Some fractional calculus findings associated with the product of incomplete ℵ-function and Srivastava polynomials

Nishant; Bhatter, Sanjay; Purohit, Sunil Dutt; Nisar, Kottakkaran Sooppy; Munjam, Shankar Rao

doi:10.2478/ijmce-2024-0008

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1

Introduction

Fractional calculus (FC) refers to the study of differential and integral operators of either real or complex order. Due to a variety of applications across several scientific disciplines and technology, FC has grown in significance as well as usage over the past four decades. Fractional operators were conceived and mathematically formalized only in recent years. The numerous properties of fractional operators have generated a great deal of interest in fractional calculus in recent years, as well as a wide range of applications, with a focus on the simulation of physical issues. Areas that have seen the largest number of applications include the formulation of constitutive equations for viscoelastic materials [1], transport processes in complex media [2], mechanics [3], non-local elasticity, plasticity [4], model-order reduction of lumped parameter systems and biomedical engineering [5, 6].

Numerous subfields of computational mathematics have found major significance in the fractional integral operator (FIO) [7], which involves a variety of special functions. Over the last five decades, several scientists like Saxena and Srivastava [8], Bhatta and Debnath [9], Saigo [10], Marichev and Kilbas [11], Ross and Miller [12], Purohit and Jangid [13], Love [14] and Ram and Kumar [15] have thoroughly investigated the characteristics, uses, and numerous extensions of several hypergeometric operators of fractional integration. Engineers, physicists, biologists and financial analysts are only some of the communities that may find several points of interest and material for further considerations in this work.

A significant number of new and recognised outcomes including Saigo FC operators and many special functions, particularly the incomplete H-function and incomplete I-function, follow as special instances of the primary discoveries. This is due to the broad scope of the Merichev-Saigo-Maeda (MSM) operators, incomplete ℵ-function, and a broad category of polynomials.

The rest of this paper is organized as follows. In section 2, the preliminaries are presented. In section 3, incomplete ℵ-functions and the Srivastava polynomial are combined, and MSM fractional order integrals of the left- and right-hand types are created. In section 4, incomplete ℵ-functions and the Srivastava polynomial are combined, and MSM fractional order derivative of the left- and right-hand types are created. In section 5, we develop the particular instances for the incomplete ℵ-functions. In section 6, the paper is completed by presenting the main contribution of the paper.

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) $γ (v, Y) = \int_{0}^{Y} u^{v - 1} e^{- u} d u, (ℜ (v) > 0; Y ≧ 0),$ \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) $Γ (v, Y) = \int_{Y}^{\infty} u^{v - 1} e^{- u} d u, (Y ≧ 0; ℜ (v) > 0 when Y = 0) .$ \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) $S_{Q}^{P} [t] = \sum_{O = 0}^{[Q P]} \frac{{(- Q)}_{P O}}{O!} A_{Q, O} t^{O},$ 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: ${(κ)}_{0} = 1 and {(κ)}_{μ} = \frac{Γ (κ + μ)}{Γ (κ)}, (μ \in ℂ),$ {(\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) $(ℐ_{0 +}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} f) (x) = \frac{x^{- ς}}{Γ (ϖ)} \int_{0}^{x} {(x - y)}^{ϖ - 1} y^{- ς^{'}} \times F_{3} (ς, ς^{'}, ϰ, ϰ^{'}; ϖ; 1 - \frac{y}{x}, 1 - \frac{x}{y}) f (y) d y,$ \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) $(ℐ_{-}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} f) (x) = \frac{x^{- ς^{'}}}{Γ (ϖ)} \int_{x}^{\infty} {(y - x)}^{ϖ - 1} y^{- ς} \times F_{3} (ς, ς^{'}, ϰ, ϰ^{'}; ϖ; 1 - \frac{x}{y}, 1 - \frac{y}{x}) f (y) d y,$ \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) $(D_{0 +}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} f) (x) = {(\frac{d}{d x})}^{α} (ℐ_{0 +}^{- ς^{'}, - ς, - ϰ^{'} + α, - ϰ, - ϖ + α} f) (x),$ \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) $(D_{-}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} f) (x) = {(- \frac{d}{d x})}^{[α]} (ℐ_{-}^{- ς^{'}, - ς, - ϰ^{'}, - ϰ + α, - ϖ + α} f) (x),$ \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 F₃ has the following definition: (9) $F_{3} (ς, ς^{'}, ϰ, ϰ^{'}; ϖ; x; y) = \sum_{i, j = 0}^{\infty} \frac{{(ς)}_{i} {(ς^{'})}_{j} {(ϰ)}_{i} {(ϰ^{'})}_{j}}{{(ϖ)}_{i + j}} \frac{x^{i} y^{j}}{i! j!},$ {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 ₂F₁( ). The left-and right-handed Saigo FIO are given the following descriptions for ϛ, ϰ, ϖ ∈ ℂ, x > 0 and ℜ(ϛ) > 0. (10) $(ℐ_{0 +}^{ς, ϰ, ϖ} f) (x) = \frac{x^{- ς - ϰ}}{Γ (ς)} \int_{0}^{x} {(x - y)}^{ς - 1}_{2} F_{1} (ς + ϰ, - ϖ; ς; 1 - \frac{y}{x}) f (y) d y,$ \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) $(ℐ_{-}^{ς, ϰ, ϖ} f) (x) = \frac{1}{Γ (ς)} \int_{x}^{\infty} {(y - x)}^{ς - 1} y^{- ς - ϰ}_{2} F_{1} (ς + ϰ, - ϖ; ς; 1 - \frac{x}{y}) f (y) d y,$ \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) $(D_{0 +}^{ς, ϰ, ϖ} f) (x) = {(\frac{d}{d x})}^{[ℜ (ς)] + 1} (ℐ_{0 +}^{- ς + [ℜ (ς)] + 1, - ϰ - [ℜ (ς)] - 1, ς + ϖ - [ℜ (ς)] - 1} f) (x),$ \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) $(D_{-}^{ς, ϰ, ϖ} f) (x) = {(- \frac{d}{d x})}^{[ℜ (ς)] + 1} (ℐ_{-}^{- ς + [ℜ (ς)] + 1, - ϰ - [ℜ (ς)] - 1, ς + ϖ} f) (x) .$ \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 = −ϛand 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]). ₂F₁ is associated with F₃ 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) $(ℐ_{0 +}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} t^{λ - 1}) (x) = x^{- ς - ς^{'} + ϖ + λ - 1} \frac{Γ (λ) Γ (- ς^{'} + ϰ^{'} + λ) Γ (- ς - ς^{'} - ϰ + ϖ + λ)}{Γ (ϰ^{'} + λ) Γ (- ς - ς^{'} + ϖ + λ) Γ (- ς^{'} - ϰ + ϖ + λ)} .$ \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) $(ℐ_{-}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} t^{- λ}) (x) = x^{- ς - ς^{'} + ϖ - λ} \frac{Γ (- ϰ + λ) Γ (ς + ς^{'} - ϖ + λ) Γ (ς + ϰ^{'} - ϖ + λ)}{Γ (λ) Γ (ς - ϰ + λ) Γ (ς + ς^{'} + ϰ^{'} - ϖ + λ)} .$ \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) $(D_{0 +}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} t^{λ - 1}) (x) = x^{ς + ς^{'} - ϖ + λ - 1} \frac{Γ (λ) Γ (ς - ϰ + λ) Γ (ς + ς^{'} + ϰ^{'} - ϖ + λ)}{Γ (- ϰ + λ) Γ (ς + ς^{'} - ϖ + λ) Γ (ς + ϰ^{'} - ϖ + λ)} .$ \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) $(D_{-}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} t^{- λ}) (x) = x^{ς + ς^{'} - ϖ - λ} \frac{Γ (ϰ^{'} + λ) Γ (- ς - ς^{'} + ϖ + λ) Γ (- ς^{'} - ϰ + ϖ + λ)}{Γ (λ) Γ (- ς^{'} + ϰ^{'} + λ) Γ (- ς - ς^{'} - ϰ + ϖ + λ)} .$ \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) $\begin{array}{l} ^{γ} ℵ_{r_{j}, s_{j}, ρ_{j}; m}^{U, V} (Z) =^{γ} ℵ_{r_{j}, s_{j}, ρ_{j}; m}^{U, V} [Z | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}] \\ = \frac{1}{2 π ι} \int_{$} Φ (q, Y) Z^{- q} d q, \end{array}$ \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) $Φ (q, Y) = \frac{γ (1 - Λ_{1} - D_{1} q; Y) \prod_{n = 1}^{U} Γ (ε_{n} + E_{n} q) \prod_{n = 2}^{V} Γ (1 - Λ_{n} - D_{n} q)}{\sum_{j = 1}^{m} ρ_{j} [\prod_{n = U + 1}^{s_{j}} Γ (1 - ε_{n j} - E_{n j} q) \prod_{n = V + 1}^{r_{j}} Γ (Λ_{n j} + D_{n j})]},$ \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) $\begin{array}{l} ^{Γ} ℵ_{r_{j}, s_{j}, ρ_{j}; m}^{U, V} (Z) =^{Γ} ℵ_{r_{j}, s_{j}, ρ_{j}; m}^{U, V} [Z | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}] \\ = \frac{1}{2 π ι} \int_{$} Ψ (q, Y) Z^{- q} d q, \end{array}$ \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) $Ψ (q, Y) = \frac{Γ (1 - Λ_{1} - D_{1} q; Y) \prod_{n = 1}^{U} Γ (ε_{n} + E_{n} q) \prod_{n = 2}^{V} Γ (1 - Λ_{n} - D_{n} q)}{\sum_{j = 1}^{m} ρ_{j} [\prod_{n = U + 1}^{s_{j}} Γ (1 - ε_{n j} - E_{n j} q) \prod_{n = V + 1}^{r_{j}} Γ (Λ_{n j} + D_{n j})]},$ \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 − 𝔇_nq) for n = 1, 2, ...,V are not perfectly matched with the gamma function poles Γ (ϵ_n + 𝔈_nq) for n = 1,2,..,U. The parameters r_j and s_j ∈ ℤ+ satisfying 0 ≤ V ≤ r_j,0 ≤ U ≤ s_j for 1 ≤ j ≤ m. The parameters 𝔇_n, 𝔈_n, 𝔈_nj, 𝔇_nj 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) $\begin{array}{l} ^{Γ} ℵ_{r_{j}, s_{j}, ρ_{j}; m}^{U, V} [Z | \begin{matrix} (Λ_{1}, D_{1} : 0), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}] \\ = ℵ_{r_{j}, s_{j}, ρ_{j}; m}^{U, V} [Z | \begin{matrix} {(Λ_{n}, D_{n})}_{1, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}] . \end{array}$ \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 ρ_j = 1 in (22) and (24), then it changes to the incomplete I-function of Bansal and Kumar [28]: (27) $\begin{array}{l} ^{γ} ℵ_{r_{j}, s_{j}, ρ_{j}; m}^{U, V} [Z | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[1 (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {[1 (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}] \\ =^{γ} I_{r_{j}, s_{j}; m}^{U, V} [Z | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {(Λ_{n j}, D_{n j})}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {(ε_{n j}, E_{n j})}_{U + 1, s_{j}} \end{matrix}], \end{array}$ \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) $\begin{array}{l} ^{Γ} ℵ_{r_{j}, s_{j}, ρ_{j}; m}^{U, V} [Z | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[1 (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {[1 (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}] \\ =^{Γ} I_{r_{j}, s_{j}; m}^{U, V} [Z | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {(Λ_{n j}, D_{n j})}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {(ε_{n j}, E_{n j})}_{U + 1, s_{j}} \end{matrix}] . \end{array}$ \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 ρ_j = 1 in (24), then it turns into the Saxena I-function [29]: (29) $\begin{array}{l} ^{Γ} ℵ_{r_{j}, s_{j}, 1; m}^{U, V} [Z | \begin{matrix} (Λ_{1}, D_{1} : 0), {(Λ_{n}, D_{n})}_{2, V}, {[1 (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {[1 (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}] \\ = I_{r_{j}, s_{j}; m}^{U, V} [Z | \begin{matrix} {(Λ_{n}, D_{n})}_{1, V}, {(Λ_{n j}, D_{n j})}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {(ε_{n j}, E_{n j})}_{U + 1, s_{j}} \end{matrix}] . \end{array}$ \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 ρ_j = 1 and m = 1 in (22) and (24), then it turns into the incomplete H-function(see [30, 31] also) of Srivastava [32]: (30) $\begin{array}{l} ^{γ} ℵ_{r_{j}, s_{j}, 1; 1}^{U, V} [Z | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[1 (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {[1 (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}] \\ = γ_{r, s}^{U, V} [Z | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, r} \\ {(ε_{n}, E_{n})}_{1, s} \end{matrix}], \end{array}$ \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) $\begin{array}{l} ^{Γ} ℵ_{r_{j}, s_{j}, 1; 1}^{U, V} [Z | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[1 (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {[1 (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}] \\ = Γ_{r, s}^{U, V} [Z | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, r} \\ {(ε_{n}, E_{n})}_{1, s} \end{matrix}] . \end{array}$ \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, ρ_j = 1, and m = 1 in (24), then it turns into the H-function of Srivastava [33]: (32) $\begin{array}{l} ^{Γ} ℵ_{r_{j}, s_{j}, 1; 1}^{U, V} [Z | \begin{matrix} (Λ_{1}, D_{1} : 0), {(Λ_{n}, D_{n})}_{2, V}, {[1 (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {[1 (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}] \\ = H_{r, s}^{U, V} [Z | \begin{matrix} {(Λ_{n}, D_{n})}_{1, r} \\ {(ε_{n}, E_{n})}_{1, s} \end{matrix}] . \end{array}$ \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,λ_k > 0 (k = 1,2,3,··· ,s), $ℜ (α) + μ min_{1 \leq j \leq U} ℜ (\frac{ε_{j}}{ϰ_{j}}) > max [0, ℜ (ς + ς^{'} + ϰ - ϖ), ℜ (ς^{'} - ϰ^{'})] .$ \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 r_j, s_j, 𝔓 ∈ ℤ⁺ satisfying 0 ≤ V ≤ r_j,0 ≤ U ≤ s_j for j = 1,2,···m, 𝔇_n, 𝔈_n, 𝔈_nj, 𝔇_nj ∈ ℝ⁺, Λ_n,ε_n,Λ_nj,ε_nj ∈ ℂ (j = 1,2,··· , r_j; n = 1,2,··· , s_j), $A_{Q_{k}, P_{k}}^{k}$ A_{{\mathfrak{Q}_k},{\kern 1pt} {\mathfrak{P}_k}}^k are real or complex numbers arbitrary constant for k = 1,2,··· ,s and ρ_i > 0 for i = 1,2,··· ,m, then the outcome shown below is accurate: (33) $\begin{array}{l} ℐ_{0 +}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} (t^{α - 1} \prod_{j = 1}^{s} S_{Q_{j}}^{P_{j}} [c_{j} t^{λ_{j}}] \\ \times^{Γ} ℵ_{r_{j}, s_{j}, ρ_{j}; m}^{U, V} [z t^{μ} | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}]) (x) \\ = x^{α - ς - ς^{'} + ϖ - 1} \sum_{k_{1} = 0}^{[Q_{1} / P_{1}]} \dots \sum_{k_{s} = 0}^{[Q_{s} / P_{s}]} \frac{{(- Q_{1})}_{P_{1} k_{1}} \dots {(- Q_{s})}_{P_{s} k_{s}}}{k_{1}! \dots k_{s}!} A_{Q_{1}, P_{1}}^{(1)} \dots A_{Q_{s}, P_{s}}^{(s)} c_{1}^{k_{1}} \dots c_{s}^{k_{s}} {(x)}^{\sum_{j = 1}^{s} λ_{j} k_{j}} \\ \times^{Γ} ℵ_{r_{j} + 3, s_{j} + 3, ρ_{j}; m}^{U, V + 3} [z x^{μ} | \begin{matrix} (1 - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 - α + ς + ς^{'} + ϰ - ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \\ {(ε_{n}, E_{n})}_{1, U}, (1 - α + ς + ς^{'} - ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \end{matrix} \\ \begin{matrix} (1 - α + ς^{'} - ϰ^{'} - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ (1 - α - ϰ^{'} - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 + ς^{'} + ϰ - ϖ - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}] . \end{array}$ \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) $\begin{array}{l} T_{1} = ℐ_{0 +}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} (t^{α - 1} \prod_{j = 1}^{s} S_{Q_{j}}^{P_{j}} [c_{j} t^{λ_{j}}] \\ \times^{Γ} ℵ_{r_{j}, s_{j}, ρ_{j}; m}^{U, V} [z t^{μ} | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}]) . \end{array}$ \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) $\begin{array}{l} T_{1} = \sum_{k_{1} = 0}^{[Q_{1} / P_{1}]} \dots \sum_{k_{s} = 0}^{[Q_{s} / P_{s}]} \frac{{(- Q_{1})}_{P_{1} k_{1}} \dots {(- Q_{s})}_{P_{s} k_{s}}}{k_{1}! \dots k_{s}!} \times A_{Q_{1}, P_{1}}^{(1)} \dots A_{Q_{s}, P_{s}}^{(s)} c_{1}^{k_{1}} \dots c_{s}^{k_{s}} \\ \times \frac{1}{2 π ι} \int_{$} Ψ (q, Y) z^{- q} (ℐ_{0 +}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} t^{α + \sum_{j = 1}^{s} λ_{j} k_{j} - μ q - 1}) (x) d q, \end{array}$ \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) $\begin{array}{l} T_{1} = \sum_{k_{1} = 0}^{[Q_{1} / P_{1}]} \dots \sum_{k_{s} = 0}^{[Q_{s} / P_{s}]} \frac{{(- Q_{1})}_{P_{1} k_{1}} \dots {(- Q_{s})}_{P_{s} k_{s}}}{k_{1}! \dots k_{s}!} \times A_{Q_{1}, P_{1}}^{(1)} \dots A_{Q_{s}, P_{s}}^{(s)} c_{1}^{k_{1}} \dots c_{s}^{k_{s}} \\ \times \frac{1}{2 π ι} \int_{$} x^{α - ς - ς^{'} + ϖ + \sum_{j = 1}^{s} λ_{j} k_{j} - 1} Ψ (q, Y) {(z x^{μ})}^{- q} \frac{Γ (α + \sum_{j = 1}^{s} λ_{j} k_{j} - μ q)}{Γ (ϰ^{'} + α + \sum_{j = 1}^{s} λ_{j} k_{j} - μ q)} \times \\ \frac{Γ (- ς^{'} + ϰ^{'} + α + \sum_{j = 1}^{s} λ_{j} k_{j} - μ q) Γ (- ς - ς^{'} - ϰ + ϖ + α + \sum_{j = 1}^{s} λ_{j} k_{j} - μ q)}{Γ (- ς - ς^{'} + ϖ + α + \sum_{j = 1}^{s} λ_{j} k_{j} - μ q) Γ (- ς^{'} - ϰ + ϖ + α + \sum_{j = 1}^{s} λ_{j} k_{j} - μ q)} d q . \end{array}$ \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,λ_k > 0 (k = 1,2,3,···, s), $ℜ (α) + μ min_{1 \leq j \leq U} ℜ (\frac{ε_{j}}{ϰ_{j}}) > max [0, ℜ (ς + ς^{'} + ϰ - ϖ), ℜ (ς^{'} - ϰ^{'})] .$ \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 r_j, s_j, 𝔓 ∈ ℤ⁺ satisfying 0 ≤ V ≤ r_j,0 ≤ U ≤ s_j for j = 1,2,···m, 𝔇_n, 𝔈_n, 𝔈_nj, 𝔇_nj ∈ ℝ⁺, Λ_n, ε_n,Λ_nj,ε_nj ∈ ℂ (j = 1,2,··· , r_j; n = 1,2,··· , s_j), $A_{Q_{k}, P_{k}}^{k}$ A_{{\mathfrak{Q}_k},{\kern 1pt} {\mathfrak{P}_k}}^k are real or complex numbers arbitrary constant for k = 1,2,··· ,s and ρ_i > 0 for i = 1,2,··· ,m, then the outcome shown below is accurate: (37) $\begin{array}{l} ℐ_{0 +}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} (t^{α - 1} \prod_{j = 1}^{s} S_{Q_{j}}^{P_{j}} [c_{j} t^{λ_{j}}] \\ ^{γ} ℵ_{r_{j}, s_{j}, ρ_{j}; m}^{U, V} [z t^{μ} | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}]) (x) \\ = x^{α - ς - ς^{'} + ϖ - 1} \sum_{k_{1} = 0}^{[Q_{1} / P_{1}]} \dots \sum_{k_{s} = 0}^{[Q_{s} / P_{s}]} \frac{{(- Q_{1})}_{P_{1} k_{1}} \dots {(- Q_{s})}_{P_{s} k_{s}}}{k_{1}! \dots k_{s}!} A_{Q_{1}, P_{1}}^{(1)} \dots A_{Q_{s}, P_{s}}^{(s)} c_{1}^{k_{1}} \dots c_{s}^{k_{s}} {(x)}^{\sum_{j = 1}^{s} λ_{j} k_{j}} \\ \times^{γ} ℵ_{r_{j} + 3, s_{j} + 3, ρ_{j}; m}^{U, V + 3} [z x^{μ} | \begin{matrix} (1 - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 - α + ς + ς^{'} + ϰ - ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \\ {(ε_{n}, E_{n})}_{1, U}, (1 - α + ς + ς^{'} - ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \end{matrix} \\ \begin{matrix} (1 - α + ς^{'} - ϰ^{'} - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ (1 - α - ϰ^{'} - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 + ς^{'} + ϰ - ϖ - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}] . \end{array}$ \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,λ_k > 0 (k = 1,2,··· ,s), $ℜ (α) + μ min_{1 \leq j \leq U} ℜ (\frac{ε_{j}}{ϰ_{j}}) > max [0, ℜ (ϰ - ϖ)] .$ \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 r_j, s_j, 𝔓 ∈ ℤ⁺ satisfying 0 ≤ V ≤ r_j, 0 U ≤ s_j for j = 1,2,···m, 𝔇_n, 𝔈_n, 𝔈_nj, 𝔇_nj ∈ ℝ⁺, Λ_n,ε_n,Λ_nj,ε_nj ∈ ℂ (j = 1,2, ··· , r_j;n = 1,2,··· , s_j), $A_{Q_{k}, P_{k}}^{k}$ A_{{\mathfrak{Q}_k},{\kern 1pt} {\mathfrak{P}_k}}^k are real or complex numbers arbitrary constant for k = 1,2,3,··· ,s and ρ_i > 0 for i = 1,2,··· ,m, then the outcome shown below is accurate: (38) $\begin{array}{l} ℐ_{0 +}^{ς, ϰ, ϖ} (t^{α - 1} \prod_{j = 1}^{s} S_{Q_{j}}^{P_{j}} [c_{j} t^{λ_{j}}] \\ ^{Γ} ℵ_{r_{j}, s_{j}, ρ_{j}; m}^{U, V} [z t^{μ} | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}]) (x) \\ = x^{α - ϰ - 1} \sum_{k_{1} = 0}^{[Q_{1} / P_{1}]} \dots \sum_{k_{s} = 0}^{[Q_{s} / P_{s}]} \frac{{(- Q_{1})}_{P_{1} k_{1}} \dots {(- Q_{s})}_{P_{s} k_{s}}}{k_{1}! \dots k_{s}!} A_{Q_{1}, P_{1}}^{(1)} \dots A_{Q_{s}, P_{s}}^{(s)} c_{1}^{k_{1}} \dots c_{s}^{k_{s}} {(x)}^{\sum_{j = 1}^{s} λ_{j} k_{j}} \\ \times^{Γ} ℵ_{r_{j} + 2, s_{j} + 3, ρ_{j}; m}^{U, V + 2} [z x^{μ} | \begin{matrix} (1 - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 - α + ϰ - ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \\ {(ε_{n}, E_{n})}_{1, U}, (1 - α - ς - ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \end{matrix} \\ \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ (1 - α + ϰ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}] . \end{array}$ \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,λ_k > 0 (_k = 1,2,3,··· ,s), $ℜ (α) - μ min_{1 \leq j \leq U} ℜ (\frac{ε_{j}}{ϰ_{j}}) < 1 + min [ℜ (- ϰ), ℜ (ς + ς^{'} - ϖ), ℜ (ς + ϰ^{'} - ϖ)] .$ \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 r_j, s_j, 𝔓 ∈ ℤ⁺ satisfying 0 ≤ V ≤ r_j,0 ≤ U ≤ s_j for j = 1,2,···m, 𝔇_n, 𝔈_n, 𝔈_nj, 𝔇_nj ∈ ℝ⁺, Λ_n,ε_n,Λ_nj,ε_nj ∈ ℂ (j = 1,2,··· , r_j;n = 1,2,··· , s_j), $A_{Q_{k}, P_{k}}^{k}$ A_{{\mathfrak{Q}_k},{\kern 1pt} {\mathfrak{P}_k}}^k are real or complex numbers arbitrary constant for k = 1,2,··· ,s and ρ_i > 0 for i = 1,2,··· ,m, then the outcome shown below is accurate: (39) $\begin{array}{l} ℐ_{-}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} (t^{α - 1} \prod_{j = 1}^{s} S_{Q_{j}}^{P_{j}} [c_{j} t^{λ_{j}}] \\ ^{Γ} ℵ_{r_{j}, s_{j}, ρ_{j}; m}^{U, V} [z t^{μ} | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}]) (x) \\ = x^{α - ς - ς^{'} + ϖ - 1} \sum_{k_{1} = 0}^{[Q_{1} / P_{1}]} \dots \sum_{k_{s} = 0}^{[Q_{s} / P_{s}]} \frac{{(- Q_{1})}_{P_{1} k_{1}} \dots {(- Q_{s})}_{P_{s} k_{s}}}{k_{1}! \dots k_{s}!} A_{Q_{1}, P_{1}}^{(1)} \dots A_{Q_{s}, P_{s}}^{(s)} c_{1}^{k_{1}} \dots c_{s}^{k_{s}} {(x)}^{\sum_{j = 1}^{s} λ_{j} k_{j}} \\ \times^{Γ} ℵ_{r_{j} + 3, s_{j} + 3, ρ_{j}; m}^{U + 3, V} [z x^{μ} | \begin{matrix} (1 - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 - α + ς + ς^{'} + ϰ^{'} - ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \\ {(ε_{n}, E_{n})}_{1, U}, (1 - α + ς + ς^{'} - ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \end{matrix} \\ \begin{matrix} (1 - α + ς - ϰ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ (1 - α - ϰ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 + ς + ϰ^{'} - ϖ - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}] . \end{array}$ \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) $\begin{array}{l} T_{2} = ℐ_{-}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} (t^{α - 1} \prod_{j = 1}^{s} S_{Q_{j}}^{P_{j}} [c_{j} t^{λ_{j}}] \\ ^{Γ} ℵ_{r_{j}, s_{j}, ρ_{j}; m}^{U, V} [z t^{μ} | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}]) . \end{array}$ \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) $\begin{array}{l} T_{2} = \sum_{k_{1} = 0}^{[Q_{1} / P_{1}]} \dots \sum_{k_{s} = 0}^{[Q_{s} / P_{s}]} \frac{{(- Q_{1})}_{P_{1} k_{1}} \dots {(- Q_{s})}_{P_{s} k_{s}}}{k_{1}! \dots k_{s}!} \times A_{Q_{1}, P_{1}}^{(1)} \dots A_{Q_{s}, P_{s}}^{(s)} c_{1}^{k_{1}} \dots c_{s}^{k_{s}} \\ \times \frac{1}{2 π ι} \int_{$} Ψ (q, Y) z^{- q} (ℐ_{-}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} t^{- (- α - \sum_{j = 1}^{s} λ_{j} k_{j} + μ q + 1)}) (x) d q, \end{array}$ \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) $\begin{array}{l} T_{2} = \sum_{k_{1} = 0}^{[Q_{1} / P_{1}]} \dots \sum_{k_{s} = 0}^{[Q_{s} / P_{s}]} \frac{{(- Q_{1})}_{P_{1} k_{1}} \dots {(- Q_{s})}_{P_{s} k_{s}}}{k_{1}! \dots k_{s}!} \times A_{Q_{1}, P_{1}}^{(1)} \dots A_{Q_{s}, P_{s}}^{(s)} c_{1}^{k_{1}} \dots c_{s}^{k_{s}} \\ \times \frac{1}{2 π ι} \int_{$} x^{α - ς - ς^{'} + ϖ + \sum_{j = 1}^{s} λ_{j} k_{j} - 1} Ψ (q, Y) {(z x^{μ})}^{- q} \frac{Γ (1 - α - ϰ - \sum_{j = 1}^{s} λ_{j} k_{j} + μ q)}{Γ (1 - α - \sum_{j = 1}^{s} λ_{j} k_{j} + μ q)} \times \\ \frac{Γ (1 + ς + ς^{'} - ϖ - α - \sum_{j = 1}^{s} λ_{j} k_{j} + μ q) Γ (1 - α + ς + ϰ^{'} - ϖ - \sum_{j = 1}^{s} λ_{j} k_{j} + μ q)}{Γ (1 - α + ς - ϰ - \sum_{j = 1}^{s} λ_{j} k_{j} + μ q) Γ (1 - α + ς + ς^{'} + ϰ^{'} - ϖ - \sum_{j = 1}^{s} λ_{j} k_{j} + μ q)} d q . \end{array}$ \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,λ_k > 0 (k = 1,2,3,··· ,s), $ℜ (α) - μ min_{1 \leq j \leq U} ℜ (\frac{ε_{j}}{ϰ_{j}}) < 1 + min [ℜ (- ϰ), ℜ (ς + ς^{'} - ϖ), ℜ (ς + ϰ^{'} - ϖ)] .$ \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 r_j, s_j, 𝔓 ∈ ℤ⁺ satisfying 0 ≤ V ≤ r_j,0 ≤ U ≤ s_j for j = 1,2,···m, 𝔇_n, 𝔈_n, 𝔈_nj, 𝔇_nj ∈ ℝ⁺, Λ_n,ε_n,Λ_nj,ε_nj ∈ ℂ (j = 1,2,··· , r_j;n = 1,2,··· , s_j), $A_{Q_{k}, P_{k}}^{k}$ A_{{\mathfrak{Q}_k},{\kern 1pt} {\mathfrak{P}_k}}^k are real or complex numbers arbitrary constant for k = 1,2,··· ,s and ρ_i > 0 for i = 1,2,··· ,m, then the outcome shown below is accurate: (43) $\begin{array}{l} ℐ_{-}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} (t^{α - 1} \prod_{j = 1}^{s} S_{Q_{j}}^{P_{j}} [c_{j} t^{λ_{j}}] \\ ^{γ} ℵ_{r_{j}, s_{j}, ρ_{j}; m}^{U, V} [z t^{μ} | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}]) (x) \\ = x^{α - ς - ς^{'} + ϖ - 1} \sum_{k_{1} = 0}^{[Q_{1} / P_{1}]} \dots \sum_{k_{s} = 0}^{[Q_{s} / P_{s}]} \frac{{(- Q_{1})}_{P_{1} k_{1}} \dots {(- Q_{s})}_{P_{s} k_{s}}}{k_{1}! \dots k_{s}!} A_{Q_{1}, P_{1}}^{(1)} \dots A_{Q_{s}, P_{s}}^{(s)} c_{1}^{k_{1}} \dots c_{s}^{k_{s}} {(x)}^{\sum_{j = 1}^{s} λ_{j} k_{j}} \\ \times^{γ} ℵ_{r_{j} + 3, s_{j} + 3, ρ_{j}; m}^{U + 3, V} [z x^{μ} | \begin{matrix} (1 - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 - α + ς + ς^{'} + ϰ^{'} - ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \\ {(ε_{n}, E_{n})}_{1, U}, (1 - α + ς + ς^{'} - ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \end{matrix} \\ \begin{matrix} (1 - α + ς - ϰ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ (1 - α - ϰ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 + ς + ϰ^{'} - ϖ - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}] . \end{array}$ \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,λ_k > 0 (k = 1,2,··· ,s), $ℜ (α) - μ min_{1 \leq j \leq U} ℜ (\frac{ε_{j}}{ϰ_{j}}) < 1 + min [ℜ (ϰ), ℜ (ϖ)] .$ \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 r_j, s_j, 𝔓 ∈ ℤ⁺ satisfying 0 ≤ V ≤ r_j,0 ≤ U ≤ s_j for j = 1,2,···m, 𝔇_n, 𝔈_n, 𝔈_nj, 𝔇_nj ∈ ℝ⁺, Λ_n,ε_n,Λ_nj,ε_nj ∈ ℂ (j = 1,2, ··· , r_j;n = 1,2,··· , s_j), $A_{Q_{k}, P_{k}}^{k}$ A_{{\mathfrak{Q}_k},{\kern 1pt} {\mathfrak{P}_k}}^k are real or complex numbers arbitrary constant for k = 1,2,··· ,s and ρ_i > 0 for i = 1,2,··· ,m, then the outcome shown below is accurate: (44) $\begin{array}{l} ℐ_{-}^{ς, ϰ, ϖ} (t^{α - 1} \prod_{j = 1}^{s} S_{Q_{j}}^{P_{j}} [c_{j} t^{λ_{j}}] \\ ^{Γ} ℵ_{r_{j}, s_{j}, ρ_{j}; m}^{U, V} [z t^{μ} | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}]) (x) \\ = x^{α - ϰ - 1} \sum_{k_{1} = 0}^{[Q_{1} / P_{1}]} \dots \sum_{k_{s} = 0}^{[Q_{s} / P_{s}]} \frac{{(- Q_{1})}_{P_{1} k_{1}} \dots {(- Q_{s})}_{P_{s} k_{s}}}{k_{1}! \dots k_{s}!} A_{Q_{1}, P_{1}}^{(1)} \dots A_{Q_{s}, P_{s}}^{(s)} c_{1}^{k_{1}} \dots c_{s}^{k_{s}} {(x)}^{\sum_{j = 1}^{s} λ_{j} k_{j}} \\ \times^{Γ} ℵ_{r_{j} + 2, s_{j} + 2, ρ_{j}; m}^{U + 2, V} [z x^{μ} | \begin{matrix} (1 - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 - α + ς + ϰ + ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \\ {(ε_{n}, E_{n})}_{1, U}, (1 + ϰ - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \end{matrix} \\ \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ (1 - α + ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}] . \end{array}$ \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 = −ϛand 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,λ_k > 0 (k = 1,2,3,··· ,s), $μ max_{1 \leq j \leq U} [\frac{- ℜ (ε_{j})}{E_{j}}] < ℜ (α) + min [0, ℜ (ς - ϰ), ℜ (ς^{'} + ϰ^{'} + ς - ϖ)] .$ \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 r_j, s_j, 𝔓 ∈ ℤ⁺ satisfying 0 ≤ V ≤ r_j,0 ≤ U ≤ s_j for j = 1,2,···m, 𝔇_n, 𝔈_n, 𝔈_nj, 𝔇_nj ∈ ℝ⁺, Λ_n,ε_n,Λ_nj,ε_nj ∈ ℂ (j = 1,2,··· , r_j;n = 1,2,··· , s_j), $A_{Q_{k}, P_{k}}^{k}$ A_{{\mathfrak{Q}_k},{\kern 1pt} {\mathfrak{P}_k}}^k are real or complex numbers arbitrary constant for k = 1,2,··· ,s and ρ_i > 0 for i = 1,2,··· ,m, then the outcome shown below is accurate: (45) $\begin{array}{l} D_{0 +}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} (t^{α - 1} \prod_{j = 1}^{s} S_{Q_{j}}^{P_{j}} [c_{j} t^{λ_{j}}] \\ ^{Γ} ℵ_{r_{j}, s_{j}, ρ_{j}; m}^{U, V} [z t^{μ} | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}]) (x) \\ = x^{α + ς + ς^{'} - ϖ - 1} \sum_{k_{1} = 0}^{[Q_{1} / P_{1}]} \dots \sum_{k_{s} = 0}^{[Q_{s} / P_{s}]} \frac{{(- Q_{1})}_{P_{1} k_{1}} \dots {(- Q_{s})}_{P_{s} k_{s}}}{k_{1}! \dots k_{s}!} A_{Q_{1}, P_{1}}^{(1)} \dots A_{Q_{s}, P_{s}}^{(s)} c_{1}^{k_{1}} \dots c_{s}^{k_{s}} {(x)}^{\sum_{j = 1}^{s} λ_{j} k_{j}} \\ \times^{Γ} ℵ_{r_{j} + 3, s_{j} + 3, ρ_{j}; m}^{U, V + 3} [z x^{μ} | \begin{matrix} (1 - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 - α - ς - ς^{'} - ϰ^{'} + ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \\ {(ε_{n}, E_{n})}_{1, U}, (1 - ς - ς^{'} + ϖ - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \end{matrix} \\ \begin{matrix} (1 - α - ς + ϰ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ (1 - α + ϰ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 - ς - ϰ^{'} + ϖ - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}] . \end{array}$ \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) $\begin{array}{l} T_{3} = D_{0 +}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} (t^{α - 1} \prod_{j = 1}^{s} S_{Q_{j}}^{P_{j}} [c_{j} t^{λ_{j}}] \\ ^{Γ} ℵ_{r_{j}, s_{j}, ρ_{j}; m}^{U, V} [z t^{μ} | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}]) (x) . \end{array}$ \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) $\begin{array}{l} T_{3} = \sum_{k_{1} = 0}^{[Q_{1} / P_{1}]} \dots \sum_{k_{s} = 0}^{[Q_{s} / P_{s}]} \frac{{(- Q_{1})}_{P_{1} k_{1}} \dots {(- Q_{s})}_{P_{s} k_{s}}}{k_{1}! \dots k_{s}!} \times A_{Q_{1}, P_{1}}^{(1)} \dots A_{Q_{s}, P_{s}}^{(s)} c_{1}^{k_{1}} \dots c_{s}^{k_{s}} \\ \times \frac{1}{2 π ι} \int_{$} Ψ (q, Y) z^{- q} (D_{0 +}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} t^{α + \sum_{j = 1}^{s} λ_{j} k_{j} - μ q - 1}) (x) d q, \end{array}$ \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) $\begin{array}{l} T_{3} = \sum_{k_{1} = 0}^{[Q_{1} / P_{1}]} \dots \sum_{k_{s} = 0}^{[Q_{s} / P_{s}]} \frac{{(- Q_{1})}_{P_{1} k_{1}} \dots {(- Q_{s})}_{P_{s} k_{s}}}{k_{1}! \dots k_{s}!} \times A_{Q_{1}, P_{1}}^{(1)} \dots A_{Q_{s}, P_{s}}^{(s)} c_{1}^{k_{1}} \dots c_{s}^{k_{s}} \\ \times \frac{1}{2 π ι} \int_{$} x^{α - ς - ς^{'} + ϖ + \sum_{j = 1}^{s} λ_{j} k_{j} - 1} Ψ (q, Y) {(z x^{μ})}^{- q} \frac{Γ (α + \sum_{j = 1}^{s} λ_{j} k_{j} - μ q)}{Γ (α - ϰ + \sum_{j = 1}^{s} λ_{j} k_{j} - μ q)} \times \\ \frac{Γ (ς - ϰ + α + \sum_{j = 1}^{s} λ_{j} k_{j} - μ q) Γ (α + ς + ς^{'} + ϰ^{'} - ϖ + \sum_{j = 1}^{s} λ_{j} k_{j} - μ q)}{Γ (α + ς + ϰ^{'} - ϖ + \sum_{j = 1}^{s} λ_{j} k_{j} - μ q) Γ (α + ς + ς^{'} - ϖ + \sum_{j = 1}^{s} λ_{j} k_{j} - μ q)} d q . \end{array}$ \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), $μ max_{1 \leq j \leq U} [\frac{- ℜ (ε_{j})}{E_{j}}] < ℜ (α) + min [0, ℜ (ς - ϰ), ℜ (ς^{'} + ϰ^{'} + ς - ϖ)] .$ \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 r_j, s_j, 𝔓 ∈ ℤ⁺ satisfying 0 ≤ V r_j,0 ≤ U ≤ s_j for j = 1,2,···m, 𝔇_n, 𝔈_n, 𝔈_nj, 𝔇_nj ∈ ℝ⁺, Λ_n,ε_n,Λ_nj,ε_nj ∈ ℂ (j = 1,2,··· , r_j;n = 1,2,··· , s_j), $A_{Q_{k}, P_{k}}^{k}$ A_{{\mathfrak{Q}_k},{\kern 1pt} {\mathfrak{P}_k}}^k are real or complex numbers arbitrary constant for k = 1,2,3,··· ,s and ρ_i > 0 for i = 1,2,··· ,m, then the outcome shown below is accurate: (49) $\begin{array}{l} D_{0 +}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} (t^{α - 1} \prod_{j = 1}^{s} S_{Q_{j}}^{P_{j}} [c_{j} t^{λ_{j}}] \\ ^{γ} ℵ_{r_{j}, s_{j}, ρ_{j}; m}^{U, V} [z t^{μ} | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}]) (x) \\ = x^{α + ς + ς^{'} - ϖ - 1} \sum_{k_{1} = 0}^{[Q_{1} / P_{1}]} \dots \sum_{k_{s} = 0}^{[Q_{s} / P_{s}]} \frac{{(- Q_{1})}_{P_{1} k_{1}} \dots {(- Q_{s})}_{P_{s} k_{s}}}{k_{1}! \dots k_{s}!} A_{Q_{1}, P_{1}}^{(1)} \dots A_{Q_{s}, P_{s}}^{(s)} c_{1}^{k_{1}} \dots c_{s}^{k_{s}} {(x)}^{\sum_{j = 1}^{s} λ_{j} k_{j}} \\ \times^{γ} ℵ_{r_{j} + 3, s_{j} + 3, ρ_{j}; m}^{U, V + 3} [z x^{μ} | \begin{matrix} (1 - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 - α - ς - ς^{'} - ϰ^{'} + ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \\ {(ε_{n}, E_{n})}_{1, U}, (1 - ς - ς^{'} + ϖ - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \end{matrix} \\ \begin{matrix} (1 - α - ς + ϰ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ (1 - α + ϰ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 - ς - ϰ^{'} + ϖ - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}] . \end{array}$ \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,λ_k > 0 (k = 1,2,3,··· ,s), $μ max_{1 \leq j \leq U} [\frac{- ℜ (ε_{j})}{E_{j}}] < ℜ (α) + min [0, ℜ (ς - ϰ), ℜ (ς - ϖ)] .$ \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 r_j, s_j, 𝔓 ∈ ℤ⁺ satisfying 0 ≤ V ≤ r_j,0 ≤ U ≤ s_j for j = 1,2,···m, 𝔇_n, 𝔈_n, 𝔈_nj, 𝔇_nj ∈ ℝ⁺, Λ_n,ε_n,Λ_nj,ε_nj ∈ ℂ (j = 1,2,··· , r_j;n = 1,2,··· , s_j), $A_{Q_{k}, P_{k}}^{k}$ A_{{\mathfrak{Q}_k},{\kern 1pt} {\mathfrak{P}_k}}^k are real or complex numbers arbitrary constant for k = 1,2,··· ,s and ρ_i > 0 for i = 1,2,··· ,m, then the outcome shown below is accurate: (50) $\begin{array}{l} D_{0 +}^{ς, ϰ, ϖ} (t^{α - 1} \prod_{j = 1}^{s} S_{Q_{j}}^{P_{j}} [c_{j} t^{λ_{j}}] \\ ^{Γ} ℵ_{r_{j}, s_{j}, ρ_{j}; m}^{U, V} [z t^{μ} | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}]) (x) \\ = x^{α + ς + ς^{'} - ϖ - 1} \sum_{k_{1} = 0}^{[Q_{1} / P_{1}]} \dots \sum_{k_{s} = 0}^{[Q_{s} / P_{s}]} \frac{{(- Q_{1})}_{P_{1} k_{1}} \dots {(- Q_{s})}_{P_{s} k_{s}}}{k_{1}! \dots k_{s}!} A_{Q_{1}, P_{1}}^{(1)} \dots A_{Q_{s}, P_{s}}^{(s)} c_{1}^{k_{1}} \dots c_{s}^{k_{s}} {(x)}^{\sum_{j = 1}^{s} λ_{j} k_{j}} \\ \times^{Γ} ℵ_{r_{j} + 2, s_{j} + 2, ρ_{j}; m}^{U, V + 2} [z x^{μ} | \begin{matrix} (1 - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 - α - ς - ϰ - ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \\ {(ε_{n}, E_{n})}_{1, U}, (1 - α - ϰ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \end{matrix} \\ \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ (1 - α - ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}] . \end{array}$ \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 = −ϛand 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,λ_k > 0 (k = 1,2,3,··· ,s), $μ min_{1 \leq j \leq V} [\frac{1 - ℜ (Λ_{j})}{D_{j}}] + 1 > ℜ (α) - min [0, ℜ (ϖ - ς - ς^{'} - V), ℜ (- ς^{'} - ϰ + ϖ), - ℜ (ϰ^{'})] .$ \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 r_j, s_j, 𝔓 ∈ ℤ⁺ satisfying 0 ≤ V ≤ r_j,0 ≤ U ≤ s_j for j = 1,2,···m, 𝔇_n, 𝔈_n, 𝔈_nj, 𝔇_nj ∈ ℝ⁺, Λ_n,ε_n,Λ_nj,ε_nj ∈ ℂ (j = 1,2,··· , r_j;n = 1,2,··· , s_j), $A_{Q_{k}, P_{k}}^{k}$ A_{{\mathfrak{Q}_k},{\kern 1pt} {\mathfrak{P}_k}}^k are real or complex numbers arbitrary constant for k = 1,2,··· ,s and ρ_i > 0 for i = 1,2,··· ,m, then the outcome shown below is accurate: (51) $\begin{array}{l} D_{-}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} (t^{α - 1} \prod_{j = 1}^{s} S_{Q_{j}}^{P_{j}} [c_{j} t^{λ_{j}}] \\ ^{Γ} ℵ_{r_{j}, s_{j}, ρ_{j}; m}^{U, V} [z t^{μ} | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}]) (x) \\ = x^{α + ς + ς^{'} - ϖ - 1} \sum_{k_{1} = 0}^{[Q_{1} / P_{1}]} \dots \sum_{k_{s} = 0}^{[Q_{s} / P_{s}]} \frac{{(- Q_{1})}_{P_{1} k_{1}} \dots {(- Q_{s})}_{P_{s} k_{s}}}{k_{1}! \dots k_{s}!} A_{Q_{1}, P_{1}}^{(1)} \dots A_{Q_{s}, P_{s}}^{(s)} c_{1}^{k_{1}} \dots c_{s}^{k_{s}} {(x)}^{\sum_{j = 1}^{s} λ_{j} k_{j}} \\ \times^{Γ} ℵ_{r_{j} + 3, s_{j} + 3, ρ_{j}; m}^{U + 3, V} [z x^{μ} | \begin{matrix} (1 - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 - α - ς^{'} + ϰ^{'} - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \\ {(ε_{n}, E_{n})}_{1, U}, (1 - ς - ς^{'} + ϖ - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \end{matrix} \\ \begin{matrix} (1 - α - ς - ς^{'} - ϰ + ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ (1 - α + ϰ^{'} - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 - ς^{'} - ϰ + ϖ - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}] . \end{array}$ \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) $\begin{array}{l} T_{4} = D_{-}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} (t^{α - 1} \prod_{j = 1}^{s} S_{Q_{j}}^{P_{j}} [c_{j} t^{λ_{j}}] \\ ^{Γ} ℵ_{r_{j}, s_{j}, ρ_{j}; m}^{U, V} [z t^{μ} | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}]) (x) . \end{array}$ \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) $\begin{array}{l} T_{3} = \sum_{k_{1} = 0}^{[Q_{1} / P_{1}]} \dots \sum_{k_{s} = 0}^{[Q_{s} / P_{s}]} \frac{{(- Q_{1})}_{P_{1} k_{1}} \dots {(- Q_{s})}_{P_{s} k_{s}}}{k_{1}! \dots k_{s}!} \times A_{Q_{1}, P_{1}}^{(1)} \dots A_{Q_{s}, P_{s}}^{(s)} c_{1}^{k_{1}} \dots c_{s}^{k_{s}} \\ \times \frac{1}{2 π ι} \int_{$} Ψ (q, Y) z^{- q} (D_{-}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} t^{- (- α - \sum_{j = 1}^{s} λ_{j} k_{j} + μ q + 1)}) (x) d q, \end{array}$ \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) $\begin{array}{l} T_{4} = \sum_{k_{1} = 0}^{[Q_{1} / P_{1}]} \dots \sum_{k_{s} = 0}^{[Q_{s} / P_{s}]} \frac{{(- Q_{1})}_{P_{1} k_{1}} \dots {(- Q_{s})}_{P_{s} k_{s}}}{k_{1}! \dots k_{s}!} \times A_{Q_{1}, P_{1}}^{(1)} \dots A_{Q_{s}, P_{s}}^{(s)} c_{1}^{k_{1}} \dots c_{s}^{k_{s}} \\ \times \frac{1}{2 π ι} \int_{$} x^{α - ς - ς^{'} + ϖ + \sum_{j = 1}^{s} λ_{j} k_{j} - 1} Ψ (q, Y) {(z x^{μ})}^{- q} \frac{Γ (1 - α + ϰ^{'} - \sum_{j = 1}^{s} λ_{j} k_{j} + μ q)}{Γ (1 - α - \sum_{j = 1}^{s} λ_{j} k_{j} + μ q)} \times \\ \frac{Γ (1 - ς - ς^{'} + ϖ - α - \sum_{j = 1}^{s} λ_{j} k_{j} + μ q) Γ (1 - α - ς^{'} - ϰ + ϖ - \sum_{j = 1}^{s} λ_{j} k_{j} + μ q)}{Γ (1 - α - ς^{'} + ϰ^{'} - ϖ - \sum_{j = 1}^{s} λ_{j} k_{j} + μ q) Γ (1 - α - ς - ς^{'} - ϰ + ϖ - \sum_{j = 1}^{s} λ_{j} k_{j} + μ q)} d q . \end{array}$ \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,λ_k > 0 (k = 1,2,3,··· ,s), $μ min_{1 \leq j \leq V} [\frac{1 - ℜ (Λ_{j})}{D_{j}}] + 1 > ℜ (α) - min [0, ℜ (ϖ - ς - ς^{'} - V), ℜ (- ς^{'} - ϰ + ϖ), - ℜ (ϰ^{'})] .$ \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 r_j, s_j, 𝔓 ∈ ℤ⁺ satisfying 0 ≤ V ≤ r_j,0 ≤ U ≤ s_j for j = 1,2,···m, 𝔇_n, 𝔈_n, 𝔈_nj, 𝔇_nj ∈ ℝ⁺, Λ_n,ε_n,Λ_nj,ε_nj ∈ ℂ (j = 1,2,··· , r_j;n = 1,2,··· , s_j), $A_{Q_{k}, P_{k}}^{k}$ A_{{\mathfrak{Q}_k},{\kern 1pt} {\mathfrak{P}_k}}^k are real or complex numbers arbitrary constant for k = 1,2,··· ,s and ρ_i > 0 for i = 1,2,··· ,m, then the outcome shown below is accurate: (55) $\begin{array}{l} D_{-}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} (t^{α - 1} \prod_{j = 1}^{s} S_{Q_{j}}^{P_{j}} [c_{j} t^{λ_{j}}] \\ ^{γ} ℵ_{r_{j}, s_{j}, ρ_{j}; m}^{U, V} [z t^{μ} | \begin{matrix} \begin{array}{l} (Λ_{1}, D_{1} : Y), (_{Λ_{n}, D_{n}) 2, V}, [_{ρ_{n} (Λ_{n j}, D_{n j})] V + 1, r_{j}} \end{array} \\ {(ε_{n}, E_{n})}_{1, U}, {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}]) (x) \\ = x^{α + ς + ς^{'} - ϖ - 1} \sum_{k_{1} = 0}^{[Q_{1} / P_{1}]} \dots \sum_{k_{s} = 0}^{[Q_{s} / P_{s}]} \frac{{(- Q_{1})}_{P_{1} k_{1}} \dots {(- Q_{s})}_{P_{s} k_{s}}}{k_{1}! \dots k_{s}!} A_{Q_{1}, P_{1}}^{(1)} \dots A_{Q_{s}, P_{s}}^{(s)} c_{1}^{k_{1}} \dots c_{s}^{k_{s}} {(x)}^{\sum_{j = 1}^{s} λ_{j} k_{j}} \\ \times^{γ} ℵ_{r_{j} + 3, s_{j} + 3, ρ_{j}; m}^{U + 3, V} [z x^{μ} | \begin{matrix} (1 - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 - α - ς^{'} + ϰ^{'} - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \\ {(ε_{n}, E_{n})}_{1, U}, (1 - ς - ς^{'} + ϖ - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \end{matrix} \\ \begin{matrix} (1 - α - ς - ς^{'} - ϰ + ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ (1 - α + ϰ^{'} - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 - ς^{'} - ϰ + ϖ - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}] . \end{array}$ \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,λ_k > 0 (k = 1,2,3,··· ,s), $μ min_{1 \leq j \leq V} [\frac{1 - ℜ (Λ_{j})}{D_{j}}] + 1 > ℜ (α) - min [0, ℜ (ϖ - ς - ς^{'} - V), ℜ (- ς^{'} - ϰ + ϖ), - ℜ (ϰ^{'})] .$ \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 r_j, s_j, 𝔓 ∈ ℤ⁺ satisfying 0 ≤ V ≤ r_j,0 ≤ U ≤ s_j for j = 1,2,···m, 𝔇_n, 𝔈_n, 𝔈_nj, 𝔇_nj ∈ ℝ⁺, Λ_n,ε_n,Λ_nj,ε_nj ∈ ℂ (j = 1,2,··· , r_j;n = 1,2,··· , s_j), $A_{Q_{k}, P_{k}}^{k}$ A_{{\mathfrak{Q}_k},{\kern 1pt} {\mathfrak{P}_k}}^k are real or complex numbers arbitrary constant for k = 1,2,··· ,s and ρ_i > 0 for i = 1,2,··· ,m, then the following result holds: (56) $\begin{array}{l} D_{-}^{ς, ϰ, ϖ} (t^{α - 1} \prod_{j = 1}^{s} S_{Q_{j}}^{P_{j}} [c_{j} t^{λ_{j}}] \\ ^{Γ} ℵ_{r_{j}, s_{j}, ρ_{j}; m}^{U, V} [z t^{μ} | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}]) (x) \\ = x^{α + ϰ - 1} \sum_{k_{1} = 0}^{[Q_{1} / P_{1}]} \dots \sum_{k_{s} = 0}^{[Q_{s} / P_{s}]} \frac{{(- Q_{1})}_{P_{1} k_{1}} \dots {(- Q_{s})}_{P_{s} k_{s}}}{k_{1}! \dots k_{s}!} A_{Q_{1}, P_{1}}^{(1)} \dots A_{Q_{s}, P_{s}}^{(s)} c_{1}^{k_{1}} \dots c_{s}^{k_{s}} {(x)}^{\sum_{j = 1}^{s} λ_{j} k_{j}} \\ \times^{Γ} ℵ_{r_{j} + 2, s_{j} + 2, ρ_{j}; m}^{U + 2, V} [z x^{μ} | \begin{matrix} (Λ_{1}, D_{1} : Y), (1 - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \\ {(ε_{n}, E_{n})}_{1, U}, (1 - ϰ - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \end{matrix} \\ \begin{matrix} (1 - α - ϰ + ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), {(Λ_{n}, D_{n})}_{2, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ (1 + ς + ϖ - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}] . \end{array}$ \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.

5

Special cases and applications

This section focuses on a few fascinating unique cases of Theorem 3. For other theorems, it is simple for us to obtain comparable findings.

(i) On setting 𝒴 = 0, in Theorem 3 and in consideration of equation (26), then incomplete ℵ-function reduce to the ℵ-function proposed by Sudland [26, 27] and we reach the following conclusion: (57) $\begin{array}{l} ℐ_{0 +}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} (t^{α - 1} \prod_{j = 1}^{s} S_{Q_{j}}^{P_{j}} [c_{j} t^{λ_{j}}] ℵ_{r_{j}, s_{j}, ⟩_{j}; m}^{U, V} [z t^{μ} | \begin{matrix} {(Λ_{n}, D_{n})}_{1, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}]) (x) \\ = x^{α - ς - ς^{'} + ϖ - 1} \sum_{k_{1} = 0}^{[Q_{1} / P_{1}]} \dots \sum_{k_{s} = 0}^{[Q_{s} / P_{s}]} \frac{{(- Q_{1})}_{P_{1} k_{1}} \dots {(- Q_{s})}_{P_{s} k_{s}}}{k_{1}! \dots k_{s}!} A_{Q_{1}, P_{1}}^{(1)} \dots A_{Q_{s}, P_{s}}^{(s)} c_{1}^{k_{1}} \dots c_{s}^{k_{s}} {(x)}^{\sum_{j = 1}^{s} λ_{j} k_{j}} \\ \times ℵ_{r_{j} + 3, s_{j} + 3, ρ_{j}; m}^{U, V + 3} [z x^{μ} | \begin{matrix} (1 - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 - α + ς + ς^{'} + ϰ - ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \\ {(ε_{n}, E_{n})}_{1, U}, (1 - α + ς + ς^{'} - ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \end{matrix} \\ \begin{matrix} (1 - α + ς^{'} - ϰ^{'} - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), {(Λ_{n}, D_{n})}_{1, V}, {[ρ_{n} (Λ_{n j}, D_{n j})]}_{V + 1, r_{j}} \\ (1 - α - ϰ^{'} - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 + ς^{'} + ϰ - ϖ - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), {[ρ_{n} (ε_{n j}, E_{n j})]}_{U + 1, s_{j}} \end{matrix}] . \end{array}$ \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}}}]\aleph _{{r_j},{\kern 1pt} {s_j},{\kern 1pt} {\rangle _j};{\kern 1pt} m}^{U,V}{\kern 1pt} \left[ {z{\kern 1pt} {t^\mu }\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]} \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 \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 _n},{\mathfrak{D}_n})}_{1,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}
(ii) Again, setting ρ_j = 1 in Theorem 3 and in consideration of equation (28), then incomplete ℵ-function reduces to the Incomplete I-function suggested by Bansal and Kumar [28] and we reach the following conclusion: (58) $\begin{array}{l} ℐ_{0 +}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} (t^{α - 1} \prod_{j = 1}^{s} S_{Q_{j}}^{P_{j}} {[c_{j} t^{λ_{j}}]}^{Γ} I_{r_{j}, s_{j}; m}^{U, V} [z t^{μ} | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {(Λ_{n j}, D_{n j})}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {(ε_{n j}, E_{n j})}_{U + 1, s_{j}} \end{matrix}]) (x) \\ = x^{α - ς - ς^{'} + ϖ - 1} \sum_{k_{1} = 0}^{[Q_{1} / P_{1}]} \dots \sum_{k_{s} = 0}^{[Q_{s} / P_{s}]} \frac{{(- Q_{1})}_{P_{1} k_{1}} \dots {(- Q_{s})}_{P_{s} k_{s}}}{k_{1}! \dots k_{s}!} A_{Q_{1}, P_{1}}^{(1)} \dots A_{Q_{s}, P_{s}}^{(s)} c_{1}^{k_{1}} \dots c_{s}^{k_{s}} {(x)}^{\sum_{j = 1}^{s} λ_{j} k_{j}} \\ \times^{Γ} I_{r_{j} + 3, s_{j} + 3; m}^{U, V + 3} [z x^{μ} | \begin{matrix} (1 - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 - α + ς + ς^{'} + ϰ - ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \\ {(ε_{n}, E_{n})}_{1, U}, (1 - α + ς + ς^{'} - ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \end{matrix} \\ \begin{matrix} (1 - α + ς^{'} - ϰ^{'} - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, V}, {(Λ_{n j}, D_{n j})}_{V + 1, r_{j}} \\ (1 - α - ϰ^{'} - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 + ς^{'} + ϰ - ϖ - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), {(ε_{n j}, E_{n j})}_{U + 1, s_{j}} \end{matrix}] . \end{array}$ \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}}}]}^\Gamma }I_{{r_j},{\kern 1pt} {s_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} {{({\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]} \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 }I_{{r_j} + 3,{\kern 1pt} {s_j} + 3{\kern 1pt} ;{\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} {{({\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} {{({\varepsilon _{nj}},{\mathfrak{E}_{nj}})}_{U + 1,{s_j}}}}\end{array}} \right].\end{array}
(iii) Next, setting 𝒴 = 0 and ρ_j = 1 in Theorem 3 and in consideration of equation (29), then incomplete ℵ-function reduce to the I-function suggested by Saxena [29] and we reach the following conclusion: (59) $\begin{array}{l} ℐ_{0 +}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} (t^{α - 1} \prod_{j = 1}^{s} S_{Q_{j}}^{P_{j}} [c_{j} t^{λ_{j}}] I_{r_{j}, s_{j}; m}^{U, V} [z t^{μ} | \begin{matrix} {(Λ_{n}, D_{n})}_{1, V}, {(Λ_{n j}, D_{n j})}_{V + 1, r_{j}} \\ {(ε_{n}, E_{n})}_{1, U}, {(ε_{n j}, E_{n j})}_{U + 1, s_{j}} \end{matrix}]) (x) \\ = x^{α - ς - ς^{'} + ϖ - 1} \sum_{k_{1} = 0}^{[Q_{1} / P_{1}]} \dots \sum_{k_{s} = 0}^{[Q_{s} / P_{s}]} \frac{{(- Q_{1})}_{P_{1} k_{1}} \dots {(- Q_{s})}_{P_{s} k_{s}}}{k_{1}! \dots k_{s}!} A_{Q_{1}, P_{1}}^{(1)} \dots A_{Q_{s}, P_{s}}^{(s)} c_{1}^{k_{1}} \dots c_{s}^{k_{s}} {(x)}^{\sum_{j = 1}^{s} λ_{j} k_{j}} \\ \times I_{r_{j} + 3, s_{j} + 3; m}^{U, V + 3} [z x^{μ} | \begin{matrix} (1 - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 - α + ς + ς^{'} + ϰ - ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \\ {(ε_{n}, E_{n})}_{1, U}, (1 - α + ς + ς^{'} - ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \end{matrix} \\ \begin{matrix} (1 - α + ς^{'} - ϰ^{'} - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), {(Λ_{n}, D_{n})}_{1, V}, {(Λ_{n j}, D_{n j})}_{V + 1, r_{j}} \\ (1 - α - ϰ^{'} - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 + ς^{'} + ϰ - ϖ - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), {(ε_{n j}, E_{n j})}_{U + 1, s_{j}} \end{matrix}] . \end{array}$ \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}}}]I_{{r_j},{\kern 1pt} {s_j};{\kern 1pt} m}^{U,V}{\kern 1pt} \left[ {z{\kern 1pt} {t^\mu }\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]} \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 I_{{r_j} + 3,{\kern 1pt} {s_j} + 3{\kern 1pt} ;{\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 _n},{\mathfrak{D}_n})}_{1,V}},{\kern 1pt} {{({\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} {{({\varepsilon _{nj}},{\mathfrak{E}_{nj}})}_{U + 1,{s_j}}}}\end{array}} \right].\end{array}
(iv) Further setting ρ_j = 1 and m = 1 in Theorem 3 and in consideration of equation (31), then incomplete ℵ-function reduce to the incomplete H-function suggested by Srivastava [32] and we reach the following conclusion: (60) $\begin{array}{l} ℐ_{0 +}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} (t^{α - 1} \prod_{j = 1}^{s} S_{Q_{j}}^{P_{j}} [c_{j} t^{λ_{j}}] Γ_{r, s}^{U, V} [z t^{μ} | \begin{matrix} (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, r} \\ {(ε_{n}, E_{n})}_{1, s} \end{matrix}]) (x) \\ = x^{α - ς - ς^{'} + ϖ - 1} \sum_{k_{1} = 0}^{[Q_{1} / P_{1}]} \dots \sum_{k_{s} = 0}^{[Q_{s} / P_{s}]} \frac{{(- Q_{1})}_{P_{1} k_{1}} \dots {(- Q_{s})}_{P_{s} k_{s}}}{k_{1}! \dots k_{s}!} A_{Q_{1}, P_{1}}^{(1)} \dots A_{Q_{s}, P_{s}}^{(s)} c_{1}^{k_{1}} \dots c_{s}^{k_{s}} {(x)}^{\sum_{j = 1}^{s} λ_{j} k_{j}} \\ \times Γ_{r + 3, s + 3}^{U, V + 3} [z x^{μ} | \begin{matrix} (1 - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 - α + ς + ς^{'} + ϰ - ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \\ {(ε_{n}, E_{n})}_{1, s}, (1 - α + ς + ς^{'} - ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \end{matrix} \\ \begin{matrix} (1 - α + ς^{'} - ϰ^{'} - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (Λ_{1}, D_{1} : Y), {(Λ_{n}, D_{n})}_{2, r} \\ (1 - α - ϰ^{'} - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 + ς^{'} + ϰ - ϖ - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ) \end{matrix}] . \end{array}$ \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}}}]\Gamma _{r,{\kern 1pt} s}^{U,V}{\kern 1pt} \left[ {z{\kern 1pt} {t^\mu }\Biggl|\begin{array}{*{20}{c}}{({\Lambda _1},{\mathfrak{D}_1}:\mathcal{Y}),{\kern 1pt} {{({\Lambda _n},{\mathfrak{D}_n})}_{2,r}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,s}}}\\ \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 _{r + 3,{\kern 1pt} s + 3}^{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,s}},(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}),{\kern 1pt} {{({\Lambda _n},{\mathfrak{D}_n})}_{2,r}}}\\{(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 )}\end{array}} \right].\end{array}
(v) Next, setting 𝒴 = 0, ρ_j = 1, and m = 1 in Theorem 3 and in consideration of equation (32), then incomplete ℵ-function reduce to the H-function suggested by Srivastava [33] and we reach the following conclusion: (61) $\begin{array}{l} ℐ_{0 +}^{ς, ς^{'}, ϰ, ϰ^{'}, ϖ} (t^{α - 1} \prod_{j = 1}^{s} S_{Q_{j}}^{P_{j}} [c_{j} t^{λ_{j}}] H_{r, s}^{U, V} [z t^{μ} | \begin{matrix} {(Λ_{n}, D_{n})}_{1, r} \\ {(ε_{n}, E_{n})}_{1, s} \end{matrix}]) (x) \\ = x^{α - ς - ς^{'} + ϖ - 1} \sum_{k_{1} = 0}^{[Q_{1} / P_{1}]} \dots \sum_{k_{s} = 0}^{[Q_{s} / P_{s}]} \frac{{(- Q_{1})}_{P_{1} k_{1}} \dots {(- Q_{s})}_{P_{s} k_{s}}}{k_{1}! \dots k_{s}!} A_{Q_{1}, P_{1}}^{(1)} \dots A_{Q_{s}, P_{s}}^{(s)} c_{1}^{k_{1}} \dots c_{s}^{k_{s}} {(x)}^{\sum_{j = 1}^{s} λ_{j} k_{j}} \\ \times H_{r + 3, s + 3}^{U, V + 3} [z x^{μ} | \begin{matrix} (1 - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 - α + ς + ς^{'} + ϰ - ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \\ {(ε_{n}, E_{n})}_{1, s}, (1 - α + ς + ς^{'} - ϖ - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), \end{matrix} \\ \begin{matrix} (1 - α + ς^{'} - ϰ^{'} - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), {(Λ_{n}, D_{n})}_{1, r} \\ (1 - α - ϰ^{'} - \sum_{j = 1}^{s} λ_{j} k_{j}, μ), (1 + ς^{'} + ϰ - ϖ - α - \sum_{j = 1}^{s} λ_{j} k_{j}, μ) \end{matrix}] . \end{array}$ \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}}}]H_{r,{\kern 1pt} s}^{U,V}\left[ {z{\kern 1pt} {t^\mu }\Biggl|\begin{array}{*{20}{c}}{{{({\Lambda _n},{\mathfrak{D}_n})}_{1,r}}}\\{{{({\varepsilon _n},{\mathfrak{E}_n})}_{1,s}}}\\ \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 H_{r + 3,{\kern 1pt} s + 3}^{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,s}},(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 _n},{\mathfrak{D}_n})}_{1,r}}}\\{(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 )}\end{array}} \right].\end{array}

Remark 10

The known results provided by Saxena and Saigo [35] are simple to be achieved if the generic class of polynomials $S_{L}^{h_{1}, \dots, h_{s}}$ S_L^{{h_1}, \cdots ,{h_s}} is restricted to unity and the incomplete ℵ-function is reduced to Fox’s H-function.
If we set incomplete ℵ-function to ℵ-function, then we may easily achieve the results that Saxena and Kumar [36] have already provided.
The known results provided by Saxena and Ram [37] are simple to be achieved if the generic class of polynomials $S_{L}^{h_{1}, \dots, h_{s}}$ S_L^{{h_1}, \cdots ,{h_s}} is restricted to unity and the incomplete ℵ-function is reduced to ℵ-function.
Theorems 4 and Theorem 5 provided by Bansal et al. [24] are simply obtained if we set the generic class of polynomials $S_{L}^{h_{1}, \dots, h_{s}}$ S_L^{{h_1}, \cdots ,{h_s}} in Theorem 3 and Theorem 6 to unity.

6

Conclusion

In the current paper, we looked into a variety of incomplete ℵ-function based FC image formulae as well as the generic class of polynomials connected to the MSM operators. The incomplete ℵ-functions are the generalized form of various other special functions. Also, Srivastava polynomial generalize various other polynomials like: Hermite polynomial, Jacobi polynomial, Laguerre polynomial, Gegenbauer polynomial, Legendre polynomial, Tchebycheff polynomial, Gould-Hopper Polynomial and several other polynomials. Additionally, the MSM fractional operators generalize Saigo, R-L and E-K FC operators. One may get a variety of image formulae that include a class of special functions by taking the mentioned fact into consideration [23, 38,39,40] as limiting instances of the primary outcomes.

7

Declarations

7.1

Conflict of interest

Authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

7.2

Funding

The authors hereby declare that there is no any funding interest.

7.3

Author’s contribution

N.-Conceptualization. S.B.-Data Curation, Writing-Review Editing. S.D.P.-Software, Writing-Original Draft. K.S.N.-Supervisor. S.R.M.-Methodology. All authors read and approved the final submitted version of this manuscript. All authors contributed equally to the manuscript and approved the final manuscript.

7.4

Acknowledgement

The authors express their sincere thanks to the editor and reviewers for their fruitful comments and suggestions that improved the quality of the manuscript.

7.5

Data availability statement

All data that support the findings of this study are included within the article.

7.6

Using of AI tools

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

Some fractional calculus findings associated with the product of incomplete ℵ-function and Srivastava polynomials

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