A study of incomplete I-functions relating to certain fractional integral operators

Abstract

The study discussed in this article is driven by the realization that many physical processes may be understood by using applications of fractional operators and special functions. In this study, we present and examine a fractional integral operator with an I-function in its kernel. This operator is used to solve several fractional differential equations (FDEs). FDE has a set of particular cases whose solutions represent different physical phenomena. Many mathematical physics, biology, engineering, and chemistry problems are identified and solved using FDE. Specifically, a few exciting relations involving the new fractional operator with incomplete I-function (IIF) in its kernel and classical Riemann Liouville fractional integral and derivative operators, the Hilfer fractional derivative operator, and the generalized composite fractional derivate (GCFD) operator are established. The discovery and investigation of several important exceptional cases follow this.

Keywords:

• Fractional integral operators
• fractional derivative operators
• IIF
• Lebesgue measurable functions

Mathematics Subject Classifications:

• 26A33
• 33C60
• 33E20
• 34A08
• 45J05

1. Introduction

Over the last four decades, mathematicians and scientists have been attracted to the fractional calculus and special functions because of their wide range of applications and significance in fields such as fluid dynamics, computer science, viscoelasticity, diffusive transport, electrical finance networks, medical science, biological science, signal processing, social sciences, control theory, ecology, environmental science, and so on [1–4].

Baleanu et al. [5] investigated the existence of solutions for four fractional integer-differential inclusions, including the Caputo–Fabrizio fractional derivative. Recently Odibat et al. [6] studied some features of the modified Caputo-type Mittag–Leffler fractional derivative operator and its related fractional integral operator and developed a novel predictor-corrector algorithm for numerical solutions of IVPs involving modified Caputo-type Mittag–Leffler fractional derivatives. In fractional calculus, Bhatter et al. [7] developed new results for the generalized extended Mittag–Leffler function, the S-function, the general class of polynomials, and the H-function. Using a novel computational method, Geo et al. [8] investigated the effect of the fractional operator in the analytical study of the solution for a suitable nonlinear fractional dynamical system describing the coronavirus (2019-nCoV) with an explanation of some interesting properties of the coronavirus infection. Baleanu et al. [9] recently investigated a new mathematical model incorporating the general form of the Caputo fractional derivative for the actual case of a cholera outbreak and compared the results to accurate data from a cholera outbreak in Yemen, finding that the fractional framework is closer to the real data than the classical framework.

Fractional calculus has been used in several scientific areas, including fuzzy control, physics, automatic control, biology, signal processing, and robotics, because it can represent systems and physical phenomena more accurately than classical integer-order calculus. Fractional derivatives are shown to give a more clear picture when applied to an engineering problem because of the broad spectrum they work in and their memory effects [6,10,11].

Jangid et al. [12] proposed the IIF and developed various integral transformations for it. A few applications are also presented. Furthermore, the incomplete H-function and IIF have been observed in multiple response-related problems, including diffusion, reaction-diffusion, electronics and connectivity, FDEs, astrophysics, nuclear physics, groundwater, statistics, and a variety of other areas of physics, biology, and probability theory, to name a few [13,14]. The dynamic integration of Fractional derivative operators and IIFs will contribute significantly to the literature of applied Mathematics for the researchers to use it in specific problems of science and engineering along with their exciting properties, as shown in this research work.

This study presents and investigates a fractional integral operator with an I function in its kernel, inspired by a recent research effort on fractional calculus and special functions [15–18]. In particular, new fractional operators with IIF in its kernel and some exciting relations involving the classical Riemann–Liouville fractional integral and derivative operators, the Hilfer fractional derivative operator and the GCFD operator are established. Many important extraordinary cases were discovered and investigated after that. This operator is used to solve multiple FDEs. FDEs consist of a set of particular circumstances whose solutions represent different physical phenomena.

The rest of the paper is organized as follows: Section 2 introduces some crucial definitions of the special function and fractional operator. Section 3 is devoted to the main findings, followed by Section 4, which covers observations of the results. Finally, the conclusion is stated in Section 5.

2. Preliminary results and definitions

This section recalls some fundamental definitions of the special functions and fractional operators.

Incomplete Gamma Function: The usual incomplete Gamma functions $\gamma (\mathfrak{c},s)$ and $\mathrm{\Gamma }(\mathfrak{c},s)$ represented by Chaudhry and Zubair [19] (1) $\gamma (\mathfrak{c},s):={\int }_0^s{\theta }^{\mathfrak{c}-1}{\mathrm{e}}^{-\theta }\mathrm{d}\theta ,\left(\mathrm{\Re }\right(\mathfrak{c})>0;s\ge 0),$(1) and (2) $\mathrm{\Gamma }(\mathfrak{c},s):={\int }_s^{\mathrm{\infty }}{\theta }^{\mathfrak{c}-1}{\mathrm{e}}^{-\theta }\mathrm{d}\theta ,\left(\mathrm{\Re }\right(\mathfrak{c})>0;s\ge 0),$(2) satisfy the subsequent rule of decomposition: (3) $\gamma (\mathfrak{c},s)+\mathrm{\Gamma }(\mathfrak{c},s):=\mathrm{\Gamma }\left(\mathfrak{c}\right),\left(\mathrm{\Re }\right(\mathfrak{c})>0),$(3) where $\mathrm{\Re }\left(\mathfrak{c}\right)$ stands for real part of the parameter $\mathfrak{c}$.

Moreover, if we set s=0, then we have $\mathrm{\Gamma }(\mathfrak{c},s)=\mathrm{\Gamma }\left(\mathfrak{c}\right)$.

I-Function: In 1997, Rathie [20] developed a generalization of the well-known H-function of one variable, introduced by Fox [21] and further explored by Braaksma [22]. Polylogarithms, the exact partition of the Gaussian model from statistical mechanics, and other particular instances of several fundamental functions are included in this generalization. Which is called I-function and is defined as follows by the Mellin-Barnes kind contour integral: (4) $I_{r,s}^{u,v}\left(\mathcal{V}\right)=I_{r,s}^{u,v}[\mathcal{V}\left|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1),\dots ,({\Psi }_r,{\zeta }_r;A_r)\\ ({\Phi }_1,{\text{β}}_1;B_1),\dots ,({\Phi }_s,{\text{β}}_s;B_s)\end{array}\right]=\frac{1}{2\pi i}{\int }_{\mathrm{\$}}\mathrm{\Psi }\left(w\right){\mathcal{V}}^w\mathrm{d}w,$(4) where (5) $\mathrm{\Psi }\left(w\right)=\frac{\prod _{i=1}^u\left\{\mathrm{\Gamma }\right({\Phi }_i-{\text{β}}_{i}w\left){\}}^{B_j}\prod _{i=1}^v\right\{\mathrm{\Gamma }(1-{\Psi }_i+{\zeta }_{i}w){\}}^{A_i}}{\prod _{i=v+1}^r\left\{\mathrm{\Gamma }\right({\Psi }_i-{\zeta }_{i}w\left){\}}^{A_i}\prod _{i=u+1}^s\right\{\mathrm{\Gamma }(1-{\Phi }_i+{\text{β}}_{i}w){\}}^{B_i}}.$(5) The appropriate conditions for the $ contour convergence described in (4(4) $I_{r,s}^{u,v}\left(\mathcal{V}\right)=I_{r,s}^{u,v}[\mathcal{V}\left|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1),\dots ,({\Psi }_r,{\zeta }_r;A_r)\\ ({\Phi }_1,{\text{β}}_1;B_1),\dots ,({\Phi }_s,{\text{β}}_s;B_s)\end{array}\right]=\frac{1}{2\pi i}{\int }_{\mathrm{\$}}\mathrm{\Psi }\left(w\right){\mathcal{V}}^w\mathrm{d}w,$(4) ) and other representations, as well as the I-function information can be viewed in [20].

The Incomplete I-Functions: Now, we present a family of the incomplete I-functions [23] ${}^{\gamma }{I}_{r,s}^{u,v}\left(\mathcal{V}\right)$ and ${}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left(\mathcal{V}\right)$, which leads to a natural generalization of a variety of I-functions: (6) $\begin{array}{rl}{}^{\gamma }{I}_{r,s}^{u,v}\left(\mathcal{V}\right)& ={}^{\gamma }{I}_{r,s}^{u,v}\left[\mathcal{V}|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),({\Psi }_2,{\zeta }_2;A_2),\dots ,({\Psi }_r,{\zeta }_r;A_r)\\ ({\Phi }_1,{\text{β}}_1;B_1),({\Phi }_2,{\text{β}}_2;B_2),\dots ,({\Phi }_s,{\text{β}}_s;B_s)\end{array}\right]\\ & =\frac{1}{2\pi i}{\int }_{\mathrm{\$}}\mathrm{\Psi }(w,Y){\mathcal{V}}^w\mathrm{d}w,\end{array}$(6) and (7) $\begin{array}{rl}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left(\mathcal{V}\right)& ={}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),({\Psi }_2,{\zeta }_2;A_2),\dots ,({\Psi }_r,{\zeta }_r;A_r)\\ ({\Phi }_1,{\text{β}}_1;B_1),({\Phi }_2,{\text{β}}_2;B_2),\dots ,({\Phi }_s,{\text{β}}_s;B_s)\end{array}\right]\\ & =\frac{1}{2\pi i}{\int }_{\mathrm{\$}}\varphi (w,Y){\mathcal{V}}^w\mathrm{d}w,\end{array}$(7) for all $\mathcal{V}\ne 0$, where (8) $\mathrm{\Psi }(w,Y)=\frac{\left\{\gamma \right(1-{\Psi }_1+{\zeta }_{1}w,Y\left){\}}^{A_1}\prod _{i=1}^u\right\{\mathrm{\Gamma }({\Phi }_i-{\text{β}}_{i}w){\}}^{B_j}\prod _{i=2}^v\left\{\mathrm{\Gamma }\right(1-{\Psi }_i+{\zeta }_{i}w){\}}^{A_i}}{\prod _{i=v+1}^r\left\{\mathrm{\Gamma }\right({\Psi }_i-{\zeta }_{i}w\left){\}}^{A_i}\prod _{i=u+1}^s\right\{\mathrm{\Gamma }(1-{\Phi }_i+{\text{β}}_{i}w){\}}^{B_i}},$(8) and (9) $\varphi (w,Y)=\frac{\left\{\mathrm{\Gamma }\right(1-{\Psi }_1+{\zeta }_{1}w,Y\left){\}}^{A_1}\prod _{i=1}^u\right\{\mathrm{\Gamma }({\Phi }_i-{\text{β}}_{i}w){\}}^{B_j}\prod _{i=2}^v\left\{\mathrm{\Gamma }\right(1-{\Psi }_i+{\zeta }_{i}w){\}}^{A_i}}{\prod _{i=v+1}^r\left\{\mathrm{\Gamma }\right({\Psi }_i-{\zeta }_{i}w\left){\}}^{A_i}\prod _{i=u+1}^s\right\{\mathrm{\Gamma }(1-{\Phi }_i+{\text{β}}_{i}w){\}}^{B_i}}.$(9) The following division relation is immediately produced by the definitions (6(6) $\begin{array}{rl}{}^{\gamma }{I}_{r,s}^{u,v}\left(\mathcal{V}\right)& ={}^{\gamma }{I}_{r,s}^{u,v}\left[\mathcal{V}|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),({\Psi }_2,{\zeta }_2;A_2),\dots ,({\Psi }_r,{\zeta }_r;A_r)\\ ({\Phi }_1,{\text{β}}_1;B_1),({\Phi }_2,{\text{β}}_2;B_2),\dots ,({\Phi }_s,{\text{β}}_s;B_s)\end{array}\right]\\ & =\frac{1}{2\pi i}{\int }_{\mathrm{\$}}\mathrm{\Psi }(w,Y){\mathcal{V}}^w\mathrm{d}w,\end{array}$(6) ) and (7(7) $\begin{array}{rl}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left(\mathcal{V}\right)& ={}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),({\Psi }_2,{\zeta }_2;A_2),\dots ,({\Psi }_r,{\zeta }_r;A_r)\\ ({\Phi }_1,{\text{β}}_1;B_1),({\Phi }_2,{\text{β}}_2;B_2),\dots ,({\Phi }_s,{\text{β}}_s;B_s)\end{array}\right]\\ & =\frac{1}{2\pi i}{\int }_{\mathrm{\$}}\varphi (w,Y){\mathcal{V}}^w\mathrm{d}w,\end{array}$(7) ) for the value of $A_1=1$: (10) ${}^{\gamma }{I}_{r,s}^{u,v}\left[\mathcal{V}\right]+{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}\right]=I_{r,s}^{u,v}\left[\mathcal{V}\right],(for{A}_1=1).$(10)

Remark 2.1

On setting Y =0, in (6(6) $\begin{array}{rl}{}^{\gamma }{I}_{r,s}^{u,v}\left(\mathcal{V}\right)& ={}^{\gamma }{I}_{r,s}^{u,v}\left[\mathcal{V}|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),({\Psi }_2,{\zeta }_2;A_2),\dots ,({\Psi }_r,{\zeta }_r;A_r)\\ ({\Phi }_1,{\text{β}}_1;B_1),({\Phi }_2,{\text{β}}_2;B_2),\dots ,({\Phi }_s,{\text{β}}_s;B_s)\end{array}\right]\\ & =\frac{1}{2\pi i}{\int }_{\mathrm{\$}}\mathrm{\Psi }(w,Y){\mathcal{V}}^w\mathrm{d}w,\end{array}$(6) ) and (7(7) $\begin{array}{rl}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left(\mathcal{V}\right)& ={}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),({\Psi }_2,{\zeta }_2;A_2),\dots ,({\Psi }_r,{\zeta }_r;A_r)\\ ({\Phi }_1,{\text{β}}_1;B_1),({\Phi }_2,{\text{β}}_2;B_2),\dots ,({\Phi }_s,{\text{β}}_s;B_s)\end{array}\right]\\ & =\frac{1}{2\pi i}{\int }_{\mathrm{\$}}\varphi (w,Y){\mathcal{V}}^w\mathrm{d}w,\end{array}$(7) ) reduce to the I-Function proposed by Rathie [20]: (11) $\begin{array}{rl}& {}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:0),({\Psi }_2,{\zeta }_2;A_2),\dots ,({\Psi }_r,{\zeta }_r;A_r)\\ ({\Phi }_1,{\text{β}}_1;B_1),({\Phi }_2,{\text{β}}_2;B_2),\dots ,({\Phi }_s,{\text{β}}_s;B_s)\end{array}\right]\\ & =I_{r,s}^{u,v}\left[\mathcal{V}|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1),\dots ,({\Psi }_r,{\zeta }_r;A_r)\\ ({\Phi }_1,{\text{β}}_1;B_1),\dots ,({\Phi }_s,{\text{β}}_s;B_s)\end{array}\right].\end{array}$(11)

Remark 2.2

Again, setting $A_i=1,B_j=1(i=1,2,\dots ,r,j=1,2,\dots ,s)$ in (6(6) $\begin{array}{rl}{}^{\gamma }{I}_{r,s}^{u,v}\left(\mathcal{V}\right)& ={}^{\gamma }{I}_{r,s}^{u,v}\left[\mathcal{V}|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),({\Psi }_2,{\zeta }_2;A_2),\dots ,({\Psi }_r,{\zeta }_r;A_r)\\ ({\Phi }_1,{\text{β}}_1;B_1),({\Phi }_2,{\text{β}}_2;B_2),\dots ,({\Phi }_s,{\text{β}}_s;B_s)\end{array}\right]\\ & =\frac{1}{2\pi i}{\int }_{\mathrm{\$}}\mathrm{\Psi }(w,Y){\mathcal{V}}^w\mathrm{d}w,\end{array}$(6) ) and (7(7) $\begin{array}{rl}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left(\mathcal{V}\right)& ={}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),({\Psi }_2,{\zeta }_2;A_2),\dots ,({\Psi }_r,{\zeta }_r;A_r)\\ ({\Phi }_1,{\text{β}}_1;B_1),({\Phi }_2,{\text{β}}_2;B_2),\dots ,({\Phi }_s,{\text{β}}_s;B_s)\end{array}\right]\\ & =\frac{1}{2\pi i}{\int }_{\mathrm{\$}}\varphi (w,Y){\mathcal{V}}^w\mathrm{d}w,\end{array}$(7) ), then it changes into the proposed Incomplete H-Function of Srivastava [24]: (12) $\begin{array}{rl}& {}^{\gamma }{I}_{r,s}^{u,v}\left[\mathcal{V}|\begin{array}{c}({\Psi }_1,{\zeta }_1;1:Y),({\Psi }_2,{\zeta }_2;1),\dots ,({\Psi }_r,{\zeta }_r;1)\\ ({\Phi }_1,{\text{β}}_1;1),({\Phi }_2,{\text{β}}_2;1),\dots ,({\Phi }_s,{\text{β}}_s;1)\end{array}\right]\\ & ={\gamma }_{r,s}^{u,v}\left[\mathcal{V}|\begin{array}{c}({\Psi }_1,{\zeta }_1:Y),({\Psi }_2,{\zeta }_2),\dots ,({\Psi }_r,{\zeta }_r)\\ ({\Phi }_1,{\text{β}}_1),({\Phi }_2,{\text{β}}_2),\dots ,({\Phi }_s,{\text{β}}_s)\end{array}\right],\end{array}$(12) and (13) $\begin{array}{rl}& {}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}|\begin{array}{c}({\Psi }_1,{\zeta }_1;1:Y),({\Psi }_2,{\zeta }_2;1),\dots ,({\Psi }_r,{\zeta }_r;1)\\ ({\Phi }_1,{\text{β}}_1;1),({\Phi }_2,{\text{β}}_2;1),\dots ,({\Phi }_s,{\text{β}}_s;1)\end{array}\right]\\ & ={\mathrm{\Gamma }}_{r,s}^{u,v}\left[\mathcal{V}|\begin{array}{c}({\Psi }_1,{\zeta }_1:Y),({\Psi }_2,{\zeta }_2),\dots ,({\Psi }_r,{\zeta }_r)\\ ({\Phi }_1,{\text{β}}_1),({\Phi }_2,{\text{β}}_2),\dots ,({\Phi }_s,{\text{β}}_s)\end{array}\right].\end{array}$(13)

Remark 2.3

Next, we take $Y=0,A_i=1,B_j=1(i=1,2,\dots ,r,j=1,2,\dots ,s)$ in (6(6) $\begin{array}{rl}{}^{\gamma }{I}_{r,s}^{u,v}\left(\mathcal{V}\right)& ={}^{\gamma }{I}_{r,s}^{u,v}\left[\mathcal{V}|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),({\Psi }_2,{\zeta }_2;A_2),\dots ,({\Psi }_r,{\zeta }_r;A_r)\\ ({\Phi }_1,{\text{β}}_1;B_1),({\Phi }_2,{\text{β}}_2;B_2),\dots ,({\Phi }_s,{\text{β}}_s;B_s)\end{array}\right]\\ & =\frac{1}{2\pi i}{\int }_{\mathrm{\$}}\mathrm{\Psi }(w,Y){\mathcal{V}}^w\mathrm{d}w,\end{array}$(6) ). The IIF convert to the Fox H-function, which is described and represented as follows (see, [25, pp. 10]): (14) $\begin{array}{rl}& {}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}|\begin{array}{c}({\Psi }_1,{\zeta }_1;1:0),({\Psi }_2,{\zeta }_2;1),\dots ,({\Psi }_r,{\zeta }_r;1)\\ ({\Phi }_1,{\text{β}}_1;1),({\Phi }_2,{\text{β}}_2;1),\dots ,({\Phi }_s,{\text{β}}_s;1)\end{array}\right]\\ & =H_{r,s}^{u,v}\left[\mathcal{V}|\begin{array}{c}({\Psi }_1,{\zeta }_1),({\Psi }_2,{\zeta }_2),\dots ,({\Psi }_r,{\zeta }_r)\\ ({\Phi }_1,{\text{β}}_1),({\Phi }_2,{\text{β}}_2),\dots ,({\Phi }_s,{\text{β}}_s)\end{array}\right].\end{array}$(14)

Riemaan-Liouville Integral: The standard definition of the Riemaan-Liouville (RL) integral as given by Kilbas et al. [26]: (15) ${}_{\alpha }{\mathcal{L}}_w^u\varphi \left(w\right)=\frac{1}{\mathrm{\Gamma }\left(u\right)}{\int }_{\alpha }^w(w-\xi {)}^{u-1}\varphi (\xi )\mathrm{d}\xi ,(\mathrm{\Re }\left(u\right)>0,w>\alpha ).$(15) Riemaan-Liouville Derivative: The standard definition of the RL fractional derivative as given by Kilbas et al. [26]: (16) $\left({}_{\alpha }{\mathcal{Q}}_w^u\varphi \right)\left(w\right)={\left(\frac{\mathrm{d}}{\mathrm{d}w}\right)}^v\left({}_{\alpha }{\mathcal{L}}^{v-u}\varphi \right)\left(w\right),\left(\mathrm{\Re }\right(u)>0,v=[\mathrm{\Re }\left(u\right)+1\left]\right),$(16) where $\left[\mathcal{Y}\right]$ is the greatest integer in the real number $\mathcal{Y}$.

Caputo Fractional Derivative: The standard definition of the Caputo fractional derivative as given by Kilbas et al. [26]: (17) $\begin{array}{rl}& {}_{\alpha }^c{\mathcal{Q}}_w^u\varphi \left(w\right)={}_{\alpha }{\mathcal{L}}_w^{v-u}{\mathcal{Q}}_w^v\varphi \left(w\right)\\ & =\frac{1}{\mathrm{\Gamma }(v-u)}{\int }_{\alpha }^w(w-\xi {)}^{v-u-1}{\varphi }^v(\xi )\mathrm{d}\xi ,(v-1<u\le v,v\in \mathit{N}).\end{array}$(17) Hilfer Fractional Derivative: The standard definition of the Hilfer fractional derivative (HFD) is given as follows by Hilfer [27]: (18) ${}_{\alpha }{\mathcal{Q}}^{u,\vartheta }\varphi \left(w\right)=\left({}_{\alpha }{\mathcal{L}}^{\vartheta (1-u)}\frac{\mathrm{d}}{\mathrm{d}w}\left({}_{\alpha }{\mathcal{L}}^{(1-\vartheta )(1-u)}\varphi \right)\right)\left(w\right).$(18) Particular if $\vartheta =0$, then Equation (18(18) ${}_{\alpha }{\mathcal{Q}}^{u,\vartheta }\varphi \left(w\right)=\left({}_{\alpha }{\mathcal{L}}^{\vartheta (1-u)}\frac{\mathrm{d}}{\mathrm{d}w}\left({}_{\alpha }{\mathcal{L}}^{(1-\vartheta )(1-u)}\varphi \right)\right)\left(w\right).$(18) ) reduced to RL operator, if $\vartheta =1$, then Equation (18(18) ${}_{\alpha }{\mathcal{Q}}^{u,\vartheta }\varphi \left(w\right)=\left({}_{\alpha }{\mathcal{L}}^{\vartheta (1-u)}\frac{\mathrm{d}}{\mathrm{d}w}\left({}_{\alpha }{\mathcal{L}}^{(1-\vartheta )(1-u)}\varphi \right)\right)\left(w\right).$(18) ) reduced to Caputo derivative.

GCFD: Mridula et al. [28] defined a generalized composite fractional derivative (GCFD) in 2014 using fractional calculus, which includes the image of the power function, the Laplace transform, and the composition of the Riemann Liouville fractional integral with the generalized composite fractional derivative. The following is the definition of the GCFD: (19) $\begin{array}{rl}& \left({}_{\alpha }{\mathcal{Q}}_w^{\mathrm{c},\mathrm{d};\vartheta }\varphi \right)\left(w\right)=\left({}_{\alpha }{\mathcal{L}}_w^{\vartheta (v-\mathrm{d})}{\mathcal{Q}}_w^v\left({}_{\alpha }{\mathcal{L}}_w^{(1-\vartheta )(v-\mathrm{c})}\varphi \right)\right)\left(w\right),\\ & (v-1<\mathrm{c},\mathrm{d}\le v,0\le \vartheta \le 1,v\in \mathit{N}).\end{array}$(19) Equation (19(19) $\begin{array}{rl}& \left({}_{\alpha }{\mathcal{Q}}_w^{\mathrm{c},\mathrm{d};\vartheta }\varphi \right)\left(w\right)=\left({}_{\alpha }{\mathcal{L}}_w^{\vartheta (v-\mathrm{d})}{\mathcal{Q}}_w^v\left({}_{\alpha }{\mathcal{L}}_w^{(1-\vartheta )(v-\mathrm{c})}\varphi \right)\right)\left(w\right),\\ & (v-1<\mathrm{c},\mathrm{d}\le v,0\le \vartheta \le 1,v\in \mathit{N}).\end{array}$(19) ) reduces to the RL operator and Caputo derivative, respectively, if $\vartheta =0$ and $\vartheta =1$. GCFD reduces to the HFD [27], in the case that c=d.

Power Function: For the RL operator (15(15) ${}_{\alpha }{\mathcal{L}}_w^u\varphi \left(w\right)=\frac{1}{\mathrm{\Gamma }\left(u\right)}{\int }_{\alpha }^w(w-\xi {)}^{u-1}\varphi (\xi )\mathrm{d}\xi ,(\mathrm{\Re }\left(u\right)>0,w>\alpha ).$(15) ), the power function formula is as follows [29] : (20) ${}_{\alpha }{\mathcal{I}}_w^u(w-a{)}^{\text{£}-1}=\frac{\mathrm{\Gamma }\left(\pounds \right)}{\mathrm{\Gamma }(\text{£}+u)}(w-\alpha {)}^{u+\text{£}-1},\left(\mathrm{\Re }\right(u)>0,\text{£}>0).$(20)

Theorem 2.4

If $\text{£}>0,v-1<\mathrm{c},\mathrm{d}\le v,v\in \mathit{N},0\le \vartheta \le 1$, then for the GCFD, the power function is described as follows [28]: (21) ${}_0{\mathcal{D}}_w^{\mathrm{c},\mathrm{d};\vartheta }(w-\alpha {)}^{\text{£}-1}=\frac{\mathrm{\Gamma }\left(\pounds \right)}{\mathrm{\Gamma }\left(\vartheta \right(\mathrm{c}-\mathrm{d})+\text{£}-\mathrm{c})}(w-\alpha {)}^{\vartheta (\mathrm{c}-\mathrm{d})-\mathrm{c}-1+\text{£}}.$(21)

3. Fractional integral operator involving IIF

We introduced an integral operator in this part, which includes the IIF in its kernel as follows: (22) $\begin{array}{rl}& \left({\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }\varphi \right)\left(\mathcal{V}\right)={\int }_{\alpha }^{\mathcal{V}}(\mathcal{V}-t{)}^{\kappa -1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}[\mathcal{V}(\mathcal{V}-t{)}^{\tau }]\varphi \left(t\right)\mathrm{d}t,\\ & \left(\mathrm{\Re }\right(\kappa )>0,w\in {\mathbb{C}}^{\ast }=\mathbb{C}-0,\mathrm{\Re }(\tau )>0,\mathrm{\Re }(\kappa )+\tau {\displaystyle \underset{1\le i\le u}{min}\mathrm{\Re }\left(\frac{{\Phi }_i}{B_i}\right)>0}).\end{array}$(22) By specializing in the IIF's parameters, one can derive several known integral operators.

Theorem 3.1

Let ϕ be a Lebesgue measurable function in the space $\mathcal{L}(\mathtt{a},\mathtt{b})$ on a finite interval $[\mathtt{a},\mathtt{b}](\mathtt{b}>\mathtt{a})$ of the real line $\mathbb{R}$ defined by the following equation, subject to the various parametric restrictions. (23) $\mathcal{L}(\mathtt{a},\mathtt{b})=\{\varphi :\Vert \varphi {\Vert }_1={\int }_{\mathtt{a}}^{\mathtt{b}}|\varphi (y)|\mathrm{d}y<\mathrm{\infty }\},$(23) then the integral operator ${\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }\varphi$ is bounded on $\mathcal{L}(\mathtt{a},\mathtt{b})$ and (24) $\Vert {\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }\varphi {\Vert }_1\leqq \mathbb{M}\cdot \Vert \varphi {\Vert }_1(0<\mathbb{M}<\mathrm{\infty }),$(24) here $\mathbb{M}$ is a constant that is defined by (25) $\mathbb{M}=\frac{1}{2\pi i}{\int }_{\mathrm{\$}}\varphi (s,Y){\mathcal{V}}^s\frac{(\mathtt{b}-t{)}^{\kappa +\tau s}}{(\kappa +\tau s)}\mathrm{d}s(0<\mathbb{M}<\mathrm{\infty }).$(25)

Proof.

If is sufficient to prove that (26) $\Vert {\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }\varphi {\Vert }_1={\int }_{\mathtt{a}}^{\mathtt{b}}\left|{\int }_{\alpha }^{\mathcal{V}}\right(\mathcal{V}-t{)}^{\kappa -1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}\right(\mathcal{V}-t{)}^{\tau }\left]\varphi \right(t\left)\mathrm{d}t\right|\mathrm{d}\mathcal{V}.$(26) With definition (7(7) $\begin{array}{rl}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left(\mathcal{V}\right)& ={}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),({\Psi }_2,{\zeta }_2;A_2),\dots ,({\Psi }_r,{\zeta }_r;A_r)\\ ({\Phi }_1,{\text{β}}_1;B_1),({\Phi }_2,{\text{β}}_2;B_2),\dots ,({\Phi }_s,{\text{β}}_s;B_s)\end{array}\right]\\ & =\frac{1}{2\pi i}{\int }_{\mathrm{\$}}\varphi (w,Y){\mathcal{V}}^w\mathrm{d}w,\end{array}$(7) ) of IIF, we apply definitions (22(22) $\begin{array}{rl}& \left({\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }\varphi \right)\left(\mathcal{V}\right)={\int }_{\alpha }^{\mathcal{V}}(\mathcal{V}-t{)}^{\kappa -1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}[\mathcal{V}(\mathcal{V}-t{)}^{\tau }]\varphi \left(t\right)\mathrm{d}t,\\ & \left(\mathrm{\Re }\right(\kappa )>0,w\in {\mathbb{C}}^{\ast }=\mathbb{C}-0,\mathrm{\Re }(\tau )>0,\mathrm{\Re }(\kappa )+\tau {\displaystyle \underset{1\le i\le u}{min}\mathrm{\Re }\left(\frac{{\Phi }_i}{B_i}\right)>0}).\end{array}$(22) ) and (23(23) $\mathcal{L}(\mathtt{a},\mathtt{b})=\{\varphi :\Vert \varphi {\Vert }_1={\int }_{\mathtt{a}}^{\mathtt{b}}|\varphi (y)|\mathrm{d}y<\mathrm{\infty }\},$(23) ).

After that, we obtained the following by rearranging the integration using the Dirichlet formula [30], (27) $\begin{array}{rl}& \Vert {\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }\varphi {\Vert }_1\le {\int }_{\mathtt{a}}^{\mathtt{b}}|\varphi \left(t\right)|\left({\int }_t^{\mathtt{b}}\right(\mathcal{V}-t{)}^{\kappa -1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}\right(\mathcal{V}-t{)}^{\tau }\left]\mathrm{d}\mathcal{V}\right)\mathrm{d}t\\ & ={\int }_{\mathtt{a}}^{\mathtt{b}}|\varphi \left(t\right)|\cdot \left|\left({\int }_0^{\mathtt{b}-t}{W}^{\kappa -1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\right[\mathcal{V}{W}^{\tau }\left]\mathrm{d}W\right)\right|\mathrm{d}t\\ & \leqq {\int }_{\mathtt{a}}^{\mathtt{b}}|\varphi \left(t\right)|\cdot \left({\int }_0^{\mathtt{b}-t}{W}^{\mathrm{\Re }\left(\kappa \right)-1}\left|{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\right[\mathcal{V}{W}^{\tau }\left]\right|\mathrm{d}W\right)\mathrm{d}t\\ & \leqq \Vert \varphi {\Vert }_1\cdot \left(\frac{1}{2\pi i}{\int }_{\mathrm{\$}}\varphi \right(s,Y\left){\mathcal{V}}^s\frac{(\mathtt{b}-t{)}^{\kappa +\tau s}}{(\kappa +\tau s)}\mathrm{d}s\right)\\ & =\mathbb{M}\cdot \Vert \varphi {\Vert }_1,\end{array}$(27) here constant $\mathbb{M}$ is defined by Equation (25(25) $\mathbb{M}=\frac{1}{2\pi i}{\int }_{\mathrm{\$}}\varphi (s,Y){\mathcal{V}}^s\frac{(\mathtt{b}-t{)}^{\kappa +\tau s}}{(\kappa +\tau s)}\mathrm{d}s(0<\mathbb{M}<\mathrm{\infty }).$(25) ).

The boundedness property of the integral operator ${\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }$, which is stated by Theorem 3.1 fully proves it.

Theorem 3.2

Let $\mathcal{V}>\alpha (\alpha \in [0,\mathrm{\infty })),u-1<\mathrm{c},\mathrm{d}\le u,0\le \vartheta \le 1,u\in N$. Also suppose that $\mathrm{\Lambda },\nu ,\mathcal{V}\in \mathbb{C},\mathrm{\Re }\left(\mathrm{\Lambda }\right)>0$. Then (28) $\begin{array}{rl}& \left({}_{\alpha }{\mathcal{Q}}_{\mathcal{V}}^{\mathrm{c},\mathrm{d};\vartheta }\left[\right(\mathcal{G}-\alpha {)}^{\mathrm{\Lambda }-1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}\right(\mathcal{G}-\alpha {)}^{\nu }\left]\right]\right)\left(\mathcal{V}\right)=(\mathcal{V}-\alpha {)}^{\mathrm{\Lambda }+\vartheta (\mathrm{c}-\mathrm{d})-\mathrm{c}-1}\\ & {}^{\mathrm{\Gamma }}{I}_{r+1,s+1}^{u,v+1}\left[\mathcal{V}\right(\mathcal{V}-\alpha {)}^{\nu }|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),(1-\mathrm{\Lambda },\nu ;1),({\Psi }_i,{\zeta }_i;A_i{)}_{2,r}\\ ({\Phi }_i,{\text{β}}_i;B_i{)}_{1,s},(1-\mathrm{\Lambda }-\vartheta (\mathrm{c}-\mathrm{d})+\mathrm{c},\nu ;1)\end{array}],\end{array}$(28) provided that every term of the claim (28(28) $\begin{array}{rl}& \left({}_{\alpha }{\mathcal{Q}}_{\mathcal{V}}^{\mathrm{c},\mathrm{d};\vartheta }\left[\right(\mathcal{G}-\alpha {)}^{\mathrm{\Lambda }-1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}\right(\mathcal{G}-\alpha {)}^{\nu }\left]\right]\right)\left(\mathcal{V}\right)=(\mathcal{V}-\alpha {)}^{\mathrm{\Lambda }+\vartheta (\mathrm{c}-\mathrm{d})-\mathrm{c}-1}\\ & {}^{\mathrm{\Gamma }}{I}_{r+1,s+1}^{u,v+1}\left[\mathcal{V}\right(\mathcal{V}-\alpha {)}^{\nu }|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),(1-\mathrm{\Lambda },\nu ;1),({\Psi }_i,{\zeta }_i;A_i{)}_{2,r}\\ ({\Phi }_i,{\text{β}}_i;B_i{)}_{1,s},(1-\mathrm{\Lambda }-\vartheta (\mathrm{c}-\mathrm{d})+\mathrm{c},\nu ;1)\end{array}],\end{array}$(28) ) exists.

Proof.

To prove the assertion (28(28) $\begin{array}{rl}& \left({}_{\alpha }{\mathcal{Q}}_{\mathcal{V}}^{\mathrm{c},\mathrm{d};\vartheta }\left[\right(\mathcal{G}-\alpha {)}^{\mathrm{\Lambda }-1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}\right(\mathcal{G}-\alpha {)}^{\nu }\left]\right]\right)\left(\mathcal{V}\right)=(\mathcal{V}-\alpha {)}^{\mathrm{\Lambda }+\vartheta (\mathrm{c}-\mathrm{d})-\mathrm{c}-1}\\ & {}^{\mathrm{\Gamma }}{I}_{r+1,s+1}^{u,v+1}\left[\mathcal{V}\right(\mathcal{V}-\alpha {)}^{\nu }|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),(1-\mathrm{\Lambda },\nu ;1),({\Psi }_i,{\zeta }_i;A_i{)}_{2,r}\\ ({\Phi }_i,{\text{β}}_i;B_i{)}_{1,s},(1-\mathrm{\Lambda }-\vartheta (\mathrm{c}-\mathrm{d})+\mathrm{c},\nu ;1)\end{array}],\end{array}$(28) ), with the help of (17(17) $\begin{array}{rl}& {}_{\alpha }^c{\mathcal{Q}}_w^u\varphi \left(w\right)={}_{\alpha }{\mathcal{L}}_w^{v-u}{\mathcal{Q}}_w^v\varphi \left(w\right)\\ & =\frac{1}{\mathrm{\Gamma }(v-u)}{\int }_{\alpha }^w(w-\xi {)}^{v-u-1}{\varphi }^v(\xi )\mathrm{d}\xi ,(v-1<u\le v,v\in \mathit{N}).\end{array}$(17) ) and (19(19) $\begin{array}{rl}& \left({}_{\alpha }{\mathcal{Q}}_w^{\mathrm{c},\mathrm{d};\vartheta }\varphi \right)\left(w\right)=\left({}_{\alpha }{\mathcal{L}}_w^{\vartheta (v-\mathrm{d})}{\mathcal{Q}}_w^v\left({}_{\alpha }{\mathcal{L}}_w^{(1-\vartheta )(v-\mathrm{c})}\varphi \right)\right)\left(w\right),\\ & (v-1<\mathrm{c},\mathrm{d}\le v,0\le \vartheta \le 1,v\in \mathit{N}).\end{array}$(19) ), we obtain $\begin{array}{rl}& \left({}_{\alpha }{\mathcal{Q}}_{\mathcal{V}}^{\mathrm{c},\mathrm{d};\vartheta }\left[\right(\mathcal{G}-\alpha {)}^{\mathrm{\Lambda }-1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}\right(\mathcal{G}-\alpha {)}^{\nu }\left]\right]\right)\left(\mathcal{V}\right)\\ & ={}_{\alpha }{\mathcal{Q}}_{\mathcal{V}}^{\mathrm{c},\mathrm{d};\vartheta }\left[\right(\mathcal{G}-\alpha {)}^{\mathrm{\Lambda }-1}\frac{1}{2\pi \iota }{\int }_{\mathrm{\$}}\varphi (\omega ,Y)\left[\mathcal{V}\right(\mathcal{G}-\alpha {)}^{\nu }{]}^{\omega }\mathrm{d}\omega ]\left(\mathcal{V}\right)\\ & =\frac{1}{2\pi \iota }{\int }_{\mathrm{\$}}\varphi (\omega ,Y){\mathcal{V}}^{\omega }\left({}_{\alpha }{\mathcal{Q}}_{\mathcal{V}}^{\mathrm{c},\mathrm{d};\vartheta }\right[(\mathcal{G}-\alpha {)}^{\mathrm{\Lambda }+\nu \omega -1}])\left(\mathcal{V}\right)\mathrm{d}\omega \\ & =\frac{1}{2\pi \iota }{\int }_{\mathrm{\$}}\varphi (\omega ,Y){\mathcal{V}}^{\omega }\frac{\mathrm{\Gamma }(\mathrm{\Lambda }+\nu \omega )}{\mathrm{\Gamma }(\mathrm{\Lambda }+\vartheta (\mathrm{c}-\mathrm{d})-\mathrm{c}+\nu \omega )}(\mathcal{V}-\alpha {)}^{\mathrm{\Lambda }+\vartheta (\mathrm{c}-\mathrm{d})-\mathrm{c}+\nu \omega -1}\mathrm{d}\omega \\ & =(\mathcal{V}-\alpha {)}^{\mathrm{\Lambda }+\vartheta (\mathrm{c}-\mathrm{d})-\mathrm{c}-1}\frac{1}{2\pi \iota }{\int }_{\mathrm{\$}}\varphi (\omega ,Y\left)\right[\mathcal{V}(\mathcal{G}-\alpha {)}^{\nu }{]}^{\omega }\\ & \frac{\mathrm{\Gamma }(\mathrm{\Lambda }+\nu \omega )}{\mathrm{\Gamma }(\mathrm{\Lambda }+\vartheta (\mathrm{c}-\mathrm{d})-\mathrm{c}+\nu \omega )}\mathrm{d}\omega \\ & =(\mathcal{V}-\alpha {)}^{\mathrm{\Lambda }+\vartheta (\mathrm{c}-\mathrm{d})-\mathrm{c}-1}\\ & {}^{\mathrm{\Gamma }}{I}_{r+1,s+1}^{u,v+1}\left[\mathcal{V}\right(\mathcal{V}-\alpha {)}^{\nu }|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),(1-\mathrm{\Lambda },\nu ;1),({\Psi }_i,{\zeta }_i;A_i{)}_{2,r}\\ ({\Phi }_i,{\text{β}}_i;B_i{)}_{1,s},(1-\mathrm{\Lambda }-\vartheta (\mathrm{c}-\mathrm{d})+\mathrm{c},\nu ;1)\end{array}].\end{array}$ Hence the proof.

Corollary 3.3

Let $\mathcal{V}>\alpha (\alpha \in [0,\mathrm{\infty })),0<\mathrm{c}\le 1,0\le \vartheta \le 1.$ Also suppose that $\mathrm{\Lambda },\nu ,\mathcal{V}\in \mathbb{C},\mathrm{\Re }\left(\mathrm{\Lambda }\right)>0$. Then (29) $\begin{array}{rl}& \left({}_{\alpha }{\mathcal{Q}}_{\mathcal{V}}^{\mathrm{c}}\left[\right(\mathcal{G}-\alpha {)}^{\mathrm{\Lambda }-1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}\right(\mathcal{G}-\alpha {)}^{\nu }\left]\right]\right)\left(\mathcal{V}\right)=(\mathcal{V}-\alpha {)}^{\mathrm{\Lambda }-\mathrm{c}-1}\\ & {}^{\mathrm{\Gamma }}{I}_{r+1,s+1}^{u,v+1}\left[\mathcal{V}\right(\mathcal{V}-\alpha {)}^{\nu }|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),(1-\mathrm{\Lambda },\nu ;1),({\Psi }_i,{\zeta }_i;A_i{)}_{2,r}\\ ({\Phi }_i,{\text{β}}_i;B_i{)}_{1,s},(1-\mathrm{\Lambda }+\mathrm{c},\nu ;1)\end{array}]\end{array}$(29) provided that every term of the claim (29(29) $\begin{array}{rl}& \left({}_{\alpha }{\mathcal{Q}}_{\mathcal{V}}^{\mathrm{c}}\left[\right(\mathcal{G}-\alpha {)}^{\mathrm{\Lambda }-1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}\right(\mathcal{G}-\alpha {)}^{\nu }\left]\right]\right)\left(\mathcal{V}\right)=(\mathcal{V}-\alpha {)}^{\mathrm{\Lambda }-\mathrm{c}-1}\\ & {}^{\mathrm{\Gamma }}{I}_{r+1,s+1}^{u,v+1}\left[\mathcal{V}\right(\mathcal{V}-\alpha {)}^{\nu }|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),(1-\mathrm{\Lambda },\nu ;1),({\Psi }_i,{\zeta }_i;A_i{)}_{2,r}\\ ({\Phi }_i,{\text{β}}_i;B_i{)}_{1,s},(1-\mathrm{\Lambda }+\mathrm{c},\nu ;1)\end{array}]\end{array}$(29) ) exists.

Proof.

Taking $\vartheta =0$ in the Equation (28(28) $\begin{array}{rl}& \left({}_{\alpha }{\mathcal{Q}}_{\mathcal{V}}^{\mathrm{c},\mathrm{d};\vartheta }\left[\right(\mathcal{G}-\alpha {)}^{\mathrm{\Lambda }-1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}\right(\mathcal{G}-\alpha {)}^{\nu }\left]\right]\right)\left(\mathcal{V}\right)=(\mathcal{V}-\alpha {)}^{\mathrm{\Lambda }+\vartheta (\mathrm{c}-\mathrm{d})-\mathrm{c}-1}\\ & {}^{\mathrm{\Gamma }}{I}_{r+1,s+1}^{u,v+1}\left[\mathcal{V}\right(\mathcal{V}-\alpha {)}^{\nu }|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),(1-\mathrm{\Lambda },\nu ;1),({\Psi }_i,{\zeta }_i;A_i{)}_{2,r}\\ ({\Phi }_i,{\text{β}}_i;B_i{)}_{1,s},(1-\mathrm{\Lambda }-\vartheta (\mathrm{c}-\mathrm{d})+\mathrm{c},\nu ;1)\end{array}],\end{array}$(28) ), GCFD is reduced to the RL derivative, and we instantly obtain the desired outcome.

Corollary 3.4

Let $\mathcal{V}>\alpha (\alpha \in [0,\mathrm{\infty })),0<\mathrm{c}\le 1,0\le \vartheta \le 1.$ Also suppose that $\mathrm{\Lambda },\nu ,\mathcal{V}\in \mathbb{C},\mathrm{\Re }\left(\mathrm{\Lambda }\right)>0$. Then (30) $\begin{array}{rl}& \left({}_{\alpha }{\mathcal{Q}}_{\mathcal{V}}^{\mathrm{c}}\left[\right(\mathcal{G}-\alpha {)}^{\mathrm{\Lambda }-1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}\right(\mathcal{G}-\alpha {)}^{\nu }\left]\right]\right)\left(\mathcal{V}\right)=(\mathcal{V}-\alpha {)}^{\mathrm{\Lambda }-\mathrm{c}-1}\\ & {}^{\mathrm{\Gamma }}{I}_{r+1,s+1}^{u,v+1}\left[\mathcal{V}\right(\mathcal{V}-\alpha {)}^{\nu }|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),(1-\mathrm{\Lambda },\nu ;1),({\Psi }_i,{\zeta }_i;A_i{)}_{2,r}\\ ({\Phi }_i,{\text{β}}_i;B_i{)}_{1,s},(1-\mathrm{\Lambda }+\mathrm{c},\nu ;1)\end{array}]\end{array}$(30) provided that every term of the claim (30(30) $\begin{array}{rl}& \left({}_{\alpha }{\mathcal{Q}}_{\mathcal{V}}^{\mathrm{c}}\left[\right(\mathcal{G}-\alpha {)}^{\mathrm{\Lambda }-1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}\right(\mathcal{G}-\alpha {)}^{\nu }\left]\right]\right)\left(\mathcal{V}\right)=(\mathcal{V}-\alpha {)}^{\mathrm{\Lambda }-\mathrm{c}-1}\\ & {}^{\mathrm{\Gamma }}{I}_{r+1,s+1}^{u,v+1}\left[\mathcal{V}\right(\mathcal{V}-\alpha {)}^{\nu }|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),(1-\mathrm{\Lambda },\nu ;1),({\Psi }_i,{\zeta }_i;A_i{)}_{2,r}\\ ({\Phi }_i,{\text{β}}_i;B_i{)}_{1,s},(1-\mathrm{\Lambda }+\mathrm{c},\nu ;1)\end{array}]\end{array}$(30) ) exists.

Proof.

Taking $\vartheta =1$ in the Equation (28(28) $\begin{array}{rl}& \left({}_{\alpha }{\mathcal{Q}}_{\mathcal{V}}^{\mathrm{c},\mathrm{d};\vartheta }\left[\right(\mathcal{G}-\alpha {)}^{\mathrm{\Lambda }-1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}\right(\mathcal{G}-\alpha {)}^{\nu }\left]\right]\right)\left(\mathcal{V}\right)=(\mathcal{V}-\alpha {)}^{\mathrm{\Lambda }+\vartheta (\mathrm{c}-\mathrm{d})-\mathrm{c}-1}\\ & {}^{\mathrm{\Gamma }}{I}_{r+1,s+1}^{u,v+1}\left[\mathcal{V}\right(\mathcal{V}-\alpha {)}^{\nu }|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),(1-\mathrm{\Lambda },\nu ;1),({\Psi }_i,{\zeta }_i;A_i{)}_{2,r}\\ ({\Phi }_i,{\text{β}}_i;B_i{)}_{1,s},(1-\mathrm{\Lambda }-\vartheta (\mathrm{c}-\mathrm{d})+\mathrm{c},\nu ;1)\end{array}],\end{array}$(28) ), GCFD is reduced to the Caputo derivative, and we instantly obtain the desired outcome.

Corollary 3.5

Let $\mathcal{V}>\alpha (\alpha \in [0,\mathrm{\infty })),0<\mathrm{c}\le 1,0\le \vartheta \le 1.$ Also suppose that $\mathrm{\Lambda },\nu ,\mathcal{V}\in \mathbb{C},\mathrm{\Re }\left(\mathrm{\Lambda }\right)>0$. Then (31) $\begin{array}{rl}& \left({}_{\alpha }{\mathcal{Q}}_{\mathcal{V}}^{\mathrm{c},\vartheta }\left[\right(\mathcal{G}-\alpha {)}^{\mathrm{\Lambda }-1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}\right(\mathcal{G}-\alpha {)}^{\nu }\left]\right]\right)\left(\mathcal{V}\right)=(\mathcal{V}-\alpha {)}^{\mathrm{\Lambda }-\mathrm{c}-1}\\ & {}^{\mathrm{\Gamma }}{I}_{r+1,s+1}^{u,v+1}\left[\mathcal{V}\right(\mathcal{V}-\alpha {)}^{\nu }|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),(1-\mathrm{\Lambda },\nu ;1),({\Psi }_i,{\zeta }_i;A_i{)}_{2,r}\\ ({\Phi }_i,{\text{β}}_i;B_i{)}_{1,s},(1-\mathrm{\Lambda }+\mathrm{c},\nu ;1)\end{array}]\end{array}$(31) provided that every term of the claim (31(31) $\begin{array}{rl}& \left({}_{\alpha }{\mathcal{Q}}_{\mathcal{V}}^{\mathrm{c},\vartheta }\left[\right(\mathcal{G}-\alpha {)}^{\mathrm{\Lambda }-1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}\right(\mathcal{G}-\alpha {)}^{\nu }\left]\right]\right)\left(\mathcal{V}\right)=(\mathcal{V}-\alpha {)}^{\mathrm{\Lambda }-\mathrm{c}-1}\\ & {}^{\mathrm{\Gamma }}{I}_{r+1,s+1}^{u,v+1}\left[\mathcal{V}\right(\mathcal{V}-\alpha {)}^{\nu }|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),(1-\mathrm{\Lambda },\nu ;1),({\Psi }_i,{\zeta }_i;A_i{)}_{2,r}\\ ({\Phi }_i,{\text{β}}_i;B_i{)}_{1,s},(1-\mathrm{\Lambda }+\mathrm{c},\nu ;1)\end{array}]\end{array}$(31) ) exists.

Proof.

Taking $\mathrm{c}=\mathrm{d}$ in the Equation (28(28) $\begin{array}{rl}& \left({}_{\alpha }{\mathcal{Q}}_{\mathcal{V}}^{\mathrm{c},\mathrm{d};\vartheta }\left[\right(\mathcal{G}-\alpha {)}^{\mathrm{\Lambda }-1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}\right(\mathcal{G}-\alpha {)}^{\nu }\left]\right]\right)\left(\mathcal{V}\right)=(\mathcal{V}-\alpha {)}^{\mathrm{\Lambda }+\vartheta (\mathrm{c}-\mathrm{d})-\mathrm{c}-1}\\ & {}^{\mathrm{\Gamma }}{I}_{r+1,s+1}^{u,v+1}\left[\mathcal{V}\right(\mathcal{V}-\alpha {)}^{\nu }|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),(1-\mathrm{\Lambda },\nu ;1),({\Psi }_i,{\zeta }_i;A_i{)}_{2,r}\\ ({\Phi }_i,{\text{β}}_i;B_i{)}_{1,s},(1-\mathrm{\Lambda }-\vartheta (\mathrm{c}-\mathrm{d})+\mathrm{c},\nu ;1)\end{array}],\end{array}$(28) ), GCFD is reduced to the HFD, and we instantly obtain the desired outcome.

Theorem 3.6

Following composition relations hold for some Lebesgue measurable function $\zeta \in \mathrm{L}(\mathrm{a},\mathrm{b})$, assuming the conditions stated in the definition (22(22) $\begin{array}{rl}& \left({\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }\varphi \right)\left(\mathcal{V}\right)={\int }_{\alpha }^{\mathcal{V}}(\mathcal{V}-t{)}^{\kappa -1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}[\mathcal{V}(\mathcal{V}-t{)}^{\tau }]\varphi \left(t\right)\mathrm{d}t,\\ & \left(\mathrm{\Re }\right(\kappa )>0,w\in {\mathbb{C}}^{\ast }=\mathbb{C}-0,\mathrm{\Re }(\tau )>0,\mathrm{\Re }(\kappa )+\tau {\displaystyle \underset{1\le i\le u}{min}\mathrm{\Re }\left(\frac{{\Phi }_i}{B_i}\right)>0}).\end{array}$(22) ): (32) ${}_{\alpha +}{\mathcal{L}}^n{\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }\zeta ={\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }{}_{\alpha +}{\mathcal{L}}^n\zeta ,$(32) and (33) ${}_{\alpha +}{\mathcal{Q}}^n{\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }\zeta ={\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }{}_{\alpha +}{\mathcal{Q}}^n\zeta .$(33)

Proof.

The left hand side of (32(32) ${}_{\alpha +}{\mathcal{L}}^n{\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }\zeta ={\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }{}_{\alpha +}{\mathcal{L}}^n\zeta ,$(32) ) is temporarily represented as θ and thereafter, making use of (15(15) ${}_{\alpha }{\mathcal{L}}_w^u\varphi \left(w\right)=\frac{1}{\mathrm{\Gamma }\left(u\right)}{\int }_{\alpha }^w(w-\xi {)}^{u-1}\varphi (\xi )\mathrm{d}\xi ,(\mathrm{\Re }\left(u\right)>0,w>\alpha ).$(15) ) and (22(22) $\begin{array}{rl}& \left({\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }\varphi \right)\left(\mathcal{V}\right)={\int }_{\alpha }^{\mathcal{V}}(\mathcal{V}-t{)}^{\kappa -1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}[\mathcal{V}(\mathcal{V}-t{)}^{\tau }]\varphi \left(t\right)\mathrm{d}t,\\ & \left(\mathrm{\Re }\right(\kappa )>0,w\in {\mathbb{C}}^{\ast }=\mathbb{C}-0,\mathrm{\Re }(\tau )>0,\mathrm{\Re }(\kappa )+\tau {\displaystyle \underset{1\le i\le u}{min}\mathrm{\Re }\left(\frac{{\Phi }_i}{B_i}\right)>0}).\end{array}$(22) ), following is obtained: (34) $\theta =\frac{1}{\mathrm{\Gamma }\left(n\right)}{\int }_{\alpha }^{\mathcal{V}}(\mathcal{V}-g{)}^{n-1}{\int }_{\alpha }^g(g-\mathcal{G}{)}^{\kappa -1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}\right(g-\mathcal{G}{)}^{\tau }\left]\zeta \right(\mathcal{G})\mathrm{d}\mathcal{G}\mathrm{d}g.$(34) Further, changes in the order of the integration (allowed for the conditions already stated). Hence the following is obtained easily: (35) $\theta =\frac{1}{\mathrm{\Gamma }\left(n\right)}{\int }_{\alpha }^{\mathcal{V}}\left[{\int }_{\mathcal{G}}^{\mathcal{V}}\right(\mathcal{V}-g{)}^{n-1}(g-\mathcal{G}{)}^{\kappa -1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}[\mathcal{V}(g-\mathcal{G}{)}^{\tau }]\mathrm{d}g]\zeta \left(\mathcal{G}\right)\mathrm{d}\mathcal{G}.$(35) Next, following equation is obtained by substituting $g-\mathcal{G}=\xi$ in expression (35(35) $\theta =\frac{1}{\mathrm{\Gamma }\left(n\right)}{\int }_{\alpha }^{\mathcal{V}}\left[{\int }_{\mathcal{G}}^{\mathcal{V}}\right(\mathcal{V}-g{)}^{n-1}(g-\mathcal{G}{)}^{\kappa -1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}[\mathcal{V}(g-\mathcal{G}{)}^{\tau }]\mathrm{d}g]\zeta \left(\mathcal{G}\right)\mathrm{d}\mathcal{G}.$(35) ), (36) $\theta =\frac{1}{\mathrm{\Gamma }\left(n\right)}{\int }_{\alpha }^{\mathcal{V}}\left[{\int }_0^{\mathcal{V}-\mathcal{G}}\right(\mathcal{V}-\mathcal{G}-\xi {)}^{n-1}{\xi }^{\kappa -1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}{\xi }^{\tau }\right]\mathrm{d}\xi ]\zeta \left(\mathcal{G}\right)\mathrm{d}\mathcal{G}.$(36) Using RL integral and after a slight adjustment in the right-hand side of (36(36) $\theta =\frac{1}{\mathrm{\Gamma }\left(n\right)}{\int }_{\alpha }^{\mathcal{V}}\left[{\int }_0^{\mathcal{V}-\mathcal{G}}\right(\mathcal{V}-\mathcal{G}-\xi {)}^{n-1}{\xi }^{\kappa -1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}{\xi }^{\tau }\right]\mathrm{d}\xi ]\zeta \left(\mathcal{G}\right)\mathrm{d}\mathcal{G}.$(36) ), then we get (37) $\begin{array}{rl}& \theta ={\int }_{\alpha }^{\mathcal{V}}(\mathcal{V}-\mathcal{G}{)}^{\kappa +n-1}\\ & {}^{\mathrm{\Gamma }}{I}_{r+1,s+1}^{u,v+1}\left[\mathcal{V}\right(\mathcal{V}-\mathcal{G}{)}^{\tau }|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),(1-\kappa ,\tau ;1),({\Psi }_i,{\zeta }_i;A_i{)}_{2,r}\\ ({\Phi }_i,{\text{β}}_i;B_i{)}_{1,s},(1-\kappa -n,\tau ;1)\end{array}]\zeta \left(\mathcal{G}\right)\mathrm{d}\mathcal{G}.\end{array}$(37) For simplicity, the right-hand side of (32(32) ${}_{\alpha +}{\mathcal{L}}^n{\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }\zeta ={\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }{}_{\alpha +}{\mathcal{L}}^n\zeta ,$(32) ) is temporarily denoted with ℧. Thereafter, making use of the definitions (15(15) ${}_{\alpha }{\mathcal{L}}_w^u\varphi \left(w\right)=\frac{1}{\mathrm{\Gamma }\left(u\right)}{\int }_{\alpha }^w(w-\xi {)}^{u-1}\varphi (\xi )\mathrm{d}\xi ,(\mathrm{\Re }\left(u\right)>0,w>\alpha ).$(15) ) and (22(22) $\begin{array}{rl}& \left({\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }\varphi \right)\left(\mathcal{V}\right)={\int }_{\alpha }^{\mathcal{V}}(\mathcal{V}-t{)}^{\kappa -1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}[\mathcal{V}(\mathcal{V}-t{)}^{\tau }]\varphi \left(t\right)\mathrm{d}t,\\ & \left(\mathrm{\Re }\right(\kappa )>0,w\in {\mathbb{C}}^{\ast }=\mathbb{C}-0,\mathrm{\Re }(\tau )>0,\mathrm{\Re }(\kappa )+\tau {\displaystyle \underset{1\le i\le u}{min}\mathrm{\Re }\left(\frac{{\Phi }_i}{B_i}\right)>0}).\end{array}$(22) ), is given as below: (38) $\text{℧}={\int }_{\alpha }^z(z-\mathcal{G}{)}^{\kappa -1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}[\mathcal{V}(z-\mathcal{G}{)}^{\tau }]\frac{1}{\mathrm{\Gamma }\left(n\right)}{\int }_{\alpha }^{\mathcal{G}}(\mathcal{G}-\mathcal{V}{)}^{n-1}\zeta (\mathcal{V})\mathrm{d}\mathcal{V}\mathrm{d}\mathcal{G}.$(38) Further, changes in the order of the integration (allowed for the conditions already stated) and substituting $z-\mathcal{G}=\xi$ in expression (38(38) $\text{℧}={\int }_{\alpha }^z(z-\mathcal{G}{)}^{\kappa -1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}[\mathcal{V}(z-\mathcal{G}{)}^{\tau }]\frac{1}{\mathrm{\Gamma }\left(n\right)}{\int }_{\alpha }^{\mathcal{G}}(\mathcal{G}-\mathcal{V}{)}^{n-1}\zeta (\mathcal{V})\mathrm{d}\mathcal{V}\mathrm{d}\mathcal{G}.$(38) ), Hence the following is obtained easily: (39) $\text{℧}=\frac{1}{\mathrm{\Gamma }\left(n\right)}{\int }_{\alpha }^z\left[{\int }_0^{z-\mathcal{V}}\right(z-\xi -\mathcal{V}{)}^{n-1}{\xi }^{\kappa -1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}{\xi }^{\tau }\right]\mathrm{d}\xi ]\zeta \left(\mathcal{V}\right)\mathrm{d}\mathcal{V}.$(39) Finally, with the help RL integral and after a slight adjustment in the right-hand side of (39(39) $\text{℧}=\frac{1}{\mathrm{\Gamma }\left(n\right)}{\int }_{\alpha }^z\left[{\int }_0^{z-\mathcal{V}}\right(z-\xi -\mathcal{V}{)}^{n-1}{\xi }^{\kappa -1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}\left[\mathcal{V}{\xi }^{\tau }\right]\mathrm{d}\xi ]\zeta \left(\mathcal{V}\right)\mathrm{d}\mathcal{V}.$(39) ), an equation familiar with expression (37(37) $\begin{array}{rl}& \theta ={\int }_{\alpha }^{\mathcal{V}}(\mathcal{V}-\mathcal{G}{)}^{\kappa +n-1}\\ & {}^{\mathrm{\Gamma }}{I}_{r+1,s+1}^{u,v+1}\left[\mathcal{V}\right(\mathcal{V}-\mathcal{G}{)}^{\tau }|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),(1-\kappa ,\tau ;1),({\Psi }_i,{\zeta }_i;A_i{)}_{2,r}\\ ({\Phi }_i,{\text{β}}_i;B_i{)}_{1,s},(1-\kappa -n,\tau ;1)\end{array}]\zeta \left(\mathcal{G}\right)\mathrm{d}\mathcal{G}.\end{array}$(37) ) is obtained. It implies the following: (40) $\begin{array}{rl}& {}_{\alpha +}{\mathcal{L}}^n\left({\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }\zeta \right)={\int }_{\alpha }^z(z-\mathcal{V}{)}^{\kappa +n-1}\\ & {}^{\mathrm{\Gamma }}{I}_{r+1,s+1}^{u,v+1}\left[\mathcal{V}\right(z-\mathcal{V}{)}^{\tau }|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),(1-\kappa ,\tau ;1),({\Psi }_i,{\zeta }_i;A_i{)}_{2,r}\\ ({\Phi }_i,{\text{β}}_i;B_i{)}_{1,s},(1-\kappa -n,\tau ;1)\end{array}]\zeta \left(\mathcal{V}\right)\mathrm{d}\mathcal{V}\\ & ={\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }\left({}_{\alpha +}{\mathcal{L}}^n\zeta \right).\end{array}$(40) This completes the proof of (32(32) ${}_{\alpha +}{\mathcal{L}}^n{\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }\zeta ={\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }{}_{\alpha +}{\mathcal{L}}^n\zeta ,$(32) ). For the proof of (33(33) ${}_{\alpha +}{\mathcal{Q}}^n{\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }\zeta ={\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }{}_{\alpha +}{\mathcal{Q}}^n\zeta .$(33) ), one can refer to the already given proof of (32(32) ${}_{\alpha +}{\mathcal{L}}^n{\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }\zeta ={\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }{}_{\alpha +}{\mathcal{L}}^n\zeta ,$(32) ), and that's why the details are not covered in this manuscript.

Theorem 3.7

For any Lebesgue measurable function $\zeta \in \mathrm{L}(\mathrm{a},\mathrm{b})$ in the constraints given in definition (22(22) $\begin{array}{rl}& \left({\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }\varphi \right)\left(\mathcal{V}\right)={\int }_{\alpha }^{\mathcal{V}}(\mathcal{V}-t{)}^{\kappa -1}{}^{\mathrm{\Gamma }}{I}_{r,s}^{u,v}[\mathcal{V}(\mathcal{V}-t{)}^{\tau }]\varphi \left(t\right)\mathrm{d}t,\\ & \left(\mathrm{\Re }\right(\kappa )>0,w\in {\mathbb{C}}^{\ast }=\mathbb{C}-0,\mathrm{\Re }(\tau )>0,\mathrm{\Re }(\kappa )+\tau {\displaystyle \underset{1\le i\le u}{min}\mathrm{\Re }\left(\frac{{\Phi }_i}{B_i}\right)>0}).\end{array}$(22) ), each of the following composition relationships: (41) ${}_{\alpha +}{\mathcal{Q}}^{\mathrm{c},\mathrm{d}}\left({\mathfrak{E}}_{\alpha +;r,s;\kappa }^{w;u,v;\tau }\zeta \right)={}_{\alpha +}{\mathcal{L}}^{(1-\mathrm{c})(1-\mathrm{d})+\mathrm{d}(1-\mathrm{c})-1}\left({\mathfrak{E}}_{\alpha +;r,s;\kappa }^{w;u,v;\tau }\zeta \right),(0<\mathrm{c}<1,0\le \mathrm{d}\le 1)$(41) holds true.

Proof.

Initially applying definition (18(18) ${}_{\alpha }{\mathcal{Q}}^{u,\vartheta }\varphi \left(w\right)=\left({}_{\alpha }{\mathcal{L}}^{\vartheta (1-u)}\frac{\mathrm{d}}{\mathrm{d}w}\left({}_{\alpha }{\mathcal{L}}^{(1-\vartheta )(1-u)}\varphi \right)\right)\left(w\right).$(18) ) in the left hand side of (41(41) ${}_{\alpha +}{\mathcal{Q}}^{\mathrm{c},\mathrm{d}}\left({\mathfrak{E}}_{\alpha +;r,s;\kappa }^{w;u,v;\tau }\zeta \right)={}_{\alpha +}{\mathcal{L}}^{(1-\mathrm{c})(1-\mathrm{d})+\mathrm{d}(1-\mathrm{c})-1}\left({\mathfrak{E}}_{\alpha +;r,s;\kappa }^{w;u,v;\tau }\zeta \right),(0<\mathrm{c}<1,0\le \mathrm{d}\le 1)$(41) ), following is obtained: (42) ${}_{\alpha +}{\mathcal{Q}}^{\mathrm{c},\mathrm{d}}\left({\mathfrak{E}}_{\alpha +;r,s;\kappa }^{w;u,v;\tau }\zeta \right)={}_{\alpha +}{\mathcal{L}}^{\mathrm{d}(1-\mathrm{c})}\frac{\mathrm{d}}{\mathrm{d}\mathcal{V}}\left({}_{\alpha +}{\mathcal{L}}^{(1-\mathrm{c})(1-\mathrm{d})}{\mathfrak{E}}_{\alpha +;r,s;\kappa }^{w;u,v;\tau }\zeta \right).$(42) The equation obtained after applying assertion (40(40) $\begin{array}{rl}& {}_{\alpha +}{\mathcal{L}}^n\left({\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }\zeta \right)={\int }_{\alpha }^z(z-\mathcal{V}{)}^{\kappa +n-1}\\ & {}^{\mathrm{\Gamma }}{I}_{r+1,s+1}^{u,v+1}\left[\mathcal{V}\right(z-\mathcal{V}{)}^{\tau }|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),(1-\kappa ,\tau ;1),({\Psi }_i,{\zeta }_i;A_i{)}_{2,r}\\ ({\Phi }_i,{\text{β}}_i;B_i{)}_{1,s},(1-\kappa -n,\tau ;1)\end{array}]\zeta \left(\mathcal{V}\right)\mathrm{d}\mathcal{V}\\ & ={\mathfrak{E}}_{{\alpha }^{+};r,s;\kappa }^{w;u,v;\tau }\left({}_{\alpha +}{\mathcal{L}}^n\zeta \right).\end{array}$(40) ) is as follows: (43) $\begin{array}{rl}& \left({}_{\alpha +}{\mathcal{L}}^{(1-\mathrm{c})(1-\mathrm{d})}{\mathfrak{E}}_{\alpha +;r,s;\kappa }^{w;u,v;\tau }\zeta \right)\left(\mathcal{V}\right)={\int }_{\alpha }^{\mathcal{V}}(\mathcal{V}-\mathcal{G}{)}^{\kappa +(1-\mathrm{c})(1-\mathrm{d})-1}\\ & {}^{\mathrm{\Gamma }}{I}_{r+1,s+1}^{u,v+1}\left[\mathcal{V}\right(\mathcal{V}-\mathcal{G}{)}^{\tau }|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),(1-\kappa ,\tau ;1),({\Psi }_i,{\zeta }_i;A_i{)}_{2,r}\\ ({\Phi }_i,{\text{β}}_i;B_i{)}_{1,s},(1-\kappa -(1-\mathrm{c})(1-\mathrm{d}),\tau ;1)\end{array}]\zeta \left(\mathcal{G}\right)\mathrm{d}\mathcal{G}.\end{array}$(43) Further obtain the first derivative w.r.t $\mathcal{V}$, of every term in the Equation (43(43) $\begin{array}{rl}& \left({}_{\alpha +}{\mathcal{L}}^{(1-\mathrm{c})(1-\mathrm{d})}{\mathfrak{E}}_{\alpha +;r,s;\kappa }^{w;u,v;\tau }\zeta \right)\left(\mathcal{V}\right)={\int }_{\alpha }^{\mathcal{V}}(\mathcal{V}-\mathcal{G}{)}^{\kappa +(1-\mathrm{c})(1-\mathrm{d})-1}\\ & {}^{\mathrm{\Gamma }}{I}_{r+1,s+1}^{u,v+1}\left[\mathcal{V}\right(\mathcal{V}-\mathcal{G}{)}^{\tau }|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),(1-\kappa ,\tau ;1),({\Psi }_i,{\zeta }_i;A_i{)}_{2,r}\\ ({\Phi }_i,{\text{β}}_i;B_i{)}_{1,s},(1-\kappa -(1-\mathrm{c})(1-\mathrm{d}),\tau ;1)\end{array}]\zeta \left(\mathcal{G}\right)\mathrm{d}\mathcal{G}.\end{array}$(43) ) (Using Leibniz Integral Rule), to obtain the following: (44) $\begin{array}{rl}& \frac{\mathrm{d}}{\mathrm{d}\mathcal{V}}\left({}_{\alpha +}{\mathcal{L}}^{(1-\mathrm{c})(1-\mathrm{d})}{\mathfrak{E}}_{\alpha +;r,s;\kappa }^{w;u,v;\tau }\zeta \right)\left(\mathcal{V}\right)={\int }_{\alpha }^{\mathcal{V}}(\mathcal{V}-\mathcal{G}{)}^{\kappa +(1-\mathrm{c})(1-\mathrm{d})-2}\\ & {}^{\mathrm{\Gamma }}{I}_{r+1,s+1}^{u,v+1}\left[\mathcal{V}\right(\mathcal{V}-\mathcal{G}{)}^{\tau }|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),(1-\kappa ,\tau ;1),({\Psi }_i,{\zeta }_i;A_i{)}_{2,r}\\ ({\Phi }_i,{\text{β}}_i;B_i{)}_{1,s},(2-\kappa -(1-\mathrm{c})(1-\mathrm{d}),\tau ;1)\end{array}]\zeta \left(\mathcal{G}\right)\mathrm{d}\mathcal{G}.\end{array}$(44) Next, the operator ${}_{\alpha +}{\mathcal{L}}^{\mathrm{d}(1-\mathrm{c})}$ is applied to Equation (44(44) $\begin{array}{rl}& \frac{\mathrm{d}}{\mathrm{d}\mathcal{V}}\left({}_{\alpha +}{\mathcal{L}}^{(1-\mathrm{c})(1-\mathrm{d})}{\mathfrak{E}}_{\alpha +;r,s;\kappa }^{w;u,v;\tau }\zeta \right)\left(\mathcal{V}\right)={\int }_{\alpha }^{\mathcal{V}}(\mathcal{V}-\mathcal{G}{)}^{\kappa +(1-\mathrm{c})(1-\mathrm{d})-2}\\ & {}^{\mathrm{\Gamma }}{I}_{r+1,s+1}^{u,v+1}\left[\mathcal{V}\right(\mathcal{V}-\mathcal{G}{)}^{\tau }|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),(1-\kappa ,\tau ;1),({\Psi }_i,{\zeta }_i;A_i{)}_{2,r}\\ ({\Phi }_i,{\text{β}}_i;B_i{)}_{1,s},(2-\kappa -(1-\mathrm{c})(1-\mathrm{d}),\tau ;1)\end{array}]\zeta \left(\mathcal{G}\right)\mathrm{d}\mathcal{G}.\end{array}$(44) ) (use the RL integral): (45) $\begin{array}{rl}& \left({}_{\alpha +}{\mathcal{L}}^{\mathrm{d}(1-\mathrm{c})}\frac{\mathrm{d}}{\mathrm{d}\mathcal{V}}\left({}_{\alpha +}{\mathcal{L}}^{(1-\mathrm{c})(1-\mathrm{d})}{\mathfrak{E}}_{\alpha +;r,s;\kappa }^{w;u,v;\tau }\zeta \right)\right(\mathcal{V}\left)\right)\left(g\right)\\ & =\frac{1}{\mathrm{\Gamma }\left(\mathrm{d}\right(1-\mathrm{c}\left)\right)}{\int }_{\alpha }^g(g-z{)}^{\mathrm{d}(1-\mathrm{c})-1}\left({\int }_{\alpha }^z\right(z-\mathcal{G}{)}^{\kappa +(1-\mathrm{c})(1-\mathrm{d})-2}\\ & {}^{\mathrm{\Gamma }}{I}_{r+1,s+1}^{u,v+1}\left[\mathcal{V}\right(z-\mathcal{G}{)}^{\tau }|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),(1-\kappa ,\tau ;1),({\Psi }_i,{\zeta }_i;A_i{)}_{2,r}\\ ({\Phi }_i,{\text{β}}_i;B_i{)}_{1,s},(2-\kappa -(1-\mathrm{c})(1-\mathrm{d}),\tau ;1)\end{array}]\zeta \left(\mathcal{G}\right)\mathrm{d}\mathcal{G}).\end{array}$(45) Further, changes in the order of the integration (allowed for the conditions already stated) and substituting $z-\mathcal{G}=\xi$ in expression (45(45) $\begin{array}{rl}& \left({}_{\alpha +}{\mathcal{L}}^{\mathrm{d}(1-\mathrm{c})}\frac{\mathrm{d}}{\mathrm{d}\mathcal{V}}\left({}_{\alpha +}{\mathcal{L}}^{(1-\mathrm{c})(1-\mathrm{d})}{\mathfrak{E}}_{\alpha +;r,s;\kappa }^{w;u,v;\tau }\zeta \right)\right(\mathcal{V}\left)\right)\left(g\right)\\ & =\frac{1}{\mathrm{\Gamma }\left(\mathrm{d}\right(1-\mathrm{c}\left)\right)}{\int }_{\alpha }^g(g-z{)}^{\mathrm{d}(1-\mathrm{c})-1}\left({\int }_{\alpha }^z\right(z-\mathcal{G}{)}^{\kappa +(1-\mathrm{c})(1-\mathrm{d})-2}\\ & {}^{\mathrm{\Gamma }}{I}_{r+1,s+1}^{u,v+1}\left[\mathcal{V}\right(z-\mathcal{G}{)}^{\tau }|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),(1-\kappa ,\tau ;1),({\Psi }_i,{\zeta }_i;A_i{)}_{2,r}\\ ({\Phi }_i,{\text{β}}_i;B_i{)}_{1,s},(2-\kappa -(1-\mathrm{c})(1-\mathrm{d}),\tau ;1)\end{array}]\zeta \left(\mathcal{G}\right)\mathrm{d}\mathcal{G}).\end{array}$(45) ), Hence the following is obtained easily: (46) $\begin{array}{rl}& \left({}_{\alpha +}{\mathcal{L}}^{\mathrm{d}(1-\mathrm{c})}\frac{\mathrm{d}}{\mathrm{d}\mathcal{V}}\left({}_{\alpha +}{\mathcal{L}}^{(1-\mathrm{c})(1-\mathrm{d})}{\mathfrak{E}}_{\alpha +;r,s;\kappa }^{w;u,v;\tau }\zeta \right)\right(\mathcal{V}\left)\right)\left(g\right)\\ & =\frac{1}{\mathrm{\Gamma }\left(\mathrm{d}\right(1-\mathrm{c}\left)\right)}{\int }_{\alpha }^g\left({\int }_0^{g-\mathcal{G}}\right(g-\xi -\mathcal{G}{)}^{\mathrm{d}(1-\mathrm{c})-1}{\xi }^{\kappa +(1-\mathrm{c})(1-\mathrm{d})-2}\\ & {}^{\mathrm{\Gamma }}{I}_{r+1,s+1}^{u,v+1}\left[\mathcal{V}{\xi }^{\tau }|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),(1-\kappa ,\tau ;1),({\Psi }_i,{\zeta }_i;A_i{)}_{2,r}\\ ({\Phi }_i,{\text{β}}_i;B_i{)}_{1,s},(2-\kappa -(1-\mathrm{c})(1-\mathrm{d}),\tau ;1)\end{array}\right]\mathrm{d}\xi )\zeta \left(\mathcal{G}\right)\mathrm{d}\mathcal{G}.\end{array}$(46) Finally, with the help RL integral and after a slight adjustment in the right-hand side of (46(46) $\begin{array}{rl}& \left({}_{\alpha +}{\mathcal{L}}^{\mathrm{d}(1-\mathrm{c})}\frac{\mathrm{d}}{\mathrm{d}\mathcal{V}}\left({}_{\alpha +}{\mathcal{L}}^{(1-\mathrm{c})(1-\mathrm{d})}{\mathfrak{E}}_{\alpha +;r,s;\kappa }^{w;u,v;\tau }\zeta \right)\right(\mathcal{V}\left)\right)\left(g\right)\\ & =\frac{1}{\mathrm{\Gamma }\left(\mathrm{d}\right(1-\mathrm{c}\left)\right)}{\int }_{\alpha }^g\left({\int }_0^{g-\mathcal{G}}\right(g-\xi -\mathcal{G}{)}^{\mathrm{d}(1-\mathrm{c})-1}{\xi }^{\kappa +(1-\mathrm{c})(1-\mathrm{d})-2}\\ & {}^{\mathrm{\Gamma }}{I}_{r+1,s+1}^{u,v+1}\left[\mathcal{V}{\xi }^{\tau }|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),(1-\kappa ,\tau ;1),({\Psi }_i,{\zeta }_i;A_i{)}_{2,r}\\ ({\Phi }_i,{\text{β}}_i;B_i{)}_{1,s},(2-\kappa -(1-\mathrm{c})(1-\mathrm{d}),\tau ;1)\end{array}\right]\mathrm{d}\xi )\zeta \left(\mathcal{G}\right)\mathrm{d}\mathcal{G}.\end{array}$(46) ), then we get: (47) $\begin{array}{rl}& \left({}_{\alpha +}{\mathcal{L}}^{\mathrm{d}(1-\mathrm{c})}\frac{\mathrm{d}}{\mathrm{d}\mathcal{V}}\left({}_{\alpha +}{\mathcal{L}}^{(1-\mathrm{c})(1-\mathrm{d})}{\mathfrak{E}}_{\alpha +;r,s;\kappa }^{w;u,v;\tau }\zeta \right)\right(\mathcal{V}\left)\right)\left(g\right)\\ & ={\int }_{\alpha }^g(g-\mathcal{G}{)}^{\kappa +(1-\mathrm{c})(1-\mathrm{d})-1+\mathrm{d}(1-\mathrm{c})-1}\\ & {}^{\mathrm{\Gamma }}{I}_{r+1,s+1}^{u,v+1}\left[\mathcal{V}{(g-\mathcal{G})}^{\tau }|\begin{array}{c}({\Psi }_1,{\zeta }_1;A_1:Y),(1-\kappa ,\tau ;1),({\Psi }_i,{\zeta }_i;A_i{)}_{2,r}\\ ({\Phi }_i,{\text{β}}_i;B_i{)}_{1,s},(2-\kappa -(1-\mathrm{c})(1-\mathrm{d})-\mathrm{d}(1-\mathrm{c}),\tau ;1)\end{array}\right]\\ & \zeta \left(\mathcal{G}\right)\mathrm{d}\mathcal{G}\\ & ={}_{\alpha +}{\mathcal{L}}^{(1-\mathrm{c})(1-\mathrm{d})+\mathrm{d}(1-\mathrm{c})-1}\left({\mathfrak{E}}_{\alpha +;r,s;\kappa }^{w;u,v;\tau }\zeta \right)\left(g\right),\end{array}$(47) that is the entire proof of Theorem 3.7.

4. Remarks

This section presents some of the already available results, which directly follow our main results. Next, we establish some interesting results obtained by specializing in the parameters of the IIF of our main result.

Remark 4.1

We acquire the results in terms of the I-function by using Y =0 in the significant findings.

Remark 4.2

If setting $A_i=1,B_j=1(i=1,2,\dots ,r,j=1,2,\dots ,s)$ in the main findings, the outcome is parallel to Harjule et al. [17].

Remark 4.3

When we putting $Y=0,A_i=1,B_j=1(i=1,2,\dots ,r,j=1,2,\dots ,s)$ in the main results, we obtain the H-function as a results.

Remark 4.4

If we substitute $u=1,v=r,s=s+1,{\text{β}}_1=1,{\Phi }_1=0,{\Psi }_i=1-{\Psi }_i,{\Phi }_i=1-{\mathrm{\Phi }}_i,\mathcal{V}(\mathcal{G}-\alpha {)}^{\nu }=-\mathcal{V}(\mathcal{G}-\alpha {)}^{\nu },A_i=1,B_i=1$ in the main results, we obtain the Fox-Wright ${}_r{\mathrm{\Psi }}_s^{\mathrm{\Gamma }}$-function.

Remark 4.5

If we substitute $u=1,v=r,s=s+1,{\text{β}}_1=1,{\Phi }_1=0,{\Psi }_i=1-{\Psi }_i,{\Phi }_i=1-{\mathrm{\Phi }}_i,\mathcal{V}(\mathcal{G}-\alpha {)}^{\nu }=-\mathcal{V}(\mathcal{G}-\alpha {)}^{\nu },A_i=1,B_i=1,{\zeta }_i=1,{\text{β}}_i=1$ in the main results, we obtain the incomplete generalized hypergeometric ${}_r{\mathrm{\Gamma }}_s$-function.

Remark 4.6

If we reduce IIF to the Mittag–Leffler type function $E_{\tau ,\kappa }^{\alpha ,\text{β}}\left(z\right)$, then the result established by Srivastava and Tomovski [31] follows as special case.

Remark 4.7

If we reduce IIF to the multi-index Mittag–Leffler type function $E_{(\tau ,\kappa {)}_{\gamma }}^{\alpha ,\text{β}}\left(z\right)$, the results obtained by Srivastava et al. [32] follow as special case.

5. Conclusions

The introduction already specified the significance of an integral operator with the IIF family as its kernel. Some interesting results have been established by integrating fractional derivative operators with that of IIF. Also, we have shown that the results derived in this paper generalize the results obtained in earlier works by Srivastava and Tomovski [31], Srivastava et al. [32], and Harjule et al. [17]. The significant result of our study is generic. Our conclusions are crucial in many different fields. With their aid, a wide range of fascinating and valuable FDE with applications in engineering, communication theory, probability theory, and science can be created. Therefore, we conclude that this research contributes to the vast and ever-changing mathematical literature of fractional calculus and special functions.

Acknowledgments

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

Disclosure statement

No potential conflict of interest was reported by the author(s).