{\kern 1pt} {\mathfrak{k}_s}}}}}{{{\mathfrak{k}_1}! \cdots {\mathfrak{k}_s}!}}{\kern 1pt} \times A_{{\mathfrak{Q}_1},{\kern 1pt} {\mathfrak{P}_1}}^{(1)} \cdots A_{{\mathfrak{Q}_s},{\kern 1pt} {\mathfrak{P}_s}}^{(s)}{\kern 1pt} {\kern 1pt} c_1^{{\mathfrak{k}_1}} \cdots c_s^{{\mathfrak{k}_s}}\\ \times \frac{1}{{2\pi \iota }}\int_\$ \;{x^{\alpha - \varsigma - {\varsigma ^\prime } + \varpi + \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} - 1}}{\kern 1pt} {\kern 1pt} \Psi (q,\mathcal{Y})\;{(z{x^\mu })^{ - q}}\frac{{\Gamma (\alpha + \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} - \mu q)}}{{\Gamma ({\varkappa ^\prime } + \alpha + \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} - \mu q)}} \times \\\frac{{\Gamma ( - {\varsigma ^\prime } + {\varkappa ^\prime } + \alpha + \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} - \mu q){\kern 1pt} {\kern 1pt} \Gamma ( - \varsigma - {\varsigma ^\prime } - \varkappa + \varpi + \alpha + \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} - \mu q)}}{{\Gamma ( - \varsigma - {\varsigma ^\prime } + \varpi + \alpha + \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} - \mu q){\kern 1pt} {\kern 1pt} \Gamma ( - {\varsigma ^\prime } - \varkappa + \varpi + \alpha + \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} - \mu q)}}dq.\end{array}
Finally, after some adjustment of terms , we obtain RHS of equation (33) .
Theorem 4
Let ϛ , ϛ ', ϰ , ϰ ', ϖ ,z , α ∈ ℂ and ℜ(ϖ ) > 0,μ > 0,λk > 0 (k = 1,2,3,···, s ),
ℜ ( α ) + μ min 1 ≤ j ≤ U ℜ ( ε j ϰ j ) > max [ 0 , ℜ ( ς + ς ' + ϰ − ϖ ) , ℜ ( ς ' − ϰ ' ) ] .
\Re (\alpha ) + \mu \mathop {\min }\limits_{1 \le j \le U} \Re \left( {\frac{{{\varepsilon _j}}}{{{\varkappa _j}}}} \right) > \max [0,\Re (\varsigma + {\varsigma ^\prime } + \varkappa - \varpi ),{\kern 1pt} \Re ({\varsigma ^\prime } - {\varkappa ^\prime })].
Further the parameters rj , s j , 𝔓 ∈ ℤ+ satisfying 0 ≤ V ≤ rj ,0 ≤ U ≤ sj for j = 1,2,···m, 𝔇n , 𝔈n , 𝔈nj , 𝔇nj ∈ ℝ+ , Λn , ε n ,Λnj ,εnj ∈ ℂ (j = 1,2,··· , rj ; n = 1,2,··· , sj ),
A Q k , P k k
A_{{\mathfrak{Q}_k},{\kern 1pt} {\mathfrak{P}_k}}^k
are real or complex numbers arbitrary constant for k = 1,2,··· ,s and ρi > 0 for i = 1,2,··· ,m, then the outcome shown below is accurate:
(37)
ℐ 0 + ς , ς ' , ϰ , ϰ ' , ϖ ( t α − 1 ∏ j = 1 s S Q j P j [ c j t λ j ] γ ℵ r j , s j , ρ j ; m U , V [ z t μ | ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( ε n , E n ) 1 , U , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] ) ( x ) = x α − ς − ς ' + ϖ − 1 ∑ k 1 = 0 [ Q 1 / P 1 ] ⋯ ∑ k s = 0 [ Q s / P s ] ( − Q 1 ) P 1 k 1 ⋯ ( − Q s ) P s k s k 1 ! ⋯ k s ! A Q 1 , P 1 ( 1 ) ⋯ A Q s , P s ( s ) c 1 k 1 ⋯ c s k s ( x )
∑ j = 1 s λ j k j × γ ℵ r j + 3 , s j + 3 , ρ j ; m U , V + 3 [ z x μ | ( 1 − α − ∑ j = 1 s λ j k j , μ ) , ( 1 − α + ς + ς ' + ϰ − ϖ − ∑ j = 1 s λ j k j , μ ) , ( ε n , E n ) 1 , U , ( 1 − α + ς + ς ' − ϖ − ∑ j = 1 s λ j k j , μ ) , ( 1 − α + ς ' − ϰ ' − ∑ j = 1 s λ j k j , μ ) , ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( 1 − α − ϰ ' − ∑ j = 1 s λ j k j , μ ) , ( 1 + ς ' + ϰ − ϖ − α − ∑ j = 1 s λ j k j , μ ) , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] .
\begin{array}{l}\mathcal{I}_{0 + }^{\varsigma ,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },\varpi }{\kern 1pt} \left( {{t^{\alpha - 1}}\prod\limits_{j = 1}^s S_{{\mathfrak{Q}_j}}^{{\mathfrak{P}_j}}[{c_j}{\kern 1pt} {t^{{\lambda _j}}}]} \right.\\\left. {^\gamma \aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V}{\kern 1pt} \left[ {z{\kern 1pt} {t^\mu }\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\\ \end{array}} \right]} \right)(x)\\ = {x^{\alpha - \varsigma - {\varsigma ^\prime } + \varpi - 1}}\sum\limits_{{\mathfrak{k}_1} = 0}^{[{\mathfrak{Q}_1}/{\mathfrak{P}_1}]} \cdots \sum\limits_{{\mathfrak{k}_s} = 0}^{[{\mathfrak{Q}_s}/{\mathfrak{P}_s}]} \frac{{{{( - {\mathfrak{Q}_1})}_{{\mathfrak{P}_1}{\kern 1pt} {\mathfrak{k}_1}}} \cdots {{( - {\mathfrak{Q}_s})}_{{\mathfrak{P}_s}{\kern 1pt} {\mathfrak{k}_s}}}}}{{{\mathfrak{k}_1}! \cdots {\mathfrak{k}_s}!}}A_{{\mathfrak{Q}_1},{\kern 1pt} {\mathfrak{P}_1}}^{(1)} \cdots A_{{\mathfrak{Q}_s},{\kern 1pt} {\mathfrak{P}_s}}^{(s)}{\kern 1pt} {\kern 1pt} c_1^{{\mathfrak{k}_1}} \cdots c_s^{{\mathfrak{k}_s}}{\kern 1pt} {(x)^{\sum\limits_{j = 1}^s {\lambda _j}{\kern 1pt} {\mathfrak{k}_j}}}\\{ \times ^\gamma }\aleph _{{r_j} + 3,{\kern 1pt} {s_j} + 3,{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V + 3}{\kern 1pt} \left[ {z{\kern 1pt} {x^\mu }\Biggl|\begin{array}{*{20}{c}}{(1 - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),(1 - \alpha + \varsigma + {\varsigma ^\prime } + \varkappa - \varpi - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} (1 - \alpha + \varsigma + {\varsigma ^\prime } - \varpi - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),}\\ \end{array}} \right.\\\;\left. {\;\;\;\begin{array}{*{20}{c}}{(1 - \alpha + {\varsigma ^\prime } - {\varkappa ^\prime } - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} ({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{(1 - \alpha - {\varkappa ^\prime } - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} (1 + {\varsigma ^\prime } + \varkappa - \varpi - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\end{array}} \right].\end{array}
Proof
Theorem 4 is proved in the same manner as Theorem 3 with the same conditions.
The following corollary is obtained regarding the Saigo FIO [10 ] in light of the equation (14) .
Corollary 5
Let ϛ, ϰ, ϖ ,z,α ∈ ℂ and ℜ(ϛ ) > 0,μ > 0,λk > 0 (k = 1,2,··· ,s ),
ℜ ( α ) + μ min 1 ≤ j ≤ U ℜ ( ε j ϰ j ) > max [ 0 , ℜ ( ϰ − ϖ ) ] .
\Re (\alpha ) + \mu \mathop {\min }\limits_{1 \le j \le U} \Re \left( {\frac{{{\varepsilon _j}}}{{{\varkappa _j}}}} \right) > \max [0,\Re (\varkappa - \varpi )].
Further the parameters rj , s j , 𝔓 ∈ ℤ+ satisfying 0 ≤ V ≤ rj , 0 U ≤ sj for j = 1,2,···m , 𝔇n , 𝔈n , 𝔈nj , 𝔇nj ∈ ℝ+ , Λn ,εn ,Λnj ,εnj ∈ ℂ (j = 1,2, ··· , rj ;n = 1,2,··· , sj ),
A Q k , P k k
A_{{\mathfrak{Q}_k},{\kern 1pt} {\mathfrak{P}_k}}^k
are real or complex numbers arbitrary constant for k = 1,2,3,··· ,s and ρi > 0 for i = 1,2,··· ,m, then the outcome shown below is accurate:
(38)
ℐ 0 + ς , ϰ , ϖ ( t α − 1 ∏ j = 1 s S Q j P j [ c j t λ j ] Γ ℵ r j , s j , ρ j ; m U , V [ z t μ | ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( ε n , E n ) 1 , U , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] ) ( x ) = x α − ϰ − 1 ∑ k 1 = 0 [ Q 1 / P 1 ] ⋯ ∑ k s = 0 [ Q s / P s ] ( − Q 1 ) P 1 k 1 ⋯ ( − Q s ) P s k s k 1 ! ⋯ k s ! A Q 1 , P 1 ( 1 ) ⋯ A Q s , P s ( s ) c 1 k 1 ⋯ c s k s ( x ) ∑ j = 1 s λ j k j × Γ ℵ r j + 2 , s j + 3 , ρ j ; m U , V + 2 [ z x μ | ( 1 − α − ∑ j = 1 s λ j k j , μ ) , ( 1 − α + ϰ − ϖ − ∑ j = 1 s λ j k j , μ ) , ( ε n , E n ) 1 , U , ( 1 − α − ς − ϖ − ∑ j = 1 s λ j k j , μ ) , ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( 1 − α + ϰ − ∑ j = 1 s λ j k j , μ ) , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] .
\begin{array}{l}\mathcal{I}_{0 + }^{\varsigma ,{\kern 1pt} \varkappa ,{\kern 1pt} \varpi }{\kern 1pt} \left( {{t^{\alpha - 1}}\prod\limits_{j = 1}^s S_{{\mathfrak{Q}_j}}^{{\mathfrak{P}_j}}[{c_j}{\kern 1pt} {t^{{\lambda _j}}}]} \right.\\\left. {^\Gamma \aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V}{\kern 1pt} \left[ {z{\kern 1pt} {t^\mu }\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\\ \end{array}} \right]} \right)(x)\\ = {x^{\alpha - \varkappa - 1}}\sum\limits_{{\mathfrak{k}_1} = 0}^{[{\mathfrak{Q}_1}/{\mathfrak{P}_1}]} \cdots \sum\limits_{{\mathfrak{k}_s} = 0}^{[{\mathfrak{Q}_s}/{\mathfrak{P}_s}]} \frac{{{{( - {\mathfrak{Q}_1})}_{{\mathfrak{P}_1}{\kern 1pt} {\mathfrak{k}_1}}} \cdots {{( - {\mathfrak{Q}_s})}_{{\mathfrak{P}_s}{\kern 1pt} {\mathfrak{k}_s}}}}}{{{\mathfrak{k}_1}! \cdots {\mathfrak{k}_s}!}}A_{{\mathfrak{Q}_1},{\kern 1pt} {\mathfrak{P}_1}}^{(1)} \cdots A_{{\mathfrak{Q}_s},{\kern 1pt} {\mathfrak{P}_s}}^{(s)}{\kern 1pt} {\kern 1pt} c_1^{{\mathfrak{k}_1}} \cdots c_s^{{\mathfrak{k}_s}}{\kern 1pt} {(x)^{\sum\limits_{j = 1}^s {\lambda _j}{\kern 1pt} {\mathfrak{k}_j}}}\\{ \times ^\Gamma }\aleph _{{r_j} + 2,{\kern 1pt} {s_j} + 3,{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V + 2}{\kern 1pt} \left[ {z{\kern 1pt} {x^\mu }\Biggl|\begin{array}{*{20}{c}}{(1 - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),(1 - \alpha + \varkappa - \varpi - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} (1 - \alpha - \varsigma - \varpi - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),}\\ \end{array}} \right.\\\left. {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{(1 - \alpha + \varkappa - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\end{array}} \right].\end{array}
The same result can be obtained concerning Saigo FIO for the lower incomplete ℵ-function.
Remark 6
By substituting ϰ = −ϛ and ϰ = 0 in Corollary 5 , respectively, we can also get findings for the fractional derivative operators of R-L and E-K.
Theorem 6
Let ϛ , ϛ ', ϰ , ϰ ', ϖ ,z ,α ∈ ℂ and ℜ(ϖ ) > 0,μ > 0,λk > 0 (k = 1,2,3,··· ,s ),
ℜ ( α ) − μ min 1 ≤ j ≤ U ℜ ( ε j ϰ j ) < 1 + min [ ℜ ( − ϰ ) , ℜ ( ς + ς ' − ϖ ) , ℜ ( ς + ϰ ' − ϖ ) ] .
\Re (\alpha ) - \mu \mathop {\min }\limits_{1 \le j \le U} \Re \left( {\frac{{{\varepsilon _j}}}{{{\varkappa _j}}}} \right) < 1 + \min [\Re ( - \varkappa ),\Re (\varsigma + {\varsigma ^\prime } - \varpi ),\Re (\varsigma + {\varkappa ^\prime } - \varpi )].
Further the parameters rj , s j , 𝔓 ∈ ℤ+ satisfying 0 ≤ V ≤ rj ,0 ≤ U ≤ sj for j = 1,2,···m , 𝔇n , 𝔈n , 𝔈nj , 𝔇nj ∈ ℝ+ , Λn ,εn ,Λnj ,εnj ∈ ℂ (j = 1,2,··· , rj ;n = 1,2,··· , sj ),
A Q k , P k k
A_{{\mathfrak{Q}_k},{\kern 1pt} {\mathfrak{P}_k}}^k
are real or complex numbers arbitrary constant for k = 1,2,··· ,s and ρi > 0 for i = 1,2,··· ,m, then the outcome shown below is accurate:
(39)
ℐ − ς , ς ' , ϰ , ϰ ' , ϖ ( t α − 1 ∏ j = 1 s S Q j P j [ c j t λ j ] Γ ℵ r j , s j , ρ j ; m U , V [ z t μ | ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( ε n , E n ) 1 , U , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] ) ( x ) = x α − ς − ς ' + ϖ − 1 ∑ k 1 = 0 [ Q 1 / P 1 ] ⋯ ∑ k s = 0 [ Q s / P s ] ( − Q 1 ) P 1 k 1 ⋯ ( − Q s ) P s k s k 1 ! ⋯ k s ! A Q 1 , P 1 ( 1 ) ⋯ A Q s , P s ( s ) c 1 k 1 ⋯ c s k s ( x ) ∑ j = 1 s λ j k j × Γ ℵ r j + 3 , s j + 3 , ρ j ; m U + 3 , V [ z x μ | ( 1 − α − ∑ j = 1 s λ j k j , μ ) , ( 1 − α + ς + ς ' + ϰ ' − ϖ − ∑ j = 1 s λ j k j , μ ) , ( ε n , E n ) 1 , U , ( 1 − α + ς + ς ' − ϖ − ∑ j = 1 s λ j k j , μ ) , ( 1 − α + ς − ϰ − ∑ j = 1 s λ j k j , μ ) , ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( 1 − α − ϰ − ∑ j = 1 s λ j k j , μ ) , ( 1 + ς + ϰ ' − ϖ − α − ∑ j = 1 s λ j k j , μ ) , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] .
\begin{array}{l}\mathcal{I}_ - ^{\varsigma ,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },\varpi }{\kern 1pt} \left( {{t^{\alpha - 1}}\prod\limits_{j = 1}^s S_{{\mathfrak{Q}_j}}^{{\mathfrak{P}_j}}[{c_j}{\kern 1pt} {t^{{\lambda _j}}}]} \right.\\\left. {^\Gamma \aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V}{\kern 1pt} \left[ {z{\kern 1pt} {t^\mu }\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\\ \end{array}} \right]} \right)(x)\\ = {x^{\alpha - \varsigma - {\varsigma ^\prime } + \varpi - 1}}\sum\limits_{{\mathfrak{k}_1} = 0}^{[{\mathfrak{Q}_1}/{\mathfrak{P}_1}]} \cdots \sum\limits_{{\mathfrak{k}_s} = 0}^{[{\mathfrak{Q}_s}/{\mathfrak{P}_s}]} \frac{{{{( - {\mathfrak{Q}_1})}_{{\mathfrak{P}_1}{\kern 1pt} {\mathfrak{k}_1}}} \cdots {{( - {\mathfrak{Q}_s})}_{{\mathfrak{P}_s}{\kern 1pt} {\mathfrak{k}_s}}}}}{{{\mathfrak{k}_1}! \cdots {\mathfrak{k}_s}!}}A_{{\mathfrak{Q}_1},{\kern 1pt} {\mathfrak{P}_1}}^{(1)} \cdots A_{{\mathfrak{Q}_s},{\kern 1pt} {\mathfrak{P}_s}}^{(s)}{\kern 1pt} {\kern 1pt} c_1^{{\mathfrak{k}_1}} \cdots c_s^{{\mathfrak{k}_s}}{\kern 1pt} {(x)^{\sum\limits_{j = 1}^s {\lambda _j}{\kern 1pt} {\mathfrak{k}_j}}}\\{ \times ^\Gamma }\aleph _{{r_j} + 3,{\kern 1pt} {s_j} + 3,{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U + 3,V}{\kern 1pt} \left[ {z{\kern 1pt} {x^\mu }\Biggl|\begin{array}{*{20}{c}}{(1 - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),(1 - \alpha + \varsigma + {\varsigma ^\prime } + {\varkappa ^\prime } - \varpi - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} (1 - \alpha + \varsigma + {\varsigma ^\prime } - \varpi - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),}\\ \end{array}} \right.\\\left. {\;\;\;\;\begin{array}{*{20}{c}}{(1 - \alpha + \varsigma - \varkappa - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} ({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{(1 - \alpha - \varkappa - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} (1 + \varsigma + {\varkappa ^\prime } - \varpi - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\end{array}} \right].\end{array}
Proof
The LHS of equation (39) is:
(40)
T 2 = ℐ − ς , ς ' , ϰ , ϰ ' , ϖ ( t α − 1 ∏ j = 1 s S Q j P j [ c j t λ j ] Γ ℵ r j , s j , ρ j ; m U , V [ z t μ | ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( ε n , E n ) 1 , U , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] ) .
\begin{array}{l}{T_2} = \mathcal{I}_ - ^{\varsigma ,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },\varpi }{\kern 1pt} \left( {{t^{\alpha - 1}}\prod\limits_{j = 1}^s S_{{\mathfrak{Q}_j}}^{{\mathfrak{P}_j}}[{c_j}{\kern 1pt} {t^{{\lambda _j}}}]} \right.\\\left. {^\Gamma \aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V}{\kern 1pt} \left[ {z{\kern 1pt} {t^\mu }\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\\ \end{array}} \right]} \right).\end{array}
Replace the incomplete ℵ-function and Srivastava polynomial by equation (24) and (4) respectively and by reversing the summation order, we discover the subsequent form:
(41)
T 2 = ∑ k 1 = 0 [ Q 1 / P 1 ] ⋯ ∑ k s = 0 [ Q s / P s ] ( − Q 1 ) P 1 k 1 ⋯ ( − Q s ) P s k s k 1 ! ⋯ k s ! × A Q 1 , P 1 ( 1 ) ⋯ A Q s , P s ( s ) c 1 k 1 ⋯ c s k s × 1 2 π ι ∫ $ Ψ ( q , Y ) z − q ( ℐ − ς , ς ' , ϰ , ϰ ' , ϖ t − ( − α − ∑ j = 1 s λ j k j + μ q + 1 ) ) ( x ) d q ,
\begin{array}{l}{T_2} = \sum\limits_{{\mathfrak{k}_1} = 0}^{[{\mathfrak{Q}_1}/{\mathfrak{P}_1}]} \cdots \sum\limits_{{\mathfrak{k}_s} = 0}^{[{\mathfrak{Q}_s}/{\mathfrak{P}_s}]} \frac{{{{( - {\mathfrak{Q}_1})}_{{\mathfrak{P}_1}{\kern 1pt} {\mathfrak{k}_1}}} \cdots {{( - {\mathfrak{Q}_s})}_{{\mathfrak{P}_s}{\kern 1pt} {\mathfrak{k}_s}}}}}{{{\mathfrak{k}_1}! \cdots {\mathfrak{k}_s}!}}{\kern 1pt} \times A_{{\mathfrak{Q}_1},{\kern 1pt} {\mathfrak{P}_1}}^{(1)} \cdots A_{{\mathfrak{Q}_s},{\kern 1pt} {\mathfrak{P}_s}}^{(s)}{\kern 1pt} {\kern 1pt} c_1^{{\mathfrak{k}_1}} \cdots c_s^{{\mathfrak{k}_s}}\\ \times \frac{1}{{2\pi \iota }}\int_\$ \;\Psi (q,\mathcal{Y})\;{z^{ - q}}\left( {\mathcal{I}_ - ^{\varsigma ,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },\varpi }{\kern 1pt} {t^{ - ( - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} + \mu q + 1)}}} \right)(x)dq,\end{array}
where Ψ (q ,𝒴) is defined in equation (25) .
Using equation (19) of Lemma 1 , we discover the subsequent form:
(42)
T 2 = ∑ k 1 = 0 [ Q 1 / P 1 ] ⋯ ∑ k s = 0 [ Q s / P s ] ( − Q 1 ) P 1 k 1 ⋯ ( − Q s ) P s k s k 1 ! ⋯ k s ! × A Q 1 , P 1 ( 1 ) ⋯ A Q s , P s ( s ) c 1 k 1 ⋯ c s k s × 1 2 π ι ∫ $ x α − ς − ς ' + ϖ + ∑ j = 1 s λ j k j − 1 Ψ ( q , Y ) ( z x μ )
− q Γ ( 1 − α − ϰ − ∑ j = 1 s λ j k j + μ q ) Γ ( 1 − α − ∑ j = 1 s λ j k j + μ q ) × Γ ( 1 + ς + ς ' − ϖ − α − ∑ j = 1 s λ j k j + μ q ) Γ ( 1 − α + ς + ϰ ' − ϖ − ∑ j = 1 s λ j k j + μ q ) Γ ( 1 − α + ς − ϰ − ∑ j = 1 s λ j k j + μ q ) Γ ( 1 − α + ς + ς ' + ϰ ' − ϖ − ∑ j = 1 s λ j k j + μ q ) d q .
\begin{array}{l}{T_2} = \sum\limits_{{\mathfrak{k}_1} = 0}^{[{\mathfrak{Q}_1}/{\mathfrak{P}_1}]} \cdots \sum\limits_{{\mathfrak{k}_s} = 0}^{[{\mathfrak{Q}_s}/{\mathfrak{P}_s}]} \frac{{{{( - {\mathfrak{Q}_1})}_{{\mathfrak{P}_1}{\kern 1pt} {\mathfrak{k}_1}}} \cdots {{( - {\mathfrak{Q}_s})}_{{\mathfrak{P}_s}{\kern 1pt} {\mathfrak{k}_s}}}}}{{{\mathfrak{k}_1}! \cdots {\mathfrak{k}_s}!}}{\kern 1pt} \times A_{{\mathfrak{Q}_1},{\kern 1pt} {\mathfrak{P}_1}}^{(1)} \cdots A_{{\mathfrak{Q}_s},{\kern 1pt} {\mathfrak{P}_s}}^{(s)}{\kern 1pt} {\kern 1pt} c_1^{{\mathfrak{k}_1}} \cdots c_s^{{\mathfrak{k}_s}}\\ \times \frac{1}{{2\pi \iota }}\int_\$ \;{x^{\alpha - \varsigma - {\varsigma ^\prime } + \varpi + \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} - 1}}{\kern 1pt} {\kern 1pt} \Psi (q,\mathcal{Y})\;{(z{x^\mu })^{ - q}}\frac{{\Gamma (1 - \alpha - \varkappa - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} + \mu q)}}{{\Gamma (1 - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} + \mu q)}} \times \\\frac{{\Gamma (1 + \varsigma + {\varsigma ^\prime } - \varpi - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} + \mu q){\kern 1pt} \Gamma (1 - \alpha + \varsigma + {\varkappa ^\prime } - \varpi - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} + \mu q)}}{{\Gamma (1 - \alpha + \varsigma - \varkappa - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} + \mu q){\kern 1pt} \Gamma (1 - \alpha + \varsigma + {\varsigma ^\prime } + {\varkappa ^\prime } - \varpi - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} + \mu q)}}dq.\end{array}
Finally, after some adjustment of terms, we obtain RHS of equation (39) .
Theorem 7
Let ϛ , ϛ ', ϰ , ϰ ', ϖ ,z ,α ∈ ℂ and ℜ(ϖ ) > 0,μ > 0,λk > 0 (k = 1,2,3,··· ,s ),
ℜ ( α ) − μ min 1 ≤ j ≤ U ℜ ( ε j ϰ j ) < 1 + min [ ℜ ( − ϰ ) , ℜ ( ς + ς ' − ϖ ) , ℜ ( ς + ϰ ' − ϖ ) ] .
\Re (\alpha ) - \mu \mathop {\min }\limits_{1 \le j \le U} \Re \left( {\frac{{{\varepsilon _j}}}{{{\varkappa _j}}}} \right) < 1 + \min [\Re ( - \varkappa ),\Re (\varsigma + {\varsigma ^\prime } - \varpi ),\Re (\varsigma + {\varkappa ^\prime } - \varpi )].
Further the parameters rj , s j , 𝔓 ∈ ℤ+ satisfying 0 ≤ V ≤ rj ,0 ≤ U ≤ sj for j = 1,2,···m , 𝔇n , 𝔈n , 𝔈nj , 𝔇nj ∈ ℝ+ , Λn ,εn ,Λnj ,εnj ∈ ℂ (j = 1,2,··· , rj ;n = 1,2,··· , sj ),
A Q k , P k k
A_{{\mathfrak{Q}_k},{\kern 1pt} {\mathfrak{P}_k}}^k
are real or complex numbers arbitrary constant for k = 1,2,··· ,s and ρi > 0 for i = 1,2,··· ,m, then the outcome shown below is accurate:
(43)
ℐ − ς , ς ' , ϰ , ϰ ' , ϖ ( t α − 1 ∏ j = 1 s S Q j P j [ c j t λ j ] γ ℵ r j , s j , ρ j ; m U , V [ z t μ | ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( ε n , E n ) 1 , U , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] ) ( x ) = x α − ς − ς ' + ϖ − 1 ∑ k 1 = 0 [ Q 1 / P 1 ] ⋯ ∑ k s = 0 [ Q s / P s ] ( − Q 1 ) P 1 k 1 ⋯ ( − Q s ) P s k s k 1 ! ⋯ k s ! A Q 1 , P 1 ( 1 ) ⋯ A Q s , P s ( s ) c 1 k 1 ⋯ c s k s ( x ) ∑ j = 1 s λ j k j × γ ℵ r j + 3 , s j + 3 , ρ j ; m U + 3 , V [ z x μ | ( 1 − α − ∑ j = 1 s λ j k j , μ ) , ( 1 − α + ς + ς ' + ϰ ' − ϖ − ∑ j = 1 s λ j k j , μ ) , ( ε n , E n ) 1 , U , ( 1 − α + ς + ς ' − ϖ − ∑ j = 1 s λ j k j , μ ) , ( 1 − α + ς − ϰ − ∑ j = 1 s λ j k j , μ ) , ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( 1 − α − ϰ − ∑ j = 1 s λ j k j , μ ) , ( 1 + ς + ϰ ' − ϖ − α − ∑ j = 1 s λ j k j , μ ) , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] .
\begin{array}{l}\mathcal{I}_ - ^{\varsigma ,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },\varpi }{\kern 1pt} \left( {{t^{\alpha - 1}}\prod\limits_{j = 1}^s S_{{\mathfrak{Q}_j}}^{{\mathfrak{P}_j}}[{c_j}{\kern 1pt} {t^{{\lambda _j}}}]} \right.\\\left. {^\gamma \aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V}{\kern 1pt} \left[ {z{\kern 1pt} {t^\mu }\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\\ \end{array}} \right]} \right)(x)\\ = {x^{\alpha - \varsigma - {\varsigma ^\prime } + \varpi - 1}}\sum\limits_{{\mathfrak{k}_1} = 0}^{[{\mathfrak{Q}_1}/{\mathfrak{P}_1}]} \cdots \sum\limits_{{\mathfrak{k}_s} = 0}^{[{\mathfrak{Q}_s}/{\mathfrak{P}_s}]} \frac{{{{( - {\mathfrak{Q}_1})}_{{\mathfrak{P}_1}{\kern 1pt} {\mathfrak{k}_1}}} \cdots {{( - {\mathfrak{Q}_s})}_{{\mathfrak{P}_s}{\kern 1pt} {\mathfrak{k}_s}}}}}{{{\mathfrak{k}_1}! \cdots {\mathfrak{k}_s}!}}A_{{\mathfrak{Q}_1},{\kern 1pt} {\mathfrak{P}_1}}^{(1)} \cdots A_{{\mathfrak{Q}_s},{\kern 1pt} {\mathfrak{P}_s}}^{(s)}{\kern 1pt} {\kern 1pt} c_1^{{\mathfrak{k}_1}} \cdots c_s^{{\mathfrak{k}_s}}{\kern 1pt} {(x)^{\sum\limits_{j = 1}^s {\lambda _j}{\kern 1pt} {\mathfrak{k}_j}}}\\{ \times ^\gamma }\aleph _{{r_j} + 3,{\kern 1pt} {s_j} + 3,{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U + 3,V}{\kern 1pt} \left[ {z{\kern 1pt} {x^\mu }\Biggl|\begin{array}{*{20}{c}}{(1 - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),(1 - \alpha + \varsigma + {\varsigma ^\prime } + {\varkappa ^\prime } - \varpi - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} (1 - \alpha + \varsigma + {\varsigma ^\prime } - \varpi - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),}\\ \end{array}} \right.\\\left. {\;\;\;\;\begin{array}{*{20}{c}}{(1 - \alpha + \varsigma - \varkappa - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} ({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{(1 - \alpha - \varkappa - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} (1 + \varsigma + {\varkappa ^\prime } - \varpi - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\end{array}} \right].\end{array}
Proof
Theorem 7 is proved in the same way as Theorem 6 with the same conditions. The following corollary is obtained regarding the Saigo FIO [10 ] in light of the equation (15) .
Corollary 8
Let ϛ, ϰ, ϖ ,z,α ∈ ℂ and ℜ(ϛ ) > 0,μ > 0,λk > 0 (k = 1,2,··· ,s ),
ℜ ( α ) − μ min 1 ≤ j ≤ U ℜ ( ε j ϰ j ) < 1 + min [ ℜ ( ϰ ) , ℜ ( ϖ ) ] .
\Re (\alpha ) - \mu \mathop {\min }\limits_{1 \le j \le U} \Re \left( {\frac{{{\varepsilon _j}}}{{{\varkappa _j}}}} \right) < 1 + \min [\Re (\varkappa ),\Re (\varpi )].
Further the parameters rj , s j , 𝔓 ∈ ℤ+ satisfying 0 ≤ V ≤ rj ,0 ≤ U ≤ sj for j = 1,2,···m , 𝔇n , 𝔈n , 𝔈nj , 𝔇nj ∈ ℝ+ , Λn ,εn ,Λnj ,εnj ∈ ℂ (j = 1,2, ··· , rj ;n = 1,2,··· , sj ),
A Q k , P k k
A_{{\mathfrak{Q}_k},{\kern 1pt} {\mathfrak{P}_k}}^k
are real or complex numbers arbitrary constant for k = 1,2,··· ,s and ρi > 0 for i = 1,2,··· ,m, then the outcome shown below is accurate:
(44)
ℐ − ς , ϰ , ϖ ( t α − 1 ∏ j = 1 s S Q j P j [ c j t λ j ] Γ ℵ r j , s j , ρ j ; m U , V [ z t μ | ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( ε n , E n ) 1 , U , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] ) ( x ) = x α − ϰ − 1 ∑ k 1 = 0 [ Q 1 / P 1 ] ⋯ ∑ k s = 0 [ Q s / P s ] ( − Q 1 ) P 1 k 1 ⋯ ( − Q s ) P s k s k 1 ! ⋯ k s ! A Q 1 , P 1 ( 1 ) ⋯ A Q s , P s ( s ) c 1 k 1 ⋯ c s k s ( x ) ∑ j = 1 s λ j k j × Γ ℵ r j + 2 , s j + 2 , ρ j ; m U + 2 , V [ z x μ | ( 1 − α − ∑ j = 1 s λ j k j , μ ) , ( 1 − α + ς + ϰ + ϖ − ∑ j = 1 s λ j k j , μ ) , ( ε n , E n ) 1 , U , ( 1 + ϰ − α − ∑ j = 1 s λ j k j , μ ) , ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( 1 − α + ϖ − ∑ j = 1 s λ j k j , μ ) , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] .
\begin{array}{l}\mathcal{I}_ - ^{\varsigma ,{\kern 1pt} \varkappa ,{\kern 1pt} \varpi }{\kern 1pt} \left( {{t^{\alpha - 1}}\prod\limits_{j = 1}^s S_{{\mathfrak{Q}_j}}^{{\mathfrak{P}_j}}[{c_j}{\kern 1pt} {t^{{\lambda _j}}}]} \right.\\\left. {^\Gamma \aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V}\left[ {z{\kern 1pt} {t^\mu }\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\\ \end{array}} \right]} \right)(x)\\ = {x^{\alpha - \varkappa - 1}}\sum\limits_{{\mathfrak{k}_1} = 0}^{[{\mathfrak{Q}_1}/{\mathfrak{P}_1}]} \cdots \sum\limits_{{\mathfrak{k}_s} = 0}^{[{\mathfrak{Q}_s}/{\mathfrak{P}_s}]} \frac{{{{( - {\mathfrak{Q}_1})}_{{\mathfrak{P}_1}{\kern 1pt} {\mathfrak{k}_1}}} \cdots {{( - {\mathfrak{Q}_s})}_{{\mathfrak{P}_s}{\kern 1pt} {\mathfrak{k}_s}}}}}{{{\mathfrak{k}_1}! \cdots {\mathfrak{k}_s}!}}A_{{\mathfrak{Q}_1},{\kern 1pt} {\mathfrak{P}_1}}^{(1)} \cdots A_{{\mathfrak{Q}_s},{\kern 1pt} {\mathfrak{P}_s}}^{(s)}{\kern 1pt} {\kern 1pt} c_1^{{\mathfrak{k}_1}} \cdots c_s^{{\mathfrak{k}_s}}{\kern 1pt} {(x)^{\sum\limits_{j = 1}^s {\lambda _j}{\kern 1pt} {\mathfrak{k}_j}}}\\{ \times ^\Gamma }\aleph _{{r_j} + 2,{\kern 1pt} {s_j} + 2,{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U + 2,V}{\kern 1pt} \left[ {z{\kern 1pt} {x^\mu }\Biggl|\begin{array}{*{20}{c}}{(1 - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),(1 - \alpha + \varsigma + \varkappa + \varpi - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} (1 + \varkappa - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),}\\ \end{array}} \right.\\\left. {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{(1 - \alpha + \varpi - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\end{array}} \right].\end{array}
The same result can be obtained concerning Saigo FIO for the lower incomplete ℵ-function.
Remark 7
By substituting ϰ = −ϛ and ϰ = 0 in Corollary 8 , respectively, we can also get findings for the fractional derivative operators of R-L and E-K.
4 Fractional derivative formulas
In this part, we create two formulas for fractional derivative that multiply incomplete ℵ-functions and the generic class of polynomials specified in (24 ) and (4 ), respectively.
Theorem 9
Let ϛ , ϛ ', ϰ , ϰ ', ϖ ,z ,α ∈ ℂ and ℜ(ϖ ) > 0,μ > 0,λk > 0 (k = 1,2,3,··· ,s ),
μ max 1 ≤ j ≤ U [ − ℜ ( ε j ) E j ] < ℜ ( α ) + min [ 0 , ℜ ( ς − ϰ ) , ℜ ( ς ' + ϰ ' + ς − ϖ ) ] .
\mu \mathop {\max }\limits_{1 \le j \le U} \left[ {\frac{{ - \Re ({\varepsilon _j})}}{{{\mathfrak{E}_j}}}} \right] < \Re (\alpha ) + \min [0,\Re (\varsigma - \varkappa ),\Re ({\varsigma ^\prime } + {\varkappa ^\prime } + \varsigma - \varpi )].
Further the parameters rj , s j , 𝔓 ∈ ℤ+ satisfying 0 ≤ V ≤ rj ,0 ≤ U ≤ sj for j = 1,2,···m , 𝔇n , 𝔈n , 𝔈nj , 𝔇nj ∈ ℝ+ , Λn ,εn ,Λnj ,εnj ∈ ℂ (j = 1,2,··· , rj ;n = 1,2,··· , sj ),
A Q k , P k k
A_{{\mathfrak{Q}_k},{\kern 1pt} {\mathfrak{P}_k}}^k
are real or complex numbers arbitrary constant for k = 1,2,··· ,s and ρi > 0 for i = 1,2,··· ,m, then the outcome shown below is accurate:
(45)
D 0 + ς , ς ' , ϰ , ϰ ' , ϖ ( t α − 1 ∏ j = 1 s S Q j P j [ c j t λ j ] Γ ℵ r j , s j , ρ j ; m U , V [ z t μ | ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( ε n , E n ) 1 , U , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] ) ( x ) = x α + ς + ς ' − ϖ − 1 ∑ k 1 = 0 [ Q 1 / P 1 ] ⋯ ∑ k s = 0 [ Q s / P s ] ( − Q 1 ) P 1 k 1 ⋯ ( − Q s ) P s k s k 1 ! ⋯ k s ! A Q 1 , P 1 ( 1 ) ⋯ A Q s , P s ( s ) c 1 k 1 ⋯ c s k s ( x ) ∑ j = 1 s λ j k j × Γ ℵ r j + 3 , s j + 3 , ρ j ; m U , V + 3 [ z x μ | ( 1 − α − ∑ j = 1 s λ j k j , μ ) , ( 1 − α − ς − ς ' − ϰ ' + ϖ − ∑ j = 1 s λ j k j , μ ) , ( ε n , E n ) 1 , U , ( 1 − ς − ς ' + ϖ − α − ∑ j = 1 s λ j k j , μ ) , ( 1 − α − ς + ϰ − ∑ j = 1 s λ j k j , μ ) , ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( 1 − α + ϰ − ∑ j = 1 s λ j k j , μ ) , ( 1 − ς − ϰ ' + ϖ − α − ∑ j = 1 s λ j k j , μ ) , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] .
\begin{array}{l}\mathcal{D}_{0 + }^{\varsigma ,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },\varpi }{\kern 1pt} \left( {{t^{\alpha - 1}}\prod\limits_{j = 1}^s S_{{\mathfrak{Q}_j}}^{{\mathfrak{P}_j}}[{c_j}{\kern 1pt} {t^{{\lambda _j}}}]} \right.\\\left. {^\Gamma \aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V}\left[ {z{\kern 1pt} {t^\mu }\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\\ \end{array}} \right]} \right)(x)\\ = {x^{\alpha + \varsigma + {\varsigma ^\prime } - \varpi - 1}}\sum\limits_{{\mathfrak{k}_1} = 0}^{[{\mathfrak{Q}_1}/{\mathfrak{P}_1}]} \cdots \sum\limits_{{\mathfrak{k}_s} = 0}^{[{\mathfrak{Q}_s}/{\mathfrak{P}_s}]} \frac{{{{( - {\mathfrak{Q}_1})}_{{\mathfrak{P}_1}{\kern 1pt} {\mathfrak{k}_1}}} \cdots {{( - {\mathfrak{Q}_s})}_{{\mathfrak{P}_s}{\kern 1pt} {\mathfrak{k}_s}}}}}{{{\mathfrak{k}_1}! \cdots {\mathfrak{k}_s}!}}A_{{\mathfrak{Q}_1},{\kern 1pt} {\mathfrak{P}_1}}^{(1)} \cdots A_{{\mathfrak{Q}_s},{\kern 1pt} {\mathfrak{P}_s}}^{(s)}{\kern 1pt} {\kern 1pt} c_1^{{\mathfrak{k}_1}} \cdots c_s^{{\mathfrak{k}_s}}{\kern 1pt} {(x)^{\sum\limits_{j = 1}^s {\lambda _j}{\kern 1pt} {\mathfrak{k}_j}}}\\{ \times ^\Gamma }\aleph _{{r_j} + 3,{\kern 1pt} {s_j} + 3,{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V + 3}{\kern 1pt} \left[ {z{\kern 1pt} {x^\mu }\Biggl|\begin{array}{*{20}{c}}{(1 - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),(1 - \alpha - \varsigma - {\varsigma ^\prime } - {\varkappa ^\prime } + \varpi - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} (1 - \varsigma - {\varsigma ^\prime } + \varpi - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),}\\ \end{array}} \right.\\\left. {\;\;\;\;\begin{array}{*{20}{c}}{(1 - \alpha - \varsigma + \varkappa - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} ({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{(1 - \alpha + \varkappa - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} (1 - \varsigma - {\varkappa ^\prime } + \varpi - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\end{array}} \right].\end{array}
Proof
The LHS of equation (45) is:
(46)
T 3 = D 0 + ς , ς ' , ϰ , ϰ ' , ϖ ( t α − 1 ∏ j = 1 s S Q j P j [ c j t λ j ] Γ ℵ r j , s j , ρ j ; m U , V [ z t μ | ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( ε n , E n ) 1 , U , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] ) ( x ) .
\begin{array}{l}{T_3} = \mathcal{D}_{0 + }^{\varsigma ,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },\varpi }{\kern 1pt} \left( {{t^{\alpha - 1}}\prod\limits_{j = 1}^s S_{{\mathfrak{Q}_j}}^{{\mathfrak{P}_j}}[{c_j}{\kern 1pt} {t^{{\lambda _j}}}]} \right.\\\left. {^\Gamma \aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V}{\kern 1pt} \left[ {z{\kern 1pt} {t^\mu }\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\\ \end{array}} \right]} \right)(x).\end{array}
Replace the incomplete ℵ-function and Srivastava polynomial by equation (24) and (4) respectively and by reversing the summation order, we discover the subsequent form:
(47)
T 3 = ∑ k 1 = 0 [ Q 1 / P 1 ] ⋯ ∑ k s = 0 [ Q s / P s ] ( − Q 1 ) P 1 k 1 ⋯ ( − Q s ) P s k s k 1 ! ⋯ k s ! × A Q 1 , P 1 ( 1 ) ⋯ A Q s , P s ( s ) c 1 k 1 ⋯ c s k s × 1 2 π ι ∫ $ Ψ ( q , Y ) z − q ( D 0 + ς , ς ' , ϰ , ϰ ' , ϖ t α + ∑ j = 1 s λ j k j − μ q − 1 ) ( x ) d q ,
\begin{array}{l}{T_3} = \sum\limits_{{\mathfrak{k}_1} = 0}^{[{\mathfrak{Q}_1}/{\mathfrak{P}_1}]} \cdots \sum\limits_{{\mathfrak{k}_s} = 0}^{[{\mathfrak{Q}_s}/{\mathfrak{P}_s}]} \frac{{{{( - {\mathfrak{Q}_1})}_{{\mathfrak{P}_1}{\kern 1pt} {\mathfrak{k}_1}}} \cdots {{( - {\mathfrak{Q}_s})}_{{\mathfrak{P}_s}{\kern 1pt} {\mathfrak{k}_s}}}}}{{{\mathfrak{k}_1}! \cdots {\mathfrak{k}_s}!}}{\kern 1pt} \times A_{{\mathfrak{Q}_1},{\kern 1pt} {\mathfrak{P}_1}}^{(1)} \cdots A_{{\mathfrak{Q}_s},{\kern 1pt} {\mathfrak{P}_s}}^{(s)}{\kern 1pt} {\kern 1pt} c_1^{{\mathfrak{k}_1}} \cdots c_s^{{\mathfrak{k}_s}}\\ \times \frac{1}{{2\pi \iota }}\int_\$ \;\Psi (q,\mathcal{Y})\;{z^{ - q}}\left( {\mathcal{D}_{0 + }^{\varsigma ,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },\varpi }{\kern 1pt} {t^{\alpha + \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} - \mu q - 1}}} \right)(x)dq,\end{array}
where Ψ (q ,𝒴) is defined in equation (25) .
Using equation (20) of Lemma 2 , we discover the subsequent form:
(48)
T 3 = ∑ k 1 = 0 [ Q 1 / P 1 ] ⋯ ∑ k s = 0 [ Q s / P s ] ( − Q 1 ) P 1 k 1 ⋯ ( − Q s ) P s k s k 1 ! ⋯ k s ! × A Q 1 , P 1 ( 1 ) ⋯ A Q s , P s ( s ) c 1 k 1 ⋯ c s k s × 1 2 π ι ∫ $ x α − ς − ς ' + ϖ + ∑ j = 1 s λ j k j − 1 Ψ ( q , Y ) ( z x μ )
− q Γ ( α + ∑ j = 1 s λ j k j − μ q ) Γ ( α − ϰ + ∑ j = 1 s λ j k j − μ q ) × Γ ( ς − ϰ + α + ∑ j = 1 s λ j k j − μ q ) Γ ( α + ς + ς ' + ϰ ' − ϖ + ∑ j = 1 s λ j k j − μ q ) Γ ( α + ς + ϰ ' − ϖ + ∑ j = 1 s λ j k j − μ q ) Γ ( α + ς + ς ' − ϖ + ∑ j = 1 s λ j k j − μ q ) d q .
\begin{array}{l}{T_3} = \sum\limits_{{\mathfrak{k}_1} = 0}^{[{\mathfrak{Q}_1}/{\mathfrak{P}_1}]} \cdots \sum\limits_{{\mathfrak{k}_s} = 0}^{[{\mathfrak{Q}_s}/{\mathfrak{P}_s}]} \frac{{{{( - {\mathfrak{Q}_1})}_{{\mathfrak{P}_1}{\kern 1pt} {\mathfrak{k}_1}}} \cdots {{( - {\mathfrak{Q}_s})}_{{\mathfrak{P}_s}{\kern 1pt} {\mathfrak{k}_s}}}}}{{{\mathfrak{k}_1}! \cdots {\mathfrak{k}_s}!}}{\kern 1pt} \times A_{{\mathfrak{Q}_1},{\kern 1pt} {\mathfrak{P}_1}}^{(1)} \cdots A_{{\mathfrak{Q}_s},{\kern 1pt} {\mathfrak{P}_s}}^{(s)}{\kern 1pt} {\kern 1pt} c_1^{{\mathfrak{k}_1}} \cdots c_s^{{\mathfrak{k}_s}}\\ \times \frac{1}{{2\pi \iota }}\int_\$ \;{x^{\alpha - \varsigma - {\varsigma ^\prime } + \varpi + \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} - 1}}{\kern 1pt} {\kern 1pt} \Psi (q,\mathcal{Y})\;{(z{x^\mu })^{ - q}}\frac{{\Gamma (\alpha + \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} - \mu q)}}{{\Gamma (\alpha - \varkappa + \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} - \mu q)}} \times \\\frac{{\Gamma (\varsigma - \varkappa + \alpha + \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} - \mu q){\kern 1pt} \Gamma (\alpha + \varsigma + {\varsigma ^\prime } + {\varkappa ^\prime } - \varpi + \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} - \mu q)}}{{\Gamma (\alpha + \varsigma + {\varkappa ^\prime } - \varpi + \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} - \mu q){\kern 1pt} \Gamma (\alpha + \varsigma + {\varsigma ^\prime } - \varpi + \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} - \mu q)}}dq.\end{array}
Finally, after some adjustment of terms, we obtain RHS of equation (45) .
Theorem 10
Let ϛ , ϛ ', ϰ , ϰ ', ϖ ,z ,α ∈ ℂ and ℜ(ϖ ) > 0,μ > 0,λ k > 0 (k = 1,2,3,··· ,s ),
μ max 1 ≤ j ≤ U [ − ℜ ( ε j ) E j ] < ℜ ( α ) + min [ 0 , ℜ ( ς − ϰ ) , ℜ ( ς ' + ϰ ' + ς − ϖ ) ] .
\mu \mathop {\max }\limits_{1 \le j \le U} \left[ {\frac{{ - \Re ({\varepsilon _j})}}{{{\mathfrak{E}_j}}}} \right] < \Re (\alpha ) + \min [0,\Re (\varsigma - \varkappa ),\Re ({\varsigma ^\prime } + {\varkappa ^\prime } + \varsigma - \varpi )].
Further the parameters rj , s j , 𝔓 ∈ ℤ+ satisfying 0 ≤ V rj ,0 ≤ U ≤ sj for j = 1,2,···m , 𝔇n , 𝔈n , 𝔈nj , 𝔇nj ∈ ℝ+ , Λn ,εn ,Λnj ,εnj ∈ ℂ (j = 1,2,··· , rj ;n = 1,2,··· , sj ),
A Q k , P k k
A_{{\mathfrak{Q}_k},{\kern 1pt} {\mathfrak{P}_k}}^k
are real or complex numbers arbitrary constant for k = 1,2,3,··· ,s and ρi > 0 for i = 1,2,··· ,m, then the outcome shown below is accurate:
(49)
D 0 + ς , ς ' , ϰ , ϰ ' , ϖ ( t α − 1 ∏ j = 1 s S Q j P j [ c j t λ j ] γ ℵ r j , s j , ρ j ; m U , V [ z t μ | ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( ε n , E n ) 1 , U , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] ) ( x ) = x α + ς + ς ' − ϖ − 1 ∑ k 1 = 0 [ Q 1 / P 1 ] ⋯ ∑ k s = 0 [ Q s / P s ] ( − Q 1 ) P 1 k 1 ⋯ ( − Q s ) P s k s k 1 ! ⋯ k s ! A Q 1 , P 1 ( 1 ) ⋯ A Q s , P s ( s ) c 1 k 1 ⋯ c s k s ( x ) ∑ j = 1 s λ j k j × γ ℵ r j + 3 , s j + 3 , ρ j ; m U , V + 3 [ z x μ | ( 1 − α − ∑ j = 1 s λ j k j , μ ) , ( 1 − α − ς − ς ' − ϰ ' + ϖ − ∑ j = 1 s λ j k j , μ ) , ( ε n , E n ) 1 , U , ( 1 − ς − ς ' + ϖ − α − ∑ j = 1 s λ j k j , μ ) , ( 1 − α − ς + ϰ − ∑ j = 1 s λ j k j , μ ) , ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( 1 − α + ϰ − ∑ j = 1 s λ j k j , μ ) , ( 1 − ς − ϰ ' + ϖ − α − ∑ j = 1 s λ j k j , μ ) , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] .
\begin{array}{l}\mathcal{D}_{0 + }^{\varsigma ,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },\varpi }{\kern 1pt} \left( {{t^{\alpha - 1}}\prod\limits_{j = 1}^s S_{{\mathfrak{Q}_j}}^{{\mathfrak{P}_j}}[{c_j}{\kern 1pt} {t^{{\lambda _j}}}]} \right.\\\left. {^\gamma \aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V}\left[ {z{\kern 1pt} {t^\mu }\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\\ \end{array}} \right]} \right)(x)\\ = {x^{\alpha + \varsigma + {\varsigma ^\prime } - \varpi - 1}}\sum\limits_{{\mathfrak{k}_1} = 0}^{[{\mathfrak{Q}_1}/{\mathfrak{P}_1}]} \cdots \sum\limits_{{\mathfrak{k}_s} = 0}^{[{\mathfrak{Q}_s}/{\mathfrak{P}_s}]} \frac{{{{( - {\mathfrak{Q}_1})}_{{\mathfrak{P}_1}{\kern 1pt} {\mathfrak{k}_1}}} \cdots {{( - {\mathfrak{Q}_s})}_{{\mathfrak{P}_s}{\kern 1pt} {\mathfrak{k}_s}}}}}{{{\mathfrak{k}_1}! \cdots {\mathfrak{k}_s}!}}A_{{\mathfrak{Q}_1},{\kern 1pt} {\mathfrak{P}_1}}^{(1)} \cdots A_{{\mathfrak{Q}_s},{\kern 1pt} {\mathfrak{P}_s}}^{(s)}{\kern 1pt} {\kern 1pt} c_1^{{\mathfrak{k}_1}} \cdots c_s^{{\mathfrak{k}_s}}{\kern 1pt} {(x)^{\sum\limits_{j = 1}^s {\lambda _j}{\kern 1pt} {\mathfrak{k}_j}}}\\{ \times ^\gamma }\aleph _{{r_j} + 3,{\kern 1pt} {s_j} + 3,{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V + 3}{\kern 1pt} \left[ {z{\kern 1pt} {x^\mu }\Biggl|\begin{array}{*{20}{c}}{(1 - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),(1 - \alpha - \varsigma - {\varsigma ^\prime } - {\varkappa ^\prime } + \varpi - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} (1 - \varsigma - {\varsigma ^\prime } + \varpi - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),}\\ \end{array}} \right.\\\left. {\;\;\;\;\begin{array}{*{20}{c}}{(1 - \alpha - \varsigma + \varkappa - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} ({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{(1 - \alpha + \varkappa - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} (1 - \varsigma - {\varkappa ^\prime } + \varpi - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\end{array}} \right].\end{array}
Proof
Theorem 10 is proved in the same way as Theorem 9 with the same conditions.
The following corollary is obtained regarding the Saigo FIO [10 ] in light of the equation (16) .
Corollary 11
Let ϛ , ϰ , ϖ ,z ,α ∈ ℂ and ℜ(ϛ ) > 0,μ > 0,λk > 0 (k = 1,2,3,··· ,s ),
μ max 1 ≤ j ≤ U [ − ℜ ( ε j ) E j ] < ℜ ( α ) + min [ 0 , ℜ ( ς − ϰ ) , ℜ ( ς − ϖ ) ] .
\mu \mathop {\max }\limits_{1 \le j \le U} \left[ {\frac{{ - \Re ({\varepsilon _j})}}{{{\mathfrak{E}_j}}}} \right] < \Re (\alpha ) + \min [0,\Re (\varsigma - \varkappa ),\Re (\varsigma - \varpi )].
Further the parameters rj , s j , 𝔓 ∈ ℤ+ satisfying 0 ≤ V ≤ rj ,0 ≤ U ≤ sj for j = 1,2,···m , 𝔇n , 𝔈n , 𝔈nj , 𝔇nj ∈ ℝ+ , Λn ,εn ,Λnj ,εnj ∈ ℂ (j = 1,2,··· , rj ;n = 1,2,··· , sj ),
A Q k , P k k
A_{{\mathfrak{Q}_k},{\kern 1pt} {\mathfrak{P}_k}}^k
are real or complex numbers arbitrary constant for k = 1,2,··· ,s and ρi > 0 for i = 1,2,··· ,m, then the outcome shown below is accurate:
(50)
D 0 + ς , ϰ , ϖ ( t α − 1 ∏ j = 1 s S Q j P j [ c j t λ j ] Γ ℵ r j , s j , ρ j ; m U , V [ z t μ | ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( ε n , E n ) 1 , U , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] ) ( x ) = x α + ς + ς ' − ϖ − 1 ∑ k 1 = 0 [ Q 1 / P 1 ] ⋯ ∑ k s = 0 [ Q s / P s ] ( − Q 1 ) P 1 k 1 ⋯ ( − Q s ) P s k s k 1 ! ⋯ k s ! A Q 1 , P 1 ( 1 ) ⋯ A Q s , P s ( s ) c 1 k 1 ⋯ c s k s ( x ) ∑ j = 1 s λ j k j × Γ ℵ r j + 2 , s j + 2 , ρ j ; m U , V + 2 [ z x μ | ( 1 − α − ∑ j = 1 s λ j k j , μ ) , ( 1 − α − ς − ϰ − ϖ − ∑ j = 1 s λ j k j , μ ) , ( ε n , E n ) 1 , U , ( 1 − α − ϰ − ∑ j = 1 s λ j k j , μ ) , ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( 1 − α − ϖ − ∑ j = 1 s λ j k j , μ ) , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] .
\begin{array}{l}\mathcal{D}_{0 + }^{\varsigma ,{\kern 1pt} \varkappa ,{\kern 1pt} \varpi }{\kern 1pt} \left( {{t^{\alpha - 1}}\prod\limits_{j = 1}^s S_{{\mathfrak{Q}_j}}^{{\mathfrak{P}_j}}[{c_j}{\kern 1pt} {t^{{\lambda _j}}}]} \right.\\\left. {^\Gamma \aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V}{\kern 1pt} \left[ {z{\kern 1pt} {t^\mu }\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\\ \end{array}} \right]} \right)(x)\\ = {x^{\alpha + \varsigma + {\varsigma ^\prime } - \varpi - 1}}\sum\limits_{{\mathfrak{k}_1} = 0}^{[{\mathfrak{Q}_1}/{\mathfrak{P}_1}]} \cdots \sum\limits_{{\mathfrak{k}_s} = 0}^{[{\mathfrak{Q}_s}/{\mathfrak{P}_s}]} \frac{{{{( - {\mathfrak{Q}_1})}_{{\mathfrak{P}_1}{\kern 1pt} {\mathfrak{k}_1}}} \cdots {{( - {\mathfrak{Q}_s})}_{{\mathfrak{P}_s}{\kern 1pt} {\mathfrak{k}_s}}}}}{{{\mathfrak{k}_1}! \cdots {\mathfrak{k}_s}!}}A_{{\mathfrak{Q}_1},{\kern 1pt} {\mathfrak{P}_1}}^{(1)} \cdots A_{{\mathfrak{Q}_s},{\kern 1pt} {\mathfrak{P}_s}}^{(s)}{\kern 1pt} {\kern 1pt} c_1^{{\mathfrak{k}_1}} \cdots c_s^{{\mathfrak{k}_s}}{\kern 1pt} {(x)^{\sum\limits_{j = 1}^s {\lambda _j}{\kern 1pt} {\mathfrak{k}_j}}}\\{ \times ^\Gamma }\aleph _{{r_j} + 2,{\kern 1pt} {s_j} + 2,{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V + 2}\left[ {z{\kern 1pt} {x^\mu }\Biggl|\begin{array}{*{20}{c}}{(1 - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),(1 - \alpha - \varsigma - \varkappa - \varpi - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} (1 - \alpha - \varkappa - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),}\\ \end{array}} \right.\\\left. {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{(1 - \alpha - \varpi - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\end{array}} \right].\end{array}
The same result can be obtained concerning Saigo fractional derivative operator for the lower incomplete ℵ-function.
Remark 8
By substituting ϰ = −ϛ and ϰ = 0 in Corollary 11 , respectively, we can also get findings for the fractional derivative operators of R-L and E-K.
Theorem 12
Let ϛ , ϛ ', ϰ , ϰ ', ϖ ,z ,α ∈ ℂ and ℜ(ϖ ) > 0,μ > 0,λk > 0 (k = 1,2,3,··· ,s ),
μ min 1 ≤ j ≤ V [ 1 − ℜ ( Λ j ) D j ] + 1 > ℜ ( α ) − min [ 0 , ℜ ( ϖ − ς − ς ' − V ) , ℜ ( − ς ' − ϰ + ϖ ) , − ℜ ( ϰ ' ) ] .
\mu \mathop {\min }\limits_{1 \le j \le V} \left[ {\frac{{1 - \Re ({\Lambda _j})}}{{{\mathfrak{D}_j}}}} \right] + 1 > \Re (\alpha ) - \min [0,{\kern 1pt} \Re (\varpi - \varsigma - {\varsigma ^\prime } - V),\Re ( - {\varsigma ^\prime } - \varkappa + \varpi ),{\kern 1pt} - \Re ({\varkappa ^\prime })].
Further the parameters rj , s j , 𝔓 ∈ ℤ+ satisfying 0 ≤ V ≤ rj ,0 ≤ U ≤ sj for j = 1,2,···m , 𝔇n , 𝔈n , 𝔈nj , 𝔇nj ∈ ℝ+ , Λn ,εn ,Λnj ,εnj ∈ ℂ (j = 1,2,··· , rj ;n = 1,2,··· , sj ),
A Q k , P k k
A_{{\mathfrak{Q}_k},{\kern 1pt} {\mathfrak{P}_k}}^k
are real or complex numbers arbitrary constant for k = 1,2,··· ,s and ρi > 0 for i = 1,2,··· ,m, then the outcome shown below is accurate:
(51)
D − ς , ς ' , ϰ , ϰ ' , ϖ ( t α − 1 ∏ j = 1 s S Q j P j [ c j t λ j ] Γ ℵ r j , s j , ρ j ; m U , V [ z t μ | ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( ε n , E n ) 1 , U , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] ) ( x ) = x α + ς + ς ' − ϖ − 1 ∑ k 1 = 0 [ Q 1 / P 1 ] ⋯ ∑ k s = 0 [ Q s / P s ] ( − Q 1 ) P 1 k 1 ⋯ ( − Q s ) P s k s k 1 ! ⋯ k s ! A Q 1 , P 1 ( 1 ) ⋯ A Q s , P s ( s ) c 1 k 1 ⋯ c s k s ( x ) ∑ j = 1 s λ j k j × Γ ℵ r j + 3 , s j + 3 , ρ j ; m U + 3 , V [ z x μ | ( 1 − α − ∑ j = 1 s λ j k j , μ ) , ( 1 − α − ς ' + ϰ ' − ∑ j = 1 s λ j k j , μ ) , ( ε n , E n ) 1 , U , ( 1 − ς − ς ' + ϖ − α − ∑ j = 1 s λ j k j , μ ) , ( 1 − α − ς − ς ' − ϰ + ϖ − ∑ j = 1 s λ j k j , μ ) , ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( 1 − α + ϰ ' − ∑ j = 1 s λ j k j , μ ) , ( 1 − ς ' − ϰ + ϖ − α − ∑ j = 1 s λ j k j , μ ) , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] .
\begin{array}{l}\mathcal{D}_ - ^{\varsigma ,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },\varpi }{\kern 1pt} \left( {{t^{\alpha - 1}}\prod\limits_{j = 1}^s S_{{\mathfrak{Q}_j}}^{{\mathfrak{P}_j}}[{c_j}{\kern 1pt} {t^{{\lambda _j}}}]} \right.\\\left. {^\Gamma \aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V}{\kern 1pt} \left[ {z{\kern 1pt} {t^\mu }\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\\ \end{array}} \right]} \right)(x)\\ = {x^{\alpha + \varsigma + {\varsigma ^\prime } - \varpi - 1}}\sum\limits_{{\mathfrak{k}_1} = 0}^{[{\mathfrak{Q}_1}/{\mathfrak{P}_1}]} \cdots \sum\limits_{{\mathfrak{k}_s} = 0}^{[{\mathfrak{Q}_s}/{\mathfrak{P}_s}]} \frac{{{{( - {\mathfrak{Q}_1})}_{{\mathfrak{P}_1}{\kern 1pt} {\mathfrak{k}_1}}} \cdots {{( - {\mathfrak{Q}_s})}_{{\mathfrak{P}_s}{\kern 1pt} {\mathfrak{k}_s}}}}}{{{\mathfrak{k}_1}! \cdots {\mathfrak{k}_s}!}}A_{{\mathfrak{Q}_1},{\kern 1pt} {\mathfrak{P}_1}}^{(1)} \cdots A_{{\mathfrak{Q}_s},{\kern 1pt} {\mathfrak{P}_s}}^{(s)}{\kern 1pt} {\kern 1pt} c_1^{{\mathfrak{k}_1}} \cdots c_s^{{\mathfrak{k}_s}}{\kern 1pt} {(x)^{\sum\limits_{j = 1}^s {\lambda _j}{\kern 1pt} {\mathfrak{k}_j}}}\\{ \times ^\Gamma }\aleph _{{r_j} + 3,{\kern 1pt} {s_j} + 3,{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U + 3,V}{\kern 1pt} \left[ {z{\kern 1pt} {x^\mu }\Biggl|\begin{array}{*{20}{c}}{(1 - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),(1 - \alpha - {\varsigma ^\prime } + {\varkappa ^\prime } - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} (1 - \varsigma - {\varsigma ^\prime } + \varpi - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),}\\ \end{array}} \right.\\\left. {\;\;\;\;\begin{array}{*{20}{c}}{(1 - \alpha - \varsigma - {\varsigma ^\prime } - \varkappa + \varpi - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} ({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{(1 - \alpha + {\varkappa ^\prime } - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} (1 - {\varsigma ^\prime } - \varkappa + \varpi - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\end{array}} \right].\end{array}
Proof
The LHS of equation (51) is:
(52)
T 4 = D − ς , ς ' , ϰ , ϰ ' , ϖ ( t α − 1 ∏ j = 1 s S Q j P j [ c j t λ j ] Γ ℵ r j , s j , ρ j ; m U , V [ z t μ | ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( ε n , E n ) 1 , U , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] ) ( x ) .
\begin{array}{l}{T_4} = \mathcal{D}_ - ^{\varsigma ,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },\varpi }{\kern 1pt} \left( {{t^{\alpha - 1}}\prod\limits_{j = 1}^s S_{{\mathfrak{Q}_j}}^{{\mathfrak{P}_j}}[{c_j}{\kern 1pt} {t^{{\lambda _j}}}]} \right.\\\left. {^\Gamma \aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V}{\kern 1pt} \left[ {z{\kern 1pt} {t^\mu }\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\\ \end{array}} \right]} \right)(x).\end{array}
Replace the incomplete ℵ-function and Srivastava polynomial by equation (24) and (4) respectively and by reversing the summation order, we discover the subsequent form:
(53)
T 3 = ∑ k 1 = 0 [ Q 1 / P 1 ] ⋯ ∑ k s = 0 [ Q s / P s ] ( − Q 1 ) P 1 k 1 ⋯ ( − Q s ) P s k s k 1 ! ⋯ k s ! × A Q 1 , P 1 ( 1 ) ⋯ A Q s , P s ( s ) c 1 k 1 ⋯ c s k s × 1 2 π ι ∫ $ Ψ ( q , Y ) z − q ( D − ς , ς ' , ϰ , ϰ ' , ϖ t − ( − α − ∑ j = 1 s λ j k j + μ q + 1 ) ) ( x ) d q ,
\begin{array}{l}{T_3} = \sum\limits_{{\mathfrak{k}_1} = 0}^{[{\mathfrak{Q}_1}/{\mathfrak{P}_1}]} \cdots \sum\limits_{{\mathfrak{k}_s} = 0}^{[{\mathfrak{Q}_s}/{\mathfrak{P}_s}]} \frac{{{{( - {\mathfrak{Q}_1})}_{{\mathfrak{P}_1}{\kern 1pt} {\mathfrak{k}_1}}} \cdots {{( - {\mathfrak{Q}_s})}_{{\mathfrak{P}_s}{\kern 1pt} {\mathfrak{k}_s}}}}}{{{\mathfrak{k}_1}! \cdots {\mathfrak{k}_s}!}}{\kern 1pt} \times A_{{\mathfrak{Q}_1},{\kern 1pt} {\mathfrak{P}_1}}^{(1)} \cdots A_{{\mathfrak{Q}_s},{\kern 1pt} {\mathfrak{P}_s}}^{(s)}{\kern 1pt} {\kern 1pt} c_1^{{\mathfrak{k}_1}} \cdots c_s^{{\mathfrak{k}_s}}\\ \times \frac{1}{{2\pi \iota }}\int_\$ \;\Psi (q,\mathcal{Y})\;{z^{ - q}}\left( {\mathcal{D}_ - ^{\varsigma ,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },\varpi }{\kern 1pt} {t^{ - ( - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} + \mu q + 1)}}} \right)(x)dq,\end{array}
where Ψ (q ,𝒴) is defined in equation (25) .
Using equation (21) of Lemma 2 , we discover the subsequent form:
(54)
T 4 = ∑ k 1 = 0 [ Q 1 / P 1 ] ⋯ ∑ k s = 0 [ Q s / P s ] ( − Q 1 ) P 1 k 1 ⋯ ( − Q s ) P s k s k 1 ! ⋯ k s ! × A Q 1 , P 1 ( 1 ) ⋯ A Q s , P s ( s ) c 1 k 1 ⋯ c s k s × 1 2 π ι ∫ $ x α − ς − ς ' + ϖ + ∑ j = 1 s λ j k j − 1 Ψ ( q , Y ) ( z x μ )
− q Γ ( 1 − α + ϰ ' − ∑ j = 1 s λ j k j + μ q ) Γ ( 1 − α − ∑ j = 1 s λ j k j + μ q ) × Γ ( 1 − ς − ς ' + ϖ − α − ∑ j = 1 s λ j k j + μ q ) Γ ( 1 − α − ς ' − ϰ + ϖ − ∑ j = 1 s λ j k j + μ q ) Γ ( 1 − α − ς ' + ϰ ' − ϖ − ∑ j = 1 s λ j k j + μ q ) Γ ( 1 − α − ς − ς ' − ϰ + ϖ − ∑ j = 1 s λ j k j + μ q ) d q .
\begin{array}{l}{T_4} = \sum\limits_{{\mathfrak{k}_1} = 0}^{[{\mathfrak{Q}_1}/{\mathfrak{P}_1}]} \cdots \sum\limits_{{\mathfrak{k}_s} = 0}^{[{\mathfrak{Q}_s}/{\mathfrak{P}_s}]} \frac{{{{( - {\mathfrak{Q}_1})}_{{\mathfrak{P}_1}{\kern 1pt} {\mathfrak{k}_1}}} \cdots {{( - {\mathfrak{Q}_s})}_{{\mathfrak{P}_s}{\kern 1pt} {\mathfrak{k}_s}}}}}{{{\mathfrak{k}_1}! \cdots {\mathfrak{k}_s}!}}{\kern 1pt} \times A_{{\mathfrak{Q}_1},{\kern 1pt} {\mathfrak{P}_1}}^{(1)} \cdots A_{{\mathfrak{Q}_s},{\kern 1pt} {\mathfrak{P}_s}}^{(s)}{\kern 1pt} {\kern 1pt} c_1^{{\mathfrak{k}_1}} \cdots c_s^{{\mathfrak{k}_s}}\\ \times \frac{1}{{2\pi \iota }}\int_\$ \;{x^{\alpha - \varsigma - {\varsigma ^\prime } + \varpi + \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} - 1}}{\kern 1pt} {\kern 1pt} \Psi (q,\mathcal{Y})\;{(z{x^\mu })^{ - q}}\frac{{\Gamma (1 - \alpha + {\varkappa ^\prime } - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} + \mu q)}}{{\Gamma (1 - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} + \mu q)}} \times \\\frac{{\Gamma (1 - \varsigma - {\varsigma ^\prime } + \varpi - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} + \mu q){\kern 1pt} \Gamma (1 - \alpha - {\varsigma ^\prime } - \varkappa + \varpi - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} + \mu q)}}{{\Gamma (1 - \alpha - {\varsigma ^\prime } + {\varkappa ^\prime } - \varpi - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} + \mu q){\kern 1pt} \Gamma (1 - \alpha - \varsigma - {\varsigma ^\prime } - \varkappa + \varpi - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j} + \mu q)}}dq.\end{array}
Finally, after some adjustment of terms, we obtain RHS of equation (51) .
Theorem 13
Let ϛ , ϛ ', ϰ , ϰ ', ϖ ,z ,α ∈ ℂ and ℜ(ϖ ) > 0,μ > 0,λk > 0 (k = 1,2,3,··· ,s ),
μ min 1 ≤ j ≤ V [ 1 − ℜ ( Λ j ) D j ] + 1 > ℜ ( α ) − min [ 0 , ℜ ( ϖ − ς − ς ' − V ) , ℜ ( − ς ' − ϰ + ϖ ) , − ℜ ( ϰ ' ) ] .
\mu \mathop {\min }\limits_{1 \le j \le V} \left[ {\frac{{1 - \Re ({\Lambda _j})}}{{{\mathfrak{D}_j}}}} \right] + 1 > \Re (\alpha ) - \min [0,{\kern 1pt} \Re (\varpi - \varsigma - {\varsigma ^\prime } - V),\Re ( - {\varsigma ^\prime } - \varkappa + \varpi ),{\kern 1pt} - \Re ({\varkappa ^\prime })].
Further the parameters rj , s j , 𝔓 ∈ ℤ+ satisfying 0 ≤ V ≤ rj ,0 ≤ U ≤ sj for j = 1,2,···m , 𝔇n , 𝔈n , 𝔈nj , 𝔇nj ∈ ℝ+ , Λn ,εn ,Λnj ,εnj ∈ ℂ (j = 1,2,··· , rj ;n = 1,2,··· , sj ),
A Q k , P k k
A_{{\mathfrak{Q}_k},{\kern 1pt} {\mathfrak{P}_k}}^k
are real or complex numbers arbitrary constant for k = 1,2,··· ,s and ρi > 0 for i = 1,2,··· ,m, then the outcome shown below is accurate:
(55)
D − ς , ς ' , ϰ , ϰ ' , ϖ ( t α − 1 ∏ j = 1 s S Q j P j [ c j t λ j ] γ ℵ r j , s j , ρ j ; m U , V [ z t μ | ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( ε n , E n ) 1 , U , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] ) ( x ) = x α + ς + ς ' − ϖ − 1 ∑ k 1 = 0 [ Q 1 / P 1 ] ⋯ ∑ k s = 0 [ Q s / P s ] ( − Q 1 ) P 1 k 1 ⋯ ( − Q s ) P s k s k 1 ! ⋯ k s ! A Q 1 , P 1 ( 1 ) ⋯ A Q s , P s ( s ) c 1 k 1 ⋯ c s k s ( x ) ∑ j = 1 s λ j k j × γ ℵ r j + 3 , s j + 3 , ρ j ; m U + 3 , V [ z x μ | ( 1 − α − ∑ j = 1 s λ j k j , μ ) , ( 1 − α − ς ' + ϰ ' − ∑ j = 1 s λ j k j , μ ) , ( ε n , E n ) 1 , U , ( 1 − ς − ς ' + ϖ − α − ∑ j = 1 s λ j k j , μ ) , ( 1 − α − ς − ς ' − ϰ + ϖ − ∑ j = 1 s λ j k j , μ ) , ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( 1 − α + ϰ ' − ∑ j = 1 s λ j k j , μ ) , ( 1 − ς ' − ϰ + ϖ − α − ∑ j = 1 s λ j k j , μ ) , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] .
\begin{array}{l}\mathcal{D}_ - ^{\varsigma ,{\kern 1pt} {\varsigma ^\prime },{\kern 1pt} \varkappa ,{\kern 1pt} {\varkappa ^\prime },\varpi }{\kern 1pt} \left( {{t^{\alpha - 1}}\prod\limits_{j = 1}^s S_{{\mathfrak{Q}_j}}^{{\mathfrak{P}_j}}[{c_j}{\kern 1pt} {t^{{\lambda _j}}}]} \right.\\\left. {^\gamma \aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V}{\kern 1pt} \left[ {z{\kern 1pt} {t^\mu }\Biggl|\begin{array}{*{20}{c}}\begin{array}{l}\\({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{({\Lambda _n},{\mathfrak{D}_n})_{2,V}},{\kern 1pt} {[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]_{V + 1,{r_j}}}\end{array}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\\ \end{array}} \right]} \right)(x)\\ = {x^{\alpha + \varsigma + {\varsigma ^\prime } - \varpi - 1}}\sum\limits_{{\mathfrak{k}_1} = 0}^{[{\mathfrak{Q}_1}/{\mathfrak{P}_1}]} \cdots \sum\limits_{{\mathfrak{k}_s} = 0}^{[{\mathfrak{Q}_s}/{\mathfrak{P}_s}]} \frac{{{{( - {\mathfrak{Q}_1})}_{{\mathfrak{P}_1}{\kern 1pt} {\mathfrak{k}_1}}} \cdots {{( - {\mathfrak{Q}_s})}_{{\mathfrak{P}_s}{\kern 1pt} {\mathfrak{k}_s}}}}}{{{\mathfrak{k}_1}! \cdots {\mathfrak{k}_s}!}}A_{{\mathfrak{Q}_1},{\kern 1pt} {\mathfrak{P}_1}}^{(1)} \cdots A_{{\mathfrak{Q}_s},{\kern 1pt} {\mathfrak{P}_s}}^{(s)}{\kern 1pt} {\kern 1pt} c_1^{{\mathfrak{k}_1}} \cdots c_s^{{\mathfrak{k}_s}}{\kern 1pt} {(x)^{\sum\limits_{j = 1}^s {\lambda _j}{\kern 1pt} {\mathfrak{k}_j}}}\\{ \times ^\gamma }\aleph _{{r_j} + 3,{\kern 1pt} {s_j} + 3,{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U + 3,V}{\kern 1pt} \left[ {z{\kern 1pt} {x^\mu }\Biggl|\begin{array}{*{20}{c}}{(1 - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),(1 - \alpha - {\varsigma ^\prime } + {\varkappa ^\prime } - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} (1 - \varsigma - {\varsigma ^\prime } + \varpi - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),}\\ \end{array}} \right.\\\left. {\;\;\;\;\begin{array}{*{20}{c}}{(1 - \alpha - \varsigma - {\varsigma ^\prime } - \varkappa + \varpi - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} ({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{(1 - \alpha + {\varkappa ^\prime } - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} (1 - {\varsigma ^\prime } - \varkappa + \varpi - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\end{array}} \right].\end{array}
Proof
Theorem 13 is proved in the same way as Theorem 12 with the same conditions.
The following corollary is obtained regarding the Saigo fractional derivative operator [10 ] in light of the equation (16) .
Corollary 14
Let ϛ , ϛ ', ϰ , ϰ ', ϖ ,z ,α ∈ ℂ and ℜ(ϖ ) > 0,μ > 0,λk > 0 (k = 1,2,3,··· ,s ),
μ min 1 ≤ j ≤ V [ 1 − ℜ ( Λ j ) D j ] + 1 > ℜ ( α ) − min [ 0 , ℜ ( ϖ − ς − ς ' − V ) , ℜ ( − ς ' − ϰ + ϖ ) , − ℜ ( ϰ ' ) ] .
\mu \mathop {\min }\limits_{1 \le j \le V} \left[ {\frac{{1 - \Re ({\Lambda _j})}}{{{\mathfrak{D}_j}}}} \right] + 1 > \Re (\alpha ) - \min [0,{\kern 1pt} \Re (\varpi - \varsigma - {\varsigma ^\prime } - V),\Re ( - {\varsigma ^\prime } - \varkappa + \varpi ),{\kern 1pt} - \Re ({\varkappa ^\prime })].
Further the parameters rj , s j , 𝔓 ∈ ℤ+ satisfying 0 ≤ V ≤ rj ,0 ≤ U ≤ sj for j = 1,2,···m , 𝔇n , 𝔈n , 𝔈nj , 𝔇nj ∈ ℝ+ , Λn ,εn ,Λnj ,εnj ∈ ℂ (j = 1,2,··· , rj ;n = 1,2,··· , sj ),
A Q k , P k k
A_{{\mathfrak{Q}_k},{\kern 1pt} {\mathfrak{P}_k}}^k
are real or complex numbers arbitrary constant for k = 1,2,··· ,s and ρi > 0 for i = 1,2,··· ,m, then the following result holds:
(56)
D − ς , ϰ , ϖ ( t α − 1 ∏ j = 1 s S Q j P j [ c j t λ j ] Γ ℵ r j , s j , ρ j ; m U , V [ z t μ | ( Λ 1 , D 1 : Y ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( ε n , E n ) 1 , U , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] ) ( x ) = x α + ϰ − 1 ∑ k 1 = 0 [ Q 1 / P 1 ] ⋯ ∑ k s = 0 [ Q s / P s ] ( − Q 1 ) P 1 k 1 ⋯ ( − Q s ) P s k s k 1 ! ⋯ k s ! A Q 1 , P 1 ( 1 ) ⋯ A Q s , P s ( s ) c 1 k 1 ⋯ c s k s ( x ) ∑ j = 1 s λ j k j × Γ ℵ r j + 2 , s j + 2 , ρ j ; m U + 2 , V [ z x μ | ( Λ 1 , D 1 : Y ) , ( 1 − α − ∑ j = 1 s λ j k j , μ ) , ( ε n , E n ) 1 , U , ( 1 − ϰ − α − ∑ j = 1 s λ j k j , μ ) , ( 1 − α − ϰ + ϖ − ∑ j = 1 s λ j k j , μ ) , ( Λ n , D n ) 2 , V , [ ρ n ( Λ n j , D n j ) ] V + 1 , r j ( 1 + ς + ϖ − α − ∑ j = 1 s λ j k j , μ ) , [ ρ n ( ε n j , E n j ) ] U + 1 , s j ] .
\begin{array}{l}\mathcal{D}_ - ^{\varsigma ,{\kern 1pt} \varkappa ,{\kern 1pt} \varpi }{\kern 1pt} \left( {{t^{\alpha - 1}}\prod\limits_{j = 1}^s S_{{\mathfrak{Q}_j}}^{{\mathfrak{P}_j}}[{c_j}{\kern 1pt} {t^{{\lambda _j}}}]} \right.\\\left. {^\Gamma \aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U,V}{\kern 1pt} \left[ {z{\kern 1pt} {t^\mu }\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\\ \end{array}} \right]} \right)(x)\\ = {x^{\alpha + \varkappa - 1}}\sum\limits_{{\mathfrak{k}_1} = 0}^{[{\mathfrak{Q}_1}/{\mathfrak{P}_1}]} \cdots \sum\limits_{{\mathfrak{k}_s} = 0}^{[{\mathfrak{Q}_s}/{\mathfrak{P}_s}]} \frac{{{{( - {\mathfrak{Q}_1})}_{{\mathfrak{P}_1}{\kern 1pt} {\mathfrak{k}_1}}} \cdots {{( - {\mathfrak{Q}_s})}_{{\mathfrak{P}_s}{\kern 1pt} {\mathfrak{k}_s}}}}}{{{\mathfrak{k}_1}! \cdots {\mathfrak{k}_s}!}}A_{{\mathfrak{Q}_1},{\kern 1pt} {\mathfrak{P}_1}}^{(1)} \cdots A_{{\mathfrak{Q}_s},{\kern 1pt} {\mathfrak{P}_s}}^{(s)}{\kern 1pt} {\kern 1pt} c_1^{{\mathfrak{k}_1}} \cdots c_s^{{\mathfrak{k}_s}}{\kern 1pt} {(x)^{\sum\limits_{j = 1}^s {\lambda _j}{\kern 1pt} {\mathfrak{k}_j}}}\\{ \times ^\Gamma }\aleph _{{r_j} + 2,{\kern 1pt} {s_j} + 2,{\kern 1pt} {\rho _j};{\kern 1pt} m}^{U + 2,V}{\kern 1pt} \left[ {z{\kern 1pt} {x^\mu }\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{\kern 1pt} (1 - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,U}},{\kern 1pt} (1 - \varkappa - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),}\\ \end{array}} \right.\\\left. {\;\;\;\;\begin{array}{*{20}{c}}{(1 - \alpha - \varkappa + \varpi - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} {{({\Lambda _n},{\mathfrak{D}_n})}_{2,V}},{\kern 1pt} {{[{\rho _n}({\Lambda _{nj}},{\mathfrak{D}_{nj}})]}_{V + 1,{r_j}}}}\\{(1 + \varsigma + \varpi - \alpha - \sum\limits_{j = 1}^s {\lambda _j}{\mathfrak{k}_j},\mu ),{\kern 1pt} {{[{\rho _n}({\varepsilon _{nj}},{\mathfrak{E}_{nj}})]}_{U + 1,{s_j}}}}\end{array}} \right].\end{array}
The same result can be obtained regarding Saigo fractional derivative operator for the lower incomplete ℵ-function.
Remark 9
By substituting ϰ = −ϛ and ϰ = 0 in Corollary 14 , respectively, we can also get findings for the fractional derivative operators of R-L and E-K.