Certain Integral and Differential Formulas Involving the Product of Srivastava’s Polynomials and Extended Wright Function

Abstract

Many authors have established various integral and differential formulas involving different special functions in recent years. In continuation, we explore some image formulas associated with the product of Srivastava’s polynomials and extended Wright function by using Marichev–Saigo–Maeda fractional integral and differential operators, Lavoie–Trottier and Oberhettinger integral operators. The obtained outcomes are in the form of the Fox–Wright function. It is worth mentioning that some interesting special cases are also discussed.

1. Introduction and Preliminaries

Numerous integral formulas including special functions have been anticipated and they are important in solving various problems in science and engineering. Many authors have established certain unified integral formulae associated with special functions [1,2,3,4,5]. A brief study of some important properties of generalized gamma and beta functions defined in the form of Fox–Wright function is presented in [6]. Certain integral formulas involving the generalized Bessel function and Bessel–Maitland function are explored in [7,8]. A new class of integrals associated with hypergeometric function is established by Rakha et al. [9]. Certain fractional calculus operators and their applications are briefly discussed by Samraiz et al. [10]. A brief discussion of generalized Mittag–Leffler function and multivariable Mittag–Leffler function via generalized fractional calculus operators is available in [11,12]. Composition formulas of various fractional calculus operators are studied in [13,14].

Many generalized special functions have been linked to various types of issues in different fields of mathematical sciences. This reason inspired many researchers to explore the field of integrals and associated generalized special functions. Several unified integral formulas derived by many authors involving different type of special functions are discussed in (see, for example, [15,16,17,18,19]). Suthar established the composition formulae for the k-fractional calculus operators associated with k-Wright function [20]. Fractional calculus and integral transforms of general class of polynomials and incomplete Fox–Wright functions is discussed by Jangid et al. [21]. Certain expressions of the Laguerre polynomial and relations of some known functions in terms of generalized Meijer G-functions are explored in [22,23].

Lavoie–Trottier integral formula involving product of Bessel function of the first kind and general class of polynomials are established by Menaria et al. [24]. Suthar et al. [25] explored the certain integral formulae involving product of Srivastava’s polynomials and generalized Bessel–Maitland function.

In continuation of the above work, we developed generalized integral formulae involving product of Srivastava’s polynomials $S_c^d\left[r\right]$ and extended Wright function $R_{\varsigma ,\tau }^{\omega ,\epsilon }\left(z\right)$. These formulae are communicated in the form of generalized Fox–Wright function. For our purpose, we start by reviewing some known functions and earlier work. Srivastava [26] established the general class of polynomials $S_c^d\left[r\right]$ in the following way.

$$\begin{array}{c}\hfill S_c^d\left[r\right]=\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{r}^k(c=0,1,2,\dots ),\end{array}$$

(1)

where d is an arbitrary positive integer and the coefficients $G_{c,k}(c,k\ge 0)$ are arbitrary constants (real or complex). The polynomial family $S_c^d\left[r\right]$ have many known polynomials as its special cases. The extended Wright function [27] is defined as follows:

$$\begin{array}{c}\hfill R_{\varsigma ,\tau }^{\omega ,\epsilon }\left(z\right)=\sum _{n=0}^{\infty }\frac{{\left(\omega \right)}_n}{{\left(\epsilon \right)}_n\mathsf{\Gamma }(\varsigma n+\tau )}\frac{z^n}{n!},\end{array}$$

(2)

where $\varsigma >-1,\tau ,\omega ,\epsilon \in \mathbb{C},\epsilon \ne 0,-1,-2,\dots$ with $z\in \mathbb{C}$. The function $R_{\varsigma ,\tau }^{\omega ,\epsilon }\left(z\right)$ is an entire function of order $\frac{1}{1+\varsigma }$.

The two (generalized Wright type) auxiliary functions for any order $\varsigma \in (0,1)$ and for all complex variable $z\ne 0$ are defined as follows:

$$\begin{array}{c}\hfill M_{-\varsigma }^{\omega ,\epsilon }=R_{-\varsigma ,1-\varsigma }^{\omega ,\epsilon }(-z)=\sum _{n=0}^{\infty }\frac{{\left(\omega \right)}_n}{{\left(\epsilon \right)}_n\mathsf{\Gamma }(1-\varsigma (n+1))}\frac{{(-1)}^n{z}^n}{n!}\end{array}$$

(3)

and

$$\begin{array}{c}\hfill F_{-\varsigma }^{\omega ,\epsilon }=R_{-\varsigma ,1-\varsigma }^{\omega ,\epsilon }(-z)=\sum _{n=0}^{\infty }\frac{{\left(\omega \right)}_n}{{\left(\epsilon \right)}_n\mathsf{\Gamma }(-\varsigma n)}\frac{{(-1)}^n{z}^n}{n!}.\end{array}$$

(4)

For $\omega =\epsilon =0$, we have the normalized Wright function [28] which is given by

$$W_{\varsigma ,\tau }\left(z\right)=\mathsf{\Gamma }\left(\tau \right)\sum _{n=0}^{\infty }\frac{z^n}{\mathsf{\Gamma }(\tau +n\varsigma )n!},\tau >-1,\varsigma \in \mathbb{C}.$$

(5)

The extended Wright function can also be expressed in the following form

$$\begin{array}{c}\hfill R_{\varsigma ,\tau }^{\omega ,\epsilon }\left(z\right)=\frac{\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}{}_1{\psi }_2\left[\begin{array}{c}(\omega ,1);\\ (\epsilon ,1),(\tau ,\varsigma );\end{array}Z\right],\end{array}$$

(6)

where ${}_u{\psi }_x$ is the Fox–Wright hypergeometric function defined in [29] as follows: For $a_i,b_j\in \mathbb{C}$, $A_i,B_j\in \mathbb{R}$$(A_i,B_j)\ne 0,$ where $i=1,2,\dots ,u;j=1,2,\dots ,x$ and $(a_i+A_{i}n),(b_j+B_{j}n)\in \mathbb{C}\backslash k{\mathbb{Z}}^{-}$,

$${}_u{\psi }_x[(a_1,A_1),\dots ,(a_u,A_u);(b_1,B_1),\dots ,(b_x,B_x);z]=\sum _{n=0}^{\infty }\frac{\mathsf{\Gamma }(a_1+n{A}_1)\dots \mathsf{\Gamma }(a_u+n{A}_u)z^n}{\mathsf{\Gamma }(b_1+n{B}_1)\dots \mathsf{\Gamma }(b_x+n{B}_x)n!},$$

(7)

with convergence condition

$$1+\sum _{j=1}^u{B}_j-\sum _{i=1}^v{A}_i>0.$$

(8)

The extended Wright function has relationship with some other special functions which are discussed below.

Relation with Mittag–Leffler function:

$$\begin{array}{c}\hfill R_{0,\tau }^{1,\epsilon }\left(z\right)=\frac{\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\tau \right)}{E}_{1,\epsilon }\left(z\right),\end{array}$$

(9)

for $\varsigma =0,\omega =1,\tau ,\epsilon \in \mathbb{C}$ and $Re\left(\epsilon \right)>0$.

Relation with Meijer G-function:

Extended Wright function and Meijer G-function are related as

$$\begin{array}{c}\hfill R_{0,\tau }^{1,\epsilon }\left(z\right)=\frac{\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}{G}_{13}^{11}\left[z|\begin{array}{c}1-\omega \\ 0,1-\tau ,1-\epsilon \end{array}\right],\end{array}$$

(10)

for $\varsigma =1$.

Relation with Fox H-function:

From the definition of Fox H-function and extended Wright function, we obtain

$$\begin{array}{c}\hfill R_{\varsigma ,\tau }^{\omega ,\epsilon }(-z)=\frac{\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}{H}_{13}^{11}\left[z|\begin{array}{c}(1-\omega ,1)\\ (0,1),(1-\tau ,\varsigma ),(1-\epsilon ,1)\end{array}\right].\end{array}$$

(11)

The Marichev–Saigo–Maeda fractional integral operators [30] for $\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda \in \mathbb{C}$ and $x>0$ are defined as follows:

$$\begin{array}{c}\hfill I_{0^{+}}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }f\left(t\right)=\frac{x^{-\xi }}{\mathsf{\Gamma }\left(\lambda \right)}{\int }_0^x{(x-t)}^{\lambda -1}{t}^{-\acute{\xi }}{F}_3\left(\xi ,\acute{\xi },\eta ,\acute{\eta };\lambda ;1-\frac{t}{x};1-\frac{x}{t}\right)f\left(t\right)dt\end{array}$$

(12)

and

$$\begin{array}{c}\hfill I_{-}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }f\left(t\right)=\frac{x^{-\acute{\xi }}}{\mathsf{\Gamma }\left(\lambda \right)}{\int }_x^{\infty }{(t-x)}^{\lambda -1}{t}^{-\xi }{F}_3\left(\xi ,\acute{\xi },\eta ,\acute{\eta };\lambda ;1-\frac{x}{t};1-\frac{t}{x}\right)f\left(t\right)dt,\end{array}$$

(13)

where $F_3$ is Appell function.

The Marichev–Saigo–Maeda fractional differential operators [31] for $\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda \in \mathbb{C}$ and $x>0$ are defined as follows:

$$\begin{array}{c}\hfill D_{0^{+}}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }f\left(t\right)={\left(\frac{d}{dt}\right)}^m\left(I_{0^{+}}^{-\acute{\xi },-\xi ,-\acute{\eta }+m,-\eta ,-\lambda +m}f\right)\left(t\right)\end{array}$$

(14)

and

$$\begin{array}{c}\hfill D_{-}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }f\left(t\right)={\left(-\frac{d}{dt}\right)}^m\left(I_{-}^{-\acute{\xi },-\xi ,-\acute{\eta },-\eta +m,-\lambda +m}f\right)\left(t\right).\end{array}$$

(15)

The Saigo fractional integral operators [32] are defined as:

For $w\in {\mathbb{R}}^{+}$, $ϵ,\varrho ,\chi \in \mathbb{C}$ with $Re\left(ϵ\right)>0$,

$$\begin{array}{c}\hfill \left(I_{0+}^{ϵ,\varrho ,\chi }f\right)\left(w\right)=\frac{w^{-ϵ-\varrho }}{\mathsf{\Gamma }\left(ϵ\right)}{\int }_0^w{(w-t)}^{ϵ-1}{\times }_2{F}_1\left(ϵ+\varrho ,-\chi ;ϵ;\left(1-\frac{t}{w}\right)\right)f\left(t\right)dt\end{array}$$

(16)

and

$$\begin{array}{c}\hfill \left(I_{-}^{ϵ,\varrho ,\chi }f\right)\left(w\right)=\frac{1}{\mathsf{\Gamma }\left(ϵ\right)}{\int }_w^{\infty }{(t-w)}^{ϵ-1}{t}^{-ϵ-\varrho }{\times }_2{F}_1\left(ϵ+\varrho ,-\chi ;ϵ;\left(1-\frac{w}{t}\right)\right)f\left(t\right)dt.\end{array}$$

(17)

Additionally, the Saigo fractional differential operators [33], for $w>0$ and $ϵ,\varrho ,\chi \in \mathbb{C}$, $Re\left(ϵ\right)>0$ are given by

$$\begin{array}{cc}\hfill \left(D_{0+}^{ϵ,\varrho ,\chi }f\right)\left(w\right)& ={\left(\frac{d}{dw}\right)}^n\left(I_{0+}^{-ϵ+n,-\varrho -n,ϵ+\chi -n}f\right)w,n=[Re\left(ϵ\right)+1]\hfill \\ & ={\left(\frac{d}{dw}\right)}^n\frac{w^{\frac{ϵ+\varrho }{k}}}{k{\mathsf{\Gamma }}_k(-ϵ+n)}{\int }_0^w{(w-t)}^{\frac{-ϵ}{k}+n-1}\hfill \\ & {\times }_2{F}_1((-ϵ-\varrho ,-\chi -ϵ+n;-ϵ+n;(1-\frac{t}{w}))f\left(t\right)dt\hfill \end{array}$$

(18)

and

$$\begin{array}{cc}\hfill \left(D_{-}^{ϵ,\varrho ,\chi }f\right)\left(w\right)& ={\left(\frac{d}{dw}\right)}^n\left(I_{-}^{-ϵ+n,-\varrho -n,ϵ+\chi }f\right)w,n=[Re\left(ϵ\right)+1]\hfill \\ & ={\left(\frac{d}{dw}\right)}^n\frac{1}{k{\mathsf{\Gamma }}_k(-ϵ+n)}{\int }_w^{\infty }{(t-w)}^{\frac{-ϵ-n}{k}-1}{t}^{\frac{ϵ+\varrho }{k}}\hfill \\ & {\times }_2{F}_1((-ϵ-\varrho ,-\chi -ϵ+n;-ϵ+n;(1-\frac{w}{t}))f\left(t\right)dt,\hfill \end{array}$$

(19)

where $\left[Re\right(ϵ\left)\right]$ is the integeral part of $Re\left(ϵ\right)$ and ${}_2{F}_1(ϵ,\varrho ,\chi ;w)$ is the hypergeometric function.

The left-sided and right-sided Riemann–Liouville fractional integral operators [33] are defined as follows:

$$I_{a+}^{\zeta }f\left(y\right)=\frac{1}{\mathsf{\Gamma }\left(\zeta \right)}{\int }_a^y{(y-t)}^{\zeta -1}f\left(t\right)dt$$

(20)

and

$$I_{b-}^{\zeta }f\left(y\right)=\frac{1}{\mathsf{\Gamma }\left(\zeta \right)}{\int }_y^b{(t-y)}^{\zeta -1}f\left(t\right)dt,$$

(21)

where $Re\left(\zeta \right)>0$.

The left and right-sided Riemann–Liouville fractional differential operators are given by

$$\left(D_{a+}^{\zeta }y\right)\left(x\right)={\left(\frac{d}{dx}\right)}^n\frac{1}{\mathsf{\Gamma }(n-\zeta )}{\int }_a^x{(x-t)}^{\zeta -n+1}y\left(t\right)dt,$$

(22)

where $n=\left[Re\right(\zeta \left)\right]+1$, $x>a$ and

$$\left(D_{b-}^{\zeta }y\right)\left(x\right)={\left(-\frac{d}{dx}\right)}^n\frac{1}{\mathsf{\Gamma }(p-\zeta )}{\int }_z^{\infty }{(t-x)}^{\zeta -n+1}y\left(t\right)dt,$$

(23)

where $n=\left[Re\right(\zeta \left)\right]+1$ and $x<b$.

Further, we will recall the Lavoie–Trottier integral formula [34] which is given by

$${\int }_0^1{y}^{ϵ-1}{(1-y)}^{2\omega -1}{\left(1-\frac{y}{3}\right)}^{2ϵ-1}{\left(1-\frac{y}{4}\right)}^{\omega -1}dy={\left(\frac{2}{3}\right)}^{ϵ-1}\frac{\mathsf{\Gamma }\left(\gamma \right)\mathsf{\Gamma }\left(\omega \right)}{\mathsf{\Gamma }(ϵ+\omega )},$$

(24)

for $Re\left(\gamma \right),Re\left(\delta \right)>0$. Additionally, we evoke Oberhettinger’s integral formula [35] given as follows:

$${\int }_0^{\infty }{y}^{\varphi -1}{(y+b+\sqrt{y^2+2by})}^{-\chi }dy=2\chi b^{-\chi }{\left(\frac{b}{2}\right)}^{\varphi }\frac{\mathsf{\Gamma }\left(2\varphi \right)\mathsf{\Gamma }(\chi -\varphi )}{\mathsf{\Gamma }(1+\chi +\varphi )},$$

(25)

where $0<Re\left(\varphi \right)<Re\left(\chi \right)$.

2. Image Formulas for Marichev–Saigo–Maeda Integral Operators Involving the Product of Srivastava’s Polynomials and Extended Wright Function

In this section, we establish the image formulas by applying Marichev–Saigo–Maeda fractional integral operators (12) and (13) to the product of Srivastava’s Polynomials (1) and extended Wright function (2).

The following formulas for a power function, under operators (12) and (13) are given in [31] are helpful to prove our main results.

$$\begin{array}{c}\hfill \left[I_{0^{+}}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }{t}^{\nu -1}\right]y=\mathsf{\Gamma }\left[\begin{array}{c}\nu ,\nu +\lambda -\xi -\acute{\xi }-\eta ,\nu -\acute{\xi }-\xi +\acute{\eta }\\ \nu +\acute{\eta },\nu +\lambda -\xi -\acute{\xi },\nu +\lambda -\acute{\xi }-\eta \end{array}\right]y^{\nu -\xi -\acute{\xi }+\lambda -1},\end{array}$$

(26)

where $Re\left(\lambda \right)>0$ and $Re\left(\nu \right)>max\{0,Re(\xi +\acute{\xi }+\eta -\lambda ),Re(\acute{\xi }-\acute{\eta })\}$.

Additionally,

$$\begin{array}{cc}\hfill & \left[I_{0^{-}}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }{t}^{\nu -1}\right]y=\mathsf{\Gamma }\left[\begin{array}{c}1+\xi +\acute{\xi }-\lambda -\nu ,1+\xi +\acute{\eta }-\lambda -\nu ,1-\eta -\nu \\ 1-\nu ,1+\xi +\acute{\xi }+\acute{\eta }-\lambda -\nu ,1+\xi -\eta -\nu \end{array}\right]\hfill \\ \hfill & \times y^{\nu +\lambda -\xi -\acute{\xi }-1},\hfill \end{array}$$

(27)

where $Re\left(\lambda \right)>0$ and $Re\left(\nu \right)<1+min\{Re(-\eta ),Re(\xi +\acute{\eta }-\lambda ),Re(\xi +\acute{\xi }-\lambda )\}$.

We use the notation

$$\mathsf{\Gamma }\left[\begin{array}{c}a,b,c\\ d,e,f\end{array}\right]=\frac{\mathsf{\Gamma }\left(a\right)\mathsf{\Gamma }\left(b\right)\mathsf{\Gamma }\left(c\right)}{\mathsf{\Gamma }\left(d\right)\mathsf{\Gamma }\left(e\right)\mathsf{\Gamma }\left(f\right)}.$$

Theorem 1.

For $\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda ,\omega \in \mathbb{C},x>0$ such that $Re\left(\lambda \right)>0,Re\left(\varsigma \right)>-1,Re(\nu +\varsigma \mu +2\tau \mu )>max\{0,Re(\xi +\acute{\xi }+\eta -\lambda ),Re(\acute{\xi }-\acute{\eta })\}$, then prove the following formula

$$\begin{array}{cc}\hfill & \left[I_{0^{+}}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }{t}^{\nu -1}{S}_c^d\left(\sigma t^{\alpha }\right)R_{\varsigma ,\tau }^{\omega ,\epsilon }\left(\delta t^{\mu }\right)\right]\left(x\right)\hfill \\ \hfill & =x^{\nu -\xi -\acute{\xi }+\lambda -1}\frac{\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\sigma x^{\alpha }\right)}^k\hfill \\ \hfill & \times {}_4{\psi }_5\left[\begin{array}{c}(\omega ,1),(\nu +\alpha k,\mu ),(\nu +\alpha k+\lambda -\xi -\acute{\xi }-\eta ,\mu ),\\ (\nu +\alpha k-\acute{\xi }+\acute{\eta },\mu );\\ (\epsilon ,1),(\tau ,\varsigma ),(\nu +\alpha k+\acute{\eta },\mu ),(\nu +\alpha k+\lambda -\xi -\acute{\xi },\mu ),\\ (\nu +\alpha k+\lambda -\acute{\xi }+\eta ,\mu );\end{array}\delta x^{\mu }\right].\hfill \end{array}$$

(28)

Proof. 

By using (1) and (2) in the left hand side of (24) and representing it with $\mathfrak{S}$. After some simplification, we obtain

$$\begin{array}{cc}\hfill & \mathfrak{S}=\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\sigma \right)}^k\sum _{n=0}^{\infty }\frac{{\left(\omega \right)}_n}{{\left(\epsilon \right)}_n\mathsf{\Gamma }(\varsigma n+\tau )}\frac{{\left(\delta t^{\mu }\right)}^n}{n!}\hfill \\ \hfill & \times \left[I_{0^{+}}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }{t}^{\nu +\alpha k+\mu n-1}\right]\left(x\right).\hfill \end{array}$$

(29)

Now, applying the integral Formula (26) to (29), we obtain

$$\begin{array}{cc}\hfill & \mathfrak{S}=\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\sigma }^k\sum _{n=0}^{\infty }\frac{{\left(\omega \right)}_n}{{\left(\epsilon \right)}_n\mathsf{\Gamma }(\varsigma n+\tau )}\frac{{\left(\delta \right)}^n}{n!}\hfill \\ \hfill & \times \mathsf{\Gamma }\left[\begin{array}{c}\nu +\alpha k+\mu n,\nu +\alpha k+\lambda -\xi -\acute{\xi }-\eta +\mu n,\\ \nu +\alpha k-\acute{\xi }+\acute{\eta }+\mu n\\ \nu +\alpha k+\acute{\eta }+\mu n,\nu +\alpha k+\lambda -\xi -\acute{\xi }+\mu n,\\ \nu +\alpha k+\lambda -\acute{\xi }+\eta +\mu n\end{array}\right]x^{\nu +\alpha k+\mu n-\xi -\acute{\xi }+\lambda -1},\hfill \end{array}$$

(30)

which further implies

$$\begin{array}{cc}\hfill & \mathfrak{S}=x^{\nu -\xi -\acute{\xi }+\lambda -1}\frac{\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\sigma x^{\alpha }\right)}^k\hfill \\ \hfill & \times \sum _{n=0}^{\infty }\mathsf{\Gamma }\left[\begin{array}{c}\omega +n,\nu +\alpha k+\mu n,\nu +\alpha k+\lambda -\xi -\acute{\xi }-\eta +\mu n,\\ \nu +\alpha k-\acute{\xi }+\acute{\eta }+\mu n\\ \epsilon +n,\tau +\varsigma n,\nu +\alpha k+\acute{\eta }+\mu n,\nu +\alpha k+\lambda -\xi -\acute{\xi }+\mu n,\\ \nu +\alpha k+\lambda -\acute{\xi }+\eta +\mu n\end{array}\right]\frac{{\left(\delta x^{\mu }\right)}^n}{n!}.\hfill \end{array}$$

(31)

Now, by using definition (7), we arrive at required formula (28).

Theorem 2.

For $\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda ,\omega \in \mathbb{C},x>0$ such that $Re\left(\lambda \right)>0,Re\left(\varsigma \right)>-1,Re(1-\beta -\nu -\varsigma \mu -2\tau \mu )<1+min\{Re(-\eta ),Re(\xi +\acute{\eta }-\lambda ),Re(\xi +\acute{\xi }-\lambda )\}$, then the following formula holds true

$$\begin{array}{cc}\hfill & \left[I_{0^{-}}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }{t}^{-\nu -\beta }{S}_c^d\left(\sigma t^{\alpha }\right)R_{\varsigma ,\tau }^{\omega ,\epsilon }\left(\delta t^{-\mu }\right)\right]\left(x\right)\hfill \\ \hfill & =x^{\nu +\alpha k-\xi -\acute{\xi }+\lambda -1}\frac{\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\sigma x^{\alpha }\right)}^k\hfill \\ \hfill & \times {}_4{\psi }_5\left[\begin{array}{c}(\omega ,1),(\xi +\acute{\xi }-\lambda +\nu +\beta -\alpha k,\mu ),(\xi +\acute{\eta }-\lambda +\nu +\beta -\alpha k,\mu ),\\ (\nu +\beta -\eta -\alpha k,\mu );\\ (\epsilon ,1),(\tau ,\varsigma ),(\nu +\beta -\alpha k,\mu ),(\xi +\acute{\xi }+\acute{\eta }-\lambda +\nu +\beta -\alpha k,\mu ),\\ (\xi -\eta +\nu +\beta -\alpha k,\mu );\end{array}\delta x^{-\mu }\right].\hfill \end{array}$$

(32)

Proof. 

By using (1) and (2) in the left hand side of (32) and representing it by $\mathfrak{V}$. After some simplification, we obtain

$$\begin{array}{cc}\hfill & \mathfrak{V}=\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\sigma \right)}^k\sum _{n=0}^{\infty }\frac{{\left(\omega \right)}_n}{{\left(\epsilon \right)}_n\mathsf{\Gamma }(\varsigma n+\tau )}\frac{{\left(\delta \right)}^n}{n!}\hfill \\ \hfill & \times \left[I_{0^{+}}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }{t}^{1-\nu -\beta +\alpha k-\mu n-1}\right]\left(x\right).\hfill \end{array}$$

(33)

Now, applying the integral Formula (27) to (33), we obtain

$$\begin{array}{cc}\hfill & \mathfrak{V}=\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\sigma }^k\sum _{n=0}^{\infty }\frac{{\left(\omega \right)}_n}{{\left(\epsilon \right)}_n\mathsf{\Gamma }(\varsigma n+\tau )}\frac{{\left(\delta \right)}^n}{n!}\hfill \\ \hfill & \times \mathsf{\Gamma }\left[\begin{array}{c}\xi +\acute{\xi }-\lambda +\nu +\beta -\alpha k+\mu n,\xi +\acute{\eta }-\lambda +\nu +\beta -\alpha k+\mu n,\\ \nu +\beta -\eta -\alpha k+\mu n\\ \nu +\beta -\alpha k+\mu n,\xi +\acute{\xi }+\acute{\eta }-\lambda +\nu +\beta -\alpha k+\mu n,\\ \xi -\eta +\nu +\beta -\alpha k+\mu n\end{array}\right]\hfill \\ \hfill & \times x^{-\nu -\beta +\alpha k-\mu n-\xi -\acute{\xi }+\lambda -1},\hfill \end{array}$$

(34)

which further implies

$$\begin{array}{cc}\hfill & \mathfrak{V}=x^{-\nu -\beta -\xi -\acute{\xi }+\lambda -1}\frac{\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\sigma x^{\alpha }\right)}^k\hfill \\ \hfill & \times \sum _{n=0}^{\infty }\mathsf{\Gamma }\left[\begin{array}{c}\omega +n,\xi +\acute{\xi }-\lambda +\nu +\beta -\alpha k+\mu n,\xi +\acute{\eta }-\lambda +\nu +\beta -\alpha k+\mu n,\\ \nu +\beta -\eta -\alpha k+\mu n\\ \epsilon +n,\tau +\varsigma n,\nu +\beta -\alpha k+\mu n,\xi +\acute{\xi }+\acute{\eta }-\lambda +\nu +\beta -\alpha k+\mu n,\\ \xi -\eta +\nu +\beta -\alpha k+\mu n\end{array}\right]\frac{{\left(\delta x^{\mu }\right)}^n}{n!}.\hfill \end{array}$$

(35)

Now, by using definition (7), we arrive at required formula (32). □

Now, we discuss some special cases regarding extended Wright function in the following corollaries.

Corollary 1.

Assume that condition of Theorem 1 is fulfilled then using generalized Wright-type Function (3), the Formula (28) reduces to the following form.

$$\begin{array}{cc}\hfill & \left[I_{0^{+}}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }{t}^{\nu -1}{S}_c^d\left(\sigma t^{\alpha }\right)M_{-\varsigma ,1-\varsigma }^{\omega ,\epsilon }(-\delta t^{\mu })\right]\left(x\right)\hfill \\ \hfill & =x^{\nu -\xi -\acute{\xi }+\lambda -1}\frac{\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\sigma x^{\alpha }\right)}^k\hfill \\ \hfill & \times {}_4{\psi }_5\left[\begin{array}{c}(\omega ,1),(\nu +\alpha k,\mu ),(\nu +\alpha k+\lambda -\xi -\acute{\xi }-\eta ,\mu ),\\ (\nu +\alpha k-\acute{\xi }+\acute{\eta },\mu );\\ (\epsilon ,1),(1-\varsigma ,-\varsigma ),(\nu +\alpha k+\acute{\eta },\mu ),(\nu +\alpha k+\lambda -\xi -\acute{\xi },\mu ),\\ (\nu +\alpha k+\lambda -\acute{\xi }+\eta ,\mu );\end{array}-\delta x^{\mu }\right].\hfill \end{array}$$

(36)

Corollary 2.

Assume that condition of Theorem 2 is fulfilled then using generalized Wright-type Function (3), the Formula (32) reduces to the following form.

$$\begin{array}{cc}\hfill & \left[I_{0^{-}}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }{t}^{-\nu -\beta }{S}_c^d\left(\sigma t^{\alpha }\right)M_{-\varsigma ,1-\varsigma }^{\omega ,\epsilon }(-\delta t^{-\mu })\right]\left(x\right)\hfill \\ \hfill & =x^{\nu -\xi -\acute{\xi }+\lambda -1}\frac{\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\sigma x^{\alpha }\right)}^k\hfill \\ \hfill & \times {}_4{\psi }_5\left[\begin{array}{c}(\omega ,1),(\xi +\acute{\xi }-\lambda +\nu +\beta -\alpha k,\mu ),(\xi +\acute{\eta }-\lambda +\nu +\beta -\alpha k,\mu ),\\ (\nu +\beta -\eta -\alpha k,\mu );\\ (\epsilon ,1),(\nu +\beta -\alpha k,\mu ),(\xi +\acute{\xi }+\acute{\eta }-\lambda +\nu +\beta -\alpha k,\mu ),\\ (1-\varsigma ,-\varsigma ),(\xi -\eta +\nu +\beta -\alpha k,\mu );\end{array}-\delta x^{-\mu }\right].\hfill \end{array}$$

(37)

Corollary 3.

Assume that condition of Theorem 1 is fulfilled and for $\varsigma \in (0,1)$ then using generalized Wright-type Function (4), the Formula (28) reduces to the following form.

$$\begin{array}{cc}\hfill & \left[I_{0^{+}}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }{t}^{\nu -1}{S}_c^d\left(\sigma t^{\alpha }\right)F_{-\varsigma }^{\omega ,\epsilon }(-\delta t^{\mu })\right]\left(x\right)\hfill \\ \hfill & =x^{\nu -\xi -\acute{\xi }+\lambda -1}\frac{\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\sigma x^{\alpha }\right)}^k\hfill \\ \hfill & \times {}_4{\psi }_5\left[\begin{array}{c}(\omega ,1),(\nu +\alpha k,\mu ),(\nu +\alpha k+\lambda -\xi -\acute{\xi }-\eta ,\mu ),\\ (\nu +\alpha k-\acute{\xi }+\acute{\eta },\mu );\\ (\epsilon ,1),(0,-\varsigma ),(\nu +\alpha k+\acute{\eta },\mu ),(\nu +\alpha k+\lambda -\xi -\acute{\xi },\mu ),\\ (\nu +\alpha k+\lambda -\acute{\xi }+\eta ,\mu );\end{array}-\delta x^{\mu }\right].\hfill \end{array}$$

(38)

Corollary 4.

Assume that condition of Theorem 2 is fulfilled then using generalized Wright-type Function (4), the Formula (32) reduces to the following form.

$$\begin{array}{cc}\hfill & \left[I_{0^{-}}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }{t}^{-\nu -\beta }{S}_c^d\left(\sigma t^{\alpha }\right)F_{-\varsigma ,0}^{\omega ,\epsilon }(-\delta t^{-\mu })\right]\left(x\right)\hfill \\ \hfill & =x^{\nu -\xi -\acute{\xi }+\lambda -1}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\sigma x^{\alpha }\right)}^k\hfill \\ \hfill & \times {}_4{\psi }_5\left[\begin{array}{c}(\omega ,1),(\xi +\acute{\xi }-\lambda +\nu +\beta -\alpha k,\mu ),(\xi +\acute{\eta }-\lambda +\nu +\beta -\alpha k,\mu ),\\ (\nu +\beta -\eta -\alpha k,\mu );\\ (\epsilon ,1),(\nu +\beta -\alpha k,\mu ),(\xi +\acute{\xi }+\acute{\eta }-\lambda +\nu +\beta -\alpha k,\mu ),\\ (0,-\varsigma ),(\xi -\eta +\nu +\beta -\alpha k,\mu );\end{array}-\delta x^{-\mu }\right].\hfill \end{array}$$

(39)

Corollary 5.

Assume that condition of Theorem 1 is fulfilled then using normalized Wright Function (5), the Formula (28) reduces to the following form.

$$\begin{array}{cc}\hfill & \left[I_{0^{+}}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }{t}^{\nu -1}{S}_c^d\left(\sigma t^{\alpha }\right)R_{\varsigma ,\tau }\left(\delta t^{\mu }\right)\right]\left(x\right)\hfill \\ \hfill & =x^{\nu -\xi -\acute{\xi }+\lambda -1}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\sigma x^{\alpha }\right)}^k\hfill \\ \hfill & \times {}_3{\psi }_4\left[\begin{array}{c}(\nu +\alpha k,\mu ),(\nu +\alpha k+\lambda -\xi -\acute{\xi }-\eta ,\mu ),\\ (\nu +\alpha k-\acute{\xi }+\acute{\eta },\mu );\\ (\tau ,\varsigma ),(\nu +\alpha k+\acute{\eta },\mu ),(\nu +\alpha k+\lambda -\xi -\acute{\xi },\mu ),\\ (\nu +\alpha k+\lambda -\acute{\xi }+\eta ,\mu );\end{array}\delta x^{\mu }\right].\hfill \end{array}$$

(40)

Corollary 6.

Assume that condition of Theorem 2 is fulfilled then using normalized Wright Function (5), the Formula (32) reduces to the following form.

$$\begin{array}{cc}\hfill & \left[I_{0^{-}}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }{t}^{-\nu -\beta }{S}_c^d\left(\sigma t^{\alpha }\right)R_{\varsigma ,\tau }\left(\delta t^{-\mu }\right)\right]\left(x\right)\hfill \\ \hfill & =x^{\nu -\xi -\acute{\xi }+\lambda -1}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\sigma x^{\alpha }\right)}^k\hfill \\ \hfill & \times {}_3{\psi }_4\left[\begin{array}{c}(\xi +\acute{\xi }-\lambda +\nu +\beta -\alpha k,\mu ),(\xi +\acute{\eta }-\lambda +\nu +\beta -\alpha k,\mu ),\\ (\nu +\beta -\eta -\alpha k,\mu );\\ (\nu +\beta -\alpha k,\mu ),(\xi +\acute{\xi }+\acute{\eta }-\lambda +\nu +\beta -\alpha k,\mu ),\\ (\tau ,\varsigma ),(\xi -\eta +\nu +\beta -\alpha k,\mu );\end{array}\delta x^{-\mu }\right].\hfill \end{array}$$

(41)

Remark 1.

Following are the special cases of results discussed above.

 (i) 

The formulas obtained reduce to formulas for Saigo’s fractional integral operators (16) and (17) for $\acute{\xi }=0$;

 (ii) 

By substituting $\eta =-\xi$ in Saigo’s fractional integral operators, we obtain the formulas for Riemann–Liouville fractional integral operators (20) and (21).

3. Image Formulas for Marichev–Saigo–Maeda Differential Operators Involving the Product of Srivastava’s Polynomials and Extended Wright Function

In this section, we establish the image formulas by applying Marichev–Saigo–Maeda fractional differential operators (14) and (15) to the product of Srivastava’s Polynomials (1) and extended Wright function (2).

The formulas for a power function, under operators (14) and (15) are given in [36] are helpful to prove our main results.

For $\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda ,\nu \in \mathbb{C}$, such that $Re\left(\xi \right)>0$ and $Re\left(\nu \right)>max\{0,Re(-\xi +\eta ),Re(-\xi -\acute{\xi }-\acute{\eta }+\lambda )\}$, we have

$$\begin{array}{c}\hfill \left[D_{0^{+}}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }{t}^{\nu -1}\right]y=\mathsf{\Gamma }\left[\begin{array}{c}\nu ,-\eta +\xi +\nu ,\xi +\acute{\xi }+\acute{\eta -\lambda +\nu }\\ -\eta +\nu ,\xi +\acute{\xi }-\lambda +\nu ,\xi +\acute{\eta }-\lambda +\nu \end{array}\right]y^{\nu +\xi +\acute{\xi }+\lambda -1}\end{array}$$

(42)

and for $\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda ,\nu \in \mathbb{C}$, such that $Re\left(\xi \right)>0$ and $Re\left(\nu \right)>max\{0,Re(-\acute{\eta }),Re(\acute{\xi }+\eta -\lambda ),Re(\xi +\acute{\xi }-\lambda )+\left[Re\left(\lambda \right)\right]+1\}$, we have

$$\begin{array}{cc}\hfill & \left[D_{0^{-}}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }{t}^{\nu -1}\right]y=\mathsf{\Gamma }\left[\begin{array}{c}\acute{\xi }+\nu ,-\xi -\acute{\xi }+\lambda +\nu ,-\acute{\xi }-\eta +\lambda +\nu \\ \nu ,-\acute{\xi }+\acute{\eta }+\nu ,-\xi -\acute{\xi }-\eta +\lambda +\nu \end{array}\right]\hfill \\ \hfill & \times y^{\xi +\acute{\xi }-\nu -\lambda }.\hfill \end{array}$$

(43)

Theorem 3.

For $\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda ,\omega \in \mathbb{C},x>0$ such that $Re\left(\lambda \right)>0,Re\left(\varsigma \right)>-1,Re\left(\xi \right)>0$ and $Re\left(\nu \right)>max\{0,Re(-\xi +\eta ),Re(-\xi -\acute{\xi }-\acute{\eta }+\lambda )\}$, then prove the following formula

$$\begin{array}{cc}\hfill & \left[D_{0^{+}}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }{t}^{\nu -1}{S}_c^d\left(\sigma t^{\alpha }\right)R_{\varsigma ,\tau }^{\omega ,\epsilon }\left(\delta t^{\mu }\right)\right]\left(x\right)\hfill \\ \hfill & =x^{\xi +\acute{\xi }-\lambda +\nu -1}\frac{\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\sigma x^{\alpha }\right)}^k\hfill \\ \hfill & \times {}_4{\psi }_5\left[\begin{array}{c}(\omega ,1),(\nu +\alpha k+\mu ),(-\eta +\xi +\nu +\alpha k+\mu ),\\ (\xi +\acute{\xi }+\acute{\eta }-\lambda \nu +\alpha k+\mu );\\ (\epsilon ,1),(\tau ,\varsigma ),(-\eta \nu +\alpha k+\mu ),(\xi +\acute{\xi }-\lambda +\nu +\alpha k+\mu ),\\ (\xi +\acute{\eta }-\lambda +\nu +\alpha k+\mu );\end{array}\delta x^{\mu }\right].\hfill \end{array}$$

(44)

Proof. 

By using (1) and (2) in the left hand side of (44) and representing it with $\mathfrak{U}$. After some simplification, we obtain

$$\begin{array}{cc}\hfill & \mathfrak{U}=\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\sigma t^{\alpha }\right)}^k\sum _{n=0}^{\infty }\frac{{\left(\omega \right)}_n}{{\left(\epsilon \right)}_n\mathsf{\Gamma }(\varsigma n+\tau )}\frac{{\left(\delta t^{\mu }\right)}^n}{n!}\hfill \\ \hfill & \times \left[D_{0^{+}}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }{t}^{\nu +\alpha k+\mu n-1}\right]\left(x\right).\hfill \end{array}$$

(45)

Now, applying the integral Formula (42) to (45), we obtain

$$\begin{array}{cc}\hfill & \mathfrak{U}=\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\sigma }^k\sum _{n=0}^{\infty }\frac{{\left(\omega \right)}_n}{{\left(\epsilon \right)}_n\mathsf{\Gamma }(\varsigma n+\tau )}\frac{{\left(\delta \right)}^n}{n!}\hfill \\ \hfill & \times \mathsf{\Gamma }\left[\begin{array}{c}\nu +\alpha k+\mu n,-\eta +\xi +\nu +\alpha k+\mu n,\xi +\acute{\xi }+\acute{\eta }-\lambda \nu +\alpha k+\mu n\\ -\eta \nu +\alpha k+\mu n,\xi +\acute{\xi }-\lambda +\nu +\alpha k+\mu n,\xi +\acute{\eta }-\lambda +\nu +\alpha k+\mu n\end{array}\right]\hfill \\ \hfill & \times x^{\xi +\acute{\xi }-\lambda +\alpha k+\mu n--1},\hfill \end{array}$$

(46)

which further implies

$$\begin{array}{cc}\hfill & \mathfrak{U}=x^{\xi +\acute{\xi }-\lambda +\nu -1}\frac{\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\sigma t^{\alpha }\right)}^k\hfill \\ \hfill & \times \sum _{n=0}^{\infty }\mathsf{\Gamma }\left[\begin{array}{c}\omega +n,\nu +\alpha k+\mu n,-\eta +\xi +\nu +\alpha k+\mu n,\\ \xi +\acute{\xi }+\acute{\eta }-\lambda \nu +\alpha k+\mu n\\ \epsilon +n,\tau +\varsigma n,-\eta \nu +\alpha k+\mu n,\xi +\acute{\xi }-\lambda +\nu +\alpha k+\mu n,\\ \xi +\acute{\eta }-\lambda +\nu +\alpha k+\mu n\end{array}\right]\frac{{\left(\delta x^{\mu }\right)}^n}{n!}.\hfill \end{array}$$

(47)

Now, by using definition (7), we arrive at required formula (44). □

Theorem 4.

For $\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda ,\omega \in \mathbb{C},x>0$ such that $Re\left(\lambda \right)>0,Re\left(\varsigma \right)>-1$,$Re\left(\xi \right)>0$ and $Re\left(\nu \right)>max\{0,Re(-\acute{\eta }),Re(\acute{\xi }+\eta -\lambda ),Re(\xi +\acute{\xi }-\lambda )+\left[Re\left(\lambda \right)\right]+1\}$, then the following formula holds true

$$\begin{array}{cc}\hfill & \left[D_{0^{-}}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }{t}^{-\nu }{S}_c^d\left(\sigma t^{\alpha }\right)R_{\varsigma ,\tau }^{\omega ,\epsilon }\left(\delta t^{-\mu }\right)\right]\left(x\right)\hfill \\ \hfill & =x^{\xi +\acute{\xi }-\lambda -\nu }\frac{\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\sigma x^{\alpha }\right)}^k\hfill \\ \hfill & \times {}_4{\psi }_5\left[\begin{array}{c}(\omega ,1),(\acute{\eta }+\nu +-\alpha k+\mu ),(-\xi -\acute{\xi }+\lambda +\nu -\alpha k+\mu ),\\ (\acute{\xi }-\eta +\lambda +\nu -\alpha k+\mu );\\ (\epsilon ,1),(\tau ,\varsigma ),(\nu -\alpha k+\mu n),(-\acute{\xi }+\acute{\eta }+\nu -\alpha k+\mu ),\\ (-\xi -\acute{\xi }-\eta +\lambda +\nu -\alpha k+\mu );\end{array}\delta x^{-\mu }\right].\hfill \end{array}$$

(48)

Proof. 

By using (1) and (2) in the left hand side of (48) and representing it by $\mathfrak{E}$. After some simplification, we obtain

$$\begin{array}{cc}\hfill & \mathfrak{E}=\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\sigma \right)}^k\sum _{n=0}^{\infty }\frac{{\left(\omega \right)}_n}{{\left(\epsilon \right)}_n\mathsf{\Gamma }(\varsigma n+\tau )}\frac{{\left(\delta \right)}^n}{n!}\hfill \\ \hfill & \times \left[D{I}_{0^{+}}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }{t}^{-\nu +\alpha k-\mu n}\right]\left(x\right).\hfill \end{array}$$

(49)

Now, applying the integral Formula (43) to (49), we obtain

$$\begin{array}{cc}\hfill & \mathfrak{V}=\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\sigma }^k\sum _{n=0}^{\infty }\frac{{\left(\omega \right)}_n}{{\left(\epsilon \right)}_n\mathsf{\Gamma }(\varsigma n+\tau )}\frac{{\left(\delta \right)}^n}{n!}\hfill \\ \hfill & \times \mathsf{\Gamma }\left[\begin{array}{c}\acute{\eta }+\nu +-\alpha k+\mu n,-\xi -\acute{\xi }+\lambda +\nu -\alpha k+\mu n,\\ \acute{\xi }-\eta +\lambda +\nu -\alpha k+\mu n\\ \nu -\alpha k+\mu n,-\acute{\xi }+\acute{\eta }+\nu -\alpha k+\mu n,\\ -\xi -\acute{\xi }-\eta +\lambda +\nu -\alpha k+\mu n\end{array}\right]\hfill \\ \hfill & \times x^{\xi +\acute{\xi }-\lambda -\nu +\alpha k-\mu n},\hfill \end{array}$$

(50)

which further implies

$$\begin{array}{cc}\hfill & \mathfrak{V}=x^{\xi +\acute{\xi }-\lambda -\nu }\frac{\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\sigma x^{\alpha }\right)}^k\hfill \\ \hfill & \times \sum _{n=0}^{\infty }\mathsf{\Gamma }\left[\begin{array}{c}\omega +n,\acute{\eta }+\nu +-\alpha k+\mu n,-\xi -\acute{\xi }+\lambda +\nu -\alpha k+\mu n,\\ \acute{\xi }-\eta +\lambda +\nu -\alpha k+\mu n\\ \epsilon +n,\tau +\varsigma n,\nu -\alpha k+\mu n,-\acute{\xi }+\acute{\eta }+\nu -\alpha k+\mu n,\\ -\xi -\acute{\xi }-\eta +\lambda +\nu -\alpha k+\mu n\end{array}\right]\frac{{\left(\delta x^{-\mu }\right)}^n}{n!}.\hfill \end{array}$$

(51)

Now, by using definition (7), we arrive at required formula (48). □

In the following corollaries, we look at some special cases involving the extended Wright function.

Corollary 7.

Assume that condition of Theorem 3 is fulfilled then using generalized Wright-type Function (3), the Formula (44) reduces to the following form.

$$\begin{array}{cc}\hfill & \left[D_{0^{+}}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }{t}^{\nu -1}{S}_c^d\left(\sigma t^{\alpha }\right)R_{\varsigma ,\tau }^{\omega ,\epsilon }(-\delta t^{\mu })\right]\left(x\right)\hfill \\ \hfill & =x^{\xi +\acute{\xi }-\lambda +\nu -1}\frac{\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\sigma x^{\alpha }\right)}^k\hfill \\ \hfill & \times {}_4{\psi }_5\left[\begin{array}{c}(\omega ,1),(\nu +\alpha k+\mu ),(-\eta +\xi +\nu +\alpha k+\mu ),\\ (\xi +\acute{\xi }+\acute{\eta }-\lambda \nu +\alpha k+\mu );\\ (\epsilon ,1),(1-\varsigma ,-\varsigma ),(-\eta \nu +\alpha k+\mu ),(\xi +\acute{\xi }-\lambda +\nu +\alpha k+\mu ),\\ (\xi +\acute{\eta }-\lambda +\nu +\alpha k+\mu );11\end{array}-\delta x^{\mu }\right].\hfill \end{array}$$

(52)

Corollary 8.

Assume that condition of Theorem 4 is fulfilled then using generalized Wright-type Function (3), the Formula (48) reduces to the following form.

$$\begin{array}{cc}\hfill & \left[D_{0^{-}}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }{t}^{-\nu }{S}_c^d\left(\sigma t^{\alpha }\right)R_{\varsigma ,\tau }^{\omega ,\epsilon }\left(\delta t^{-\mu }\right)\right]\left(x\right)\hfill \\ \hfill & =x^{\xi +\acute{\xi }-\lambda -\nu }\frac{\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\sigma x^{\alpha }\right)}^k\hfill \\ \hfill & \times {}_4{\psi }_5\left[\begin{array}{c}(\omega ,1),(\acute{\eta }+\nu +-\alpha k+\mu ),(-\xi -\acute{\xi }+\lambda +\nu -\alpha k+\mu ),\\ (\acute{\xi }-\eta +\lambda +\nu -\alpha k+\mu );\\ (\epsilon ,1),(1-\varsigma ,-\varsigma ),(\nu -\alpha k+\mu n),(-\acute{\xi }+\acute{\eta }+\nu -\alpha k+\mu ),\\ (-\xi -\acute{\xi }-\eta +\lambda +\nu -\alpha k+\mu );\end{array}-\delta x^{-\mu }\right].\hfill \end{array}$$

(53)

Corollary 9.

Assume that condition of Theorem 3 is fulfilled and for $\varsigma \in (0,1)$ then using generalized Wright-type Function (4), the Formula (44) reduces to the following form.

$$\begin{array}{cc}\hfill & \left[D_{0^{+}}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }{t}^{\nu -1}{S}_c^d\left(\sigma t^{\alpha }\right)F_{-\varsigma }^{\omega ,\epsilon }(-\delta t^{\mu })\right]\left(x\right)\hfill \\ \hfill & =x^{\xi +\acute{\xi }-\lambda +\nu -1}\frac{\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\sigma x^{\alpha }\right)}^k\hfill \\ \hfill & \times {}_4{\psi }_5\left[\begin{array}{c}(\omega ,1),(\nu +\alpha k+\mu ),(-\eta +\xi +\nu +\alpha k+\mu ),\\ (\xi +\acute{\xi }+\acute{\eta }-\lambda \nu +\alpha k+\mu );\\ (\epsilon ,1),(0,-\varsigma ),(-\eta \nu +\alpha k+\mu ),(\xi +\acute{\xi }-\lambda +\nu +\alpha k+\mu ),\\ (\xi +\acute{\eta }-\lambda +\nu +\alpha k+\mu );11\end{array}-\delta x^{\mu }\right].\hfill \end{array}$$

(54)

Corollary 10.

Assume that condition of Theorem 4 is fulfilled then using generalized Wright-type Function (4), the Formula (48) reduces to the following form.

$$\begin{array}{cc}\hfill & \left[D_{0^{-}}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }{t}^{-\nu }{S}_c^d\left(\sigma t^{\alpha }\right)F_{-\varsigma }^{\omega ,\epsilon }\left(\delta t^{-\mu }\right)\right]\left(x\right)\hfill \\ \hfill & =x^{\xi +\acute{\xi }-\lambda -\nu }\frac{\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\sigma x^{\alpha }\right)}^k\hfill \\ \hfill & \times {}_4{\psi }_5\left[\begin{array}{c}(\omega ,1),(\acute{\eta }+\nu +-\alpha k+\mu ),(-\xi -\acute{\xi }+\lambda +\nu -\alpha k+\mu ),\\ (\acute{\xi }-\eta +\lambda +\nu -\alpha k+\mu );\\ (\epsilon ,1),(0,-\varsigma ),(\nu -\alpha k+\mu n),(-\acute{\xi }+\acute{\eta }+\nu -\alpha k+\mu ),\\ (-\xi -\acute{\xi }-\eta +\lambda +\nu -\alpha k+\mu );\end{array}-\delta x^{-\mu }\right].\hfill \end{array}$$

(55)

Corollary 11.

Assume that condition of Theorem 3 is fulfilled then using normalized Wright Function (5), the Formula (44) reduces to the following form.

$$\begin{array}{cc}\hfill & \left[D_{0^{+}}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }{t}^{\nu -1}{S}_c^d\left(\sigma t^{\alpha }\right)R_{\varsigma ,\tau }\left(\delta t^{\mu }\right)\right]\left(x\right)\hfill \\ \hfill & =x^{\xi +\acute{\xi }-\lambda +\nu -1}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\sigma x^{\alpha }\right)}^k\hfill \\ \hfill & \times {}_4{\psi }_5\left[\begin{array}{c}(\nu +\alpha k+\mu ),(-\eta +\xi +\nu +\alpha k+\mu ),\\ (\xi +\acute{\xi }+\acute{\eta }-\lambda \nu +\alpha k+\mu );\\ (\tau ,\varsigma ),(-\eta \nu +\alpha k+\mu ),(\xi +\acute{\xi }-\lambda +\nu +\alpha k+\mu ),\\ (\xi +\acute{\eta }-\lambda +\nu +\alpha k+\mu );\end{array}\delta x^{\mu }\right].\hfill \end{array}$$

(56)

Corollary 12.

Assume that condition of Theorem 2 is fulfilled then using normalized Wright Function (5), the Formula (32) reduces to the following form.

$$\begin{array}{cc}\hfill & \left[D_{0^{-}}^{\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda }{t}^{-\nu }{S}_c^d\left(\sigma t^{\alpha }\right)R_{\varsigma ,\tau }\left(\delta t^{-\mu }\right)\right]\left(x\right)\hfill \\ \hfill & =x^{\xi +\acute{\xi }-\lambda -\nu }\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\sigma x^{\alpha }\right)}^k\hfill \\ \hfill & \times {}_3{\psi }_4\left[\begin{array}{c}(\acute{\eta }+\nu +-\alpha k+\mu ),(-\xi -\acute{\xi }+\lambda +\nu -\alpha k+\mu ),\\ (\acute{\xi }-\eta +\lambda +\nu -\alpha k+\mu );\\ (\tau ,\varsigma ),(\nu -\alpha k+\mu n),(-\acute{\xi }+\acute{\eta }+\nu -\alpha k+\mu ),\\ (-\xi -\acute{\xi }-\eta +\lambda +\nu -\alpha k+\mu );\end{array}\delta x^{-\mu }\right].\hfill \end{array}$$

(57)

Remark 2.

Following are the special cases of results discussed above.

 (i) 

The formulas obtained reduce to formulas for Saigo’s fractional differential operators (18) and (19) for $\acute{\xi }=0$;

 (ii) 

By substituting $\eta =-\xi$ in Saigo’s fractional differential operators, we obtain the formulas for Riemann–Liouville fractional differential operators (22) and (23).

4. Lavoie–Trottier Integral Formulas Involving Product of Srivastava’s Polynomials and Extended Wright Function

In this section, we develop two extended integral formulas, involving the product of Srivastava polynomial (1) and extended Wright function (2).

Theorem 5.

For $\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda ,\omega \in \mathbb{C}$, $Re\left(ϵ\right)>0,Re\left(\omega \right)>0$ and $y>0$, the following formula holds true

$$\begin{array}{cc}\hfill & {\int }_0^1{y}^{ϵ-1}{(1-y)}^{2\omega -1}{\left(1-\frac{y}{3}\right)}^{2ϵ-1}{\left(1-\frac{y}{4}\right)}^{\omega -1}{S}_c^d\left(v\left(1-\frac{y}{4}\right){(1-y)}^2\right)\hfill \\ \hfill & \times R_{\varsigma ,\tau }^{\omega ,\epsilon }\left(v\left(1-\frac{y}{4}\right){(1-y)}^2\right)dy={\left(\frac{2}{3}\right)}^{2ϵ}\frac{\mathsf{\Gamma }\left(ϵ\right)\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{y}^k\hfill \\ \hfill & \times {}_2{\psi }_3\left[\begin{array}{c}(\omega ,1),(\omega +k,1);\\ (\epsilon ,1),(\tau ,\varsigma ),(ϵ+\omega +k,1);\end{array}v\right].\hfill \end{array}$$

(58)

Proof. 

By using (1) and (2) in left hand side of (58) and denoting it by $\mathfrak{H}$. After some simplification, we obtain

$$\begin{array}{cc}\hfill & \mathfrak{H}=\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{v}^k\sum _{n=0}^{\infty }\frac{{\left(\omega \right)}_n}{{\left(\epsilon \right)}_n\mathsf{\Gamma }(\varsigma n+\tau )}\frac{v^n}{n!}\hfill \\ \hfill & \times {\int }_0^1{y}^{ϵ-1}{(1-y)}^{2(\omega +k+n)-1}{\left(1-\frac{y}{3}\right)}^{2ϵ-1}{\left(1-\frac{y}{4}\right)}^{\omega +k+n}dy.\hfill \end{array}$$

(59)

Now, applying the integral Formula (24) to (59), we obtain

$$\begin{array}{cc}\hfill & \mathfrak{H}=\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{v}^k\sum _{n=0}^{\infty }\frac{{\left(\omega \right)}_n}{{\left(\epsilon \right)}_n\mathsf{\Gamma }(\varsigma n+\tau )}\frac{v^n}{n!}\hfill \\ \hfill & \times {\left(\frac{2}{3}\right)}^{2ϵ}\frac{\mathsf{\Gamma }\left(ϵ\right)\mathsf{\Gamma }(\omega +k+n)}{\mathsf{\Gamma }(ϵ+\omega +k+n)},\hfill \end{array}$$

(60)

which further implies

$$\begin{array}{cc}\hfill & \mathfrak{H}={\left(\frac{2}{3}\right)}^{2ϵ}\frac{\mathsf{\Gamma }\left(ϵ\right)\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{y}^k\hfill \\ \hfill & \times \sum _{n=0}^{\infty }\frac{\mathsf{\Gamma }(\omega +n)\mathsf{\Gamma }(\omega +k+n)}{\mathsf{\Gamma }(\epsilon +n)\mathsf{\Gamma }(\tau +\varsigma n)\mathsf{\Gamma }(ϵ+\omega +k+n)}\frac{v^n}{n!}.\hfill \end{array}$$

(61)

Now, by using definition (7), we arrive at the proof of the theorem. □

Theorem 6.

For $\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda ,\omega \in \mathbb{C}$, $Re\left(ϵ\right)>0,Re\left(\omega \right)>0$ and $y>0$, the following formula holds true

$$\begin{array}{cc}\hfill & {\int }_0^1{y}^{ϵ-1}{(1-y)}^{2\omega -1}{\left(1-\frac{y}{3}\right)}^{2ϵ-1}{\left(1-\frac{y}{4}\right)}^{\omega -1}{S}_c^d\left(vy\left(1-\frac{y}{4}\right){(1-y)}^2\right)\hfill \\ \hfill & \times R_{\varsigma ,\tau }^{\omega ,\epsilon }\left(vy\left(1-\frac{y}{4}\right){(1-y)}^2\right)dy={\left(\frac{2}{3}\right)}^{2ϵ}\frac{\mathsf{\Gamma }\left(ϵ\right)\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\frac{4v}{9}\right)}^k\hfill \\ \hfill & \times {}_2{\psi }_3\left[\begin{array}{c}(\omega ,1),(ϵ+k,1);\\ (\epsilon ,1),(\tau ,\varsigma ),(ϵ+\omega +k,1);\end{array}\frac{4v}{9}\right].\hfill \end{array}$$

(62)

Proof. 

By using (1) and (2) in the left hand side of (62) and representing it by $\mathfrak{J}$. After some simplification, we obtain

$$\begin{array}{cc}\hfill & \mathfrak{J}=\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{v}^k\sum _{n=0}^{\infty }\frac{{\left(\omega \right)}_n}{{\left(\epsilon \right)}_n\mathsf{\Gamma }(\varsigma n+\tau )}\frac{v^n}{n!}\hfill \\ \hfill & \times {\int }_0^1{y}^{ϵ+k+n-1}{(1-y)}^{2\omega -1}{\left(1-\frac{y}{3}\right)}^{2(ϵ+k+n)-1}{\left(1-\frac{y}{4}\right)}^{\omega }dy.\hfill \end{array}$$

(63)

Now, applying the integral Formula (24) to (63), we obtain

$$\begin{array}{cc}\hfill & \mathfrak{J}=\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{v}^k\sum _{n=0}^{\infty }\frac{{\left(\omega \right)}_n}{{\left(\epsilon \right)}_n\mathsf{\Gamma }(\varsigma n+\tau )}\frac{v^n}{n!}\hfill \\ \hfill & \times {\left(\frac{2}{3}\right)}^{2(ϵ+k+n)}\frac{\mathsf{\Gamma }\left(\omega \right)\mathsf{\Gamma }(ϵ+k+n)}{\mathsf{\Gamma }(ϵ+\omega +k+n)}\hfill \end{array}$$

(64)

which further implies

$$\begin{array}{cc}\hfill & \mathfrak{J}={\left(\frac{2}{3}\right)}^{2ϵ}\frac{\mathsf{\Gamma }\left(\omega \right)\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}^k{\left(\frac{4v}{9}\right)}^k\hfill \\ \hfill & \times \sum _{n=0}^{\infty }\frac{\mathsf{\Gamma }(\omega +n)\mathsf{\Gamma }(\omega +k+n)}{\mathsf{\Gamma }(\epsilon +n)\mathsf{\Gamma }(\tau +\varsigma n)\mathsf{\Gamma }(ϵ+\omega +k+n)}\frac{{\left(\frac{4v}{9}\right)}^n}{n!}.\hfill \end{array}$$

(65)

Now, by using definition (7), we arrive at the proof of the theorem. □

In next corollaries, we discuss some special cases of extended Wright function.

Corollary 13.

Assume that condition of Theorem 5 is fulfilled then using generalized Wright-type Function (3), the Formula (58) reduces to the following form.

$$\begin{array}{cc}\hfill & {\int }_0^1{y}^{ϵ-1}{(1-y)}^{2\omega -1}{\left(1-\frac{y}{3}\right)}^{2ϵ-1}{\left(1-\frac{y}{4}\right)}^{\omega -1}{S}_c^d\left(v\left(1-\frac{y}{4}\right){(1-y)}^2\right)\hfill \\ \hfill & \times M_{-\varsigma ,1-\varsigma }^{\omega ,\epsilon }\left(-v\left(1-\frac{y}{4}\right){(1-y)}^2\right)dy={\left(\frac{2}{3}\right)}^{2ϵ}\frac{\mathsf{\Gamma }\left(ϵ\right)\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{y}^k\hfill \\ \hfill & \times {}_2{\psi }_3\left[\begin{array}{c}(\omega ,1),(\omega +k,1);\\ (\epsilon ,1),(1-\varsigma ,-\varsigma ),(ϵ+\omega +k,1);\end{array}-v\right].\hfill \end{array}$$

(66)

Corollary 14.

Assume that condition of Theorem 6 is fulfilled then using generalized Wright-type Function (3), the Formula (62) reduces to the following form.

$$\begin{array}{cc}\hfill & {\int }_0^1{y}^{ϵ-1}{(1-y)}^{2\omega -1}{\left(1-\frac{y}{3}\right)}^{2ϵ-1}{\left(1-\frac{y}{4}\right)}^{\omega -1}{S}_c^d\left(vy\left(1-\frac{y}{4}\right){(1-y)}^2\right)\hfill \\ \hfill & \times M_{-\varsigma ,1-\varsigma }^{\omega ,\epsilon }\left(-vy\left(1-\frac{y}{4}\right){(1-y)}^2\right)dy={\left(\frac{2}{3}\right)}^{2ϵ}\frac{\mathsf{\Gamma }\left(ϵ\right)\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\frac{4v}{9}\right)}^k\hfill \\ \hfill & \times {}_2{\psi }_3\left[\begin{array}{c}(\omega ,1),(ϵ+k,1);\\ (\epsilon ,1),(1-\varsigma ,-\varsigma ),(ϵ+\omega +k,1);\end{array}\frac{-4v}{9}\right].\hfill \end{array}$$

(67)

Corollary 15.

Assume that condition of Theorem 5 is fulfilled then using generalized Wright-type Function (4), the Formula (58) reduces to the following form.

$$\begin{array}{cc}\hfill & {\int }_0^1{y}^{ϵ-1}{(1-y)}^{2\omega -1}{\left(1-\frac{y}{3}\right)}^{2ϵ-1}{\left(1-\frac{y}{4}\right)}^{\omega -1}{S}_c^d\left(v\left(1-\frac{y}{4}\right){(1-y)}^2\right)\hfill \\ \hfill & \times F_{-\varsigma }^{\omega ,\epsilon }\left(-v(1-\frac{y}{4}){(1-y)}^2\right)dy={\left(\frac{2}{3}\right)}^{2ϵ}\frac{\mathsf{\Gamma }\left(ϵ\right)\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{y}^k\hfill \\ \hfill & \times {}_2{\psi }_3\left[\begin{array}{c}(\omega ,1),(\omega +k,1);\\ (\epsilon ,1),(0,-\varsigma ),(ϵ+\omega +k,1);\end{array}-v\right].\hfill \end{array}$$

(68)

Corollary 16.

Assume that condition of Theorem 6 is fulfilled then using generalized Wright-type Function (4), the Formula (62) reduces to the following form.

$$\begin{array}{cc}\hfill & {\int }_0^1{y}^{ϵ-1}{(1-y)}^{2\omega -1}{\left(1-\frac{y}{3}\right)}^{2ϵ-1}{\left(1-\frac{y}{4}\right)}^{\omega -1}{S}_c^d\left(vy\left(1-\frac{y}{4}\right){(1-y)}^2\right)\hfill \\ \hfill & \times F_{-\varsigma }^{\omega ,\epsilon }\left(-vy\left(1-\frac{y}{4}\right){(1-y)}^2\right)dy={\left(\frac{2}{3}\right)}^{2ϵ}\frac{\mathsf{\Gamma }\left(ϵ\right)\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\frac{4v}{9}\right)}^k\hfill \\ \hfill & \times {}_2{\psi }_3\left[\begin{array}{c}(\omega ,1),(ϵ+k,1);\\ (\epsilon ,1),(0,-\varsigma ),(ϵ+\omega +k,1);\end{array}\frac{-4v}{9}\right].\hfill \end{array}$$

(69)

Corollary 17.

Assume that condition of Theorem 5 is fulfilled then using normalized Wright Function (5), the Formula (58) reduces to the following form.

$$\begin{array}{cc}\hfill & {\int }_0^1{y}^{ϵ-1}{(1-y)}^{2\omega -1}{\left(1-\frac{y}{3}\right)}^{2ϵ-1}{\left(1-\frac{y}{4}\right)}^{\omega -1}{S}_c^d\left(v\left(1-\frac{y}{4}\right){(1-y)}^2\right)\hfill \\ \hfill & \times R_{\varsigma ,\tau }\left(v(1-\frac{y}{4}){(1-y)}^2\right)dy={\left(\frac{2}{3}\right)}^{2ϵ}\mathsf{\Gamma }\left(ϵ\right)\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{y}^k\hfill \\ \hfill & \times {}_1{\psi }_2\left[\begin{array}{c}(\omega +k,1);\\ (\tau ,\varsigma ),(ϵ+\omega +k,1);\end{array}v\right].\hfill \end{array}$$

(70)

Corollary 18.

Assume that condition of Theorem 6 is fulfilled then using normalized Wright Function (5), the Formula (62) reduces to the following form.

$$\begin{array}{cc}\hfill & {\int }_0^1{y}^{ϵ-1}{(1-y)}^{2\omega -1}{\left(1-\frac{y}{3}\right)}^{2ϵ-1}{\left(1-\frac{y}{4}\right)}^{\omega -1}{S}_c^d\left(vy\left(1-\frac{y}{4}\right){(1-y)}^2\right)\hfill \\ \hfill & \times R_{\varsigma ,\tau }\left(vy\left(1-\frac{y}{4}\right){(1-y)}^2\right)dy={\left(\frac{2}{3}\right)}^{2ϵ}\mathsf{\Gamma }\left(ϵ\right)\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\frac{4v}{9}\right)}^k\hfill \\ \hfill & \times {}_1{\psi }_2\left[\begin{array}{c}(ϵ+k,1);\\ (\tau ,\varsigma ),(ϵ+\omega +k,1);\end{array}\frac{4v}{9}\right].\hfill \end{array}$$

(71)

5. Oberhettinger Integral Formulas Involving Product of Srivastava’s Polynomials and Extended Wright Function

The Oberhettinger’s integral formulas involving the product of Srivastava’s Polynomials (1) and extended Wright function (2) are established in this section.

$${\int }_0^{\infty }{y}^{\varphi -1}{(y+b+\sqrt{y^2+2by})}^{-\chi }dy=2\chi b^{-\chi }{\left(\frac{b}{2}\right)}^{\varphi }\frac{\mathsf{\Gamma }\left(2\varphi \right)\mathsf{\Gamma }(\chi -\varphi )}{\mathsf{\Gamma }(1+\chi +\varphi )},$$

(72)

where $0<Re\left(\varphi \right)<Re\left(\chi \right)$.

Theorem 7.

For $y>0$, $\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda ,\omega \in \mathbb{C}$, $0<Re\left(\varphi \right)<Re\left(\chi \right)$ the following formula holds true

$$\begin{array}{cc}\hfill & {\int }_0^{\infty }{y}^{\varphi -1}{(y+b+\sqrt{y^2+2by})}^{-\chi }{S}_c^d\left(\frac{v}{(y+b+\sqrt{y^2+2by})}\right)\hfill \\ \hfill & \times R_{\varsigma ,\tau }^{\omega ,\epsilon }\left(\frac{v}{(y+b+\sqrt{y^2+2by})}\right)dy\hfill \\ \hfill & =b^{\varphi -\chi }{2}^{1-\varphi }\frac{\mathsf{\Gamma }\left(2\varphi \right)\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\frac{v}{b}\right)}^k\hfill \\ \hfill & \times {}_3{\psi }_4\left[\begin{array}{c}(\omega ,1),(\chi +k+1,1),(\chi +\varphi +k,1);\\ (\epsilon ,1),(\tau ,\varsigma ),(\chi +k,1),(1+\chi +\varphi +k,1);\end{array}|\frac{v}{b}\right].\hfill \end{array}$$

(73)

Proof. 

By using (1) and (2) in the left hand side of (73) and representing it by $\mathfrak{U}$. After some simplification, we obtain

$$\begin{array}{cc}\hfill & \mathfrak{U}=\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{v}^k\sum _{n=0}^{\infty }\frac{{\left(\omega \right)}_n}{{\left(\epsilon \right)}_n\mathsf{\Gamma }(\varsigma n+\tau )}\frac{v^n}{n!}\hfill \\ \hfill & \times {\int }_0^{\infty }{y}^{\varphi -1}{(y+b+\sqrt{y^2+2by})}^{-(\chi +k+n)}dy.\hfill \end{array}$$

(74)

Now, applying the integral Formula (27) to (74), we obtain

$$\begin{array}{cc}\hfill & \mathfrak{U}=\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{v}^k\sum _{n=0}^{\infty }\frac{{\left(\omega \right)}_n}{{\left(\epsilon \right)}_n\mathsf{\Gamma }(\varsigma n+\tau )}\frac{v^n}{n!}\hfill \\ \hfill & \times 2(\chi +k+n)b^{-(\chi +k+n)}{\left(\frac{b}{2}\right)}^{\varphi }\frac{\mathsf{\Gamma }\left(2\varphi \right)\mathsf{\Gamma }(\chi -\varphi +k+n)}{\mathsf{\Gamma }(1+\chi +\varphi +k+n)},\hfill \end{array}$$

(75)

which implies

$$\begin{array}{cc}\hfill & \mathfrak{U}=b^{\varphi -\chi }{2}^{1-\varphi )}\frac{\mathsf{\Gamma }\left(2\varphi \right)\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(vb\right)}^k\hfill \\ \hfill & \times \frac{\mathsf{\Gamma }(\omega +n)\mathsf{\Gamma }(\chi +k+1+n)\mathsf{\Gamma }(\chi -\varphi +k+n)}{\mathsf{\Gamma }(\epsilon +n)\mathsf{\Gamma }(\tau +\varsigma n)\mathsf{\Gamma }(1+\chi +\varphi +k+n)}\frac{{\left(\frac{v}{b}\right)}^n}{n!}.\hfill \end{array}$$

(76)

In accordance with definition (7), we arrive at the formula (73). □

Theorem 8.

For $y>0$, $\xi ,\acute{\xi },\eta ,\acute{\eta },\lambda ,\omega \in \mathbb{C}$ $0<Re\left(\varphi \right)<Re\left(\chi \right)$ the following formula holds true

$$\begin{array}{cc}\hfill & {\int }_0^{\infty }{y}^{\varphi -1}{(y+b+\sqrt{y^2+2by})}^{-\chi }{S}_c^d\left(\frac{vy}{(y+b+\sqrt{y^2+2by})}\right)\hfill \\ \hfill & \times R_{\varsigma ,\tau }^{\omega ,\epsilon }\left(\frac{vy}{(y+b+\sqrt{y^2+2by})}\right)dy\hfill \\ \hfill & =b^{\varphi -\chi }{2}^{1-\varphi }\frac{\mathsf{\Gamma }(\chi -\varphi )\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\frac{v}{2}\right)}^k\hfill \\ \hfill & \times {}_3{\psi }_4\left[\begin{array}{c}(\omega ,1),(\chi +k+1,1),(2\varphi +2k,2);\\ (\epsilon ,1),(\tau ,\varsigma ),(\chi +k,1),(1+\chi +\varphi +2k,2);\end{array}\frac{v}{2}\right].\hfill \end{array}$$

(77)

Proof. 

By using (1) and (2) in the left hand side of (77) and representing it $\mathfrak{E}$. On some simplification, we obtain

$$\begin{array}{cc}\hfill & \mathfrak{E}=\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{v}^k\sum _{n=0}^{\infty }\frac{{\left(\omega \right)}_n}{{\left(\epsilon \right)}_n\mathsf{\Gamma }(\varsigma n+\tau )}\frac{v^n}{n!}\hfill \\ \hfill & \times {\int }_0^{\infty }{y}^{\varphi +k+n-1}{(y+b+\sqrt{y^2+2by})}^{-(\chi +k+n)}dy.\hfill \end{array}$$

(78)

Now, applying the integral Formula (27) to (78), we obtain

$$\begin{array}{cc}\hfill & \mathfrak{E}=\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{v}^k\sum _{n=0}^{\infty }\frac{{\left(\omega \right)}_n}{{\left(\epsilon \right)}_n\mathsf{\Gamma }(\varsigma n+\tau )}\frac{v^n}{n!}\hfill \\ \hfill & \times 2(\chi +k+n)b^{-(\chi +k+n)}{\left(\frac{b}{2}\right)}^{\varphi +k+n}\frac{\mathsf{\Gamma }(2\varphi +2k+2n)\mathsf{\Gamma }(\chi -\varphi )}{\mathsf{\Gamma }(1+\chi +\varphi +2k+2n)},\hfill \end{array}$$

(79)

which implies

$$\begin{array}{cc}\hfill & \mathfrak{E}=b^{\varphi -\chi }{2}^{1-\varphi }\frac{\mathsf{\Gamma }(\chi -\varphi )\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\frac{v}{2}\right)}^k\hfill \\ \hfill & \times \frac{\mathsf{\Gamma }(\omega +n)\mathsf{\Gamma }(\chi +k+1+n)\mathsf{\Gamma }(2\varphi +2k+2n)}{\mathsf{\Gamma }(\epsilon +n)\mathsf{\Gamma }(\tau +\varsigma n)\mathsf{\Gamma }(chi+k+n)\mathsf{\Gamma }(1+\chi +\varphi +2kk+2n)}\frac{{\left(\frac{v}{2}\right)}^n}{n!}.\hfill \end{array}$$

(80)

In accordance with definition (7), we arrive at the formula (77). □

In the following corollaries, we present some special cases involving the extended Wright function.

Corollary 19.

Assume that condition of Theorem 7 is fulfilled then using generalized Wright-type Function (3), the Formula (73) reduces to the following form.

$$\begin{array}{cc}\hfill & {\int }_0^{\infty }{y}^{\varphi -1}{(y+b+\sqrt{y^2+2by})}^{-\chi }{S}_c^d\left(\frac{v}{(y+b+\sqrt{y^2+2by})}\right)\hfill \\ \hfill & \times M_{-\varsigma ,1-\varsigma }^{\omega ,\epsilon }\left(\frac{-v}{(y+b+\sqrt{y^2+2by})}\right)dy\hfill \\ \hfill & =b^{\varphi -\chi }{2}^{1-\varphi }\frac{\mathsf{\Gamma }\left(2\varphi \right)\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\frac{v}{b}\right)}^k\hfill \\ \hfill & \times {}_3{\psi }_4\left[\begin{array}{c}(\omega ,1),(\chi +k+1,1),(\chi +\varphi +k,1);\\ (\epsilon ,1),(1-\varsigma ,-\varsigma ),(\chi +k,1),(1+\chi +\varphi +k,1);\end{array}\frac{-v}{b}\right].\hfill \end{array}$$

(81)

Corollary 20.

Assume that condition of Theorem 8 is fulfilled then using generalized Wright-type Function (3), the Formula (77) reduces to the following form.

$$\begin{array}{cc}\hfill & {\int }_0^{\infty }{y}^{\varphi -1}{(y+b+\sqrt{y^2+2by})}^{-\chi }{S}_c^d\left(\frac{vy}{(y+b+\sqrt{y^2+2by})}\right)\hfill \\ \hfill & \times M_{-\varsigma ,1-\varsigma }^{\omega ,\epsilon }\left(\frac{-vy}{(y+b+\sqrt{y^2+2by})}\right)dy\hfill \\ \hfill & =b^{\varphi -\chi }{2}^{1-\varphi }\frac{\mathsf{\Gamma }(\chi -\varphi )\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\frac{v}{2}\right)}^k\hfill \\ \hfill & \times {}_3{\psi }_4\left[\begin{array}{c}(\omega ,1),(\chi +k+1,1),(2\varphi +2k,2);\\ (\epsilon ,1),(1-\varsigma ,-\varsigma ),(\chi +k,1),(1+\chi +\varphi +2k,2);\end{array}\frac{-v}{2}\right].\hfill \end{array}$$

(82)

Corollary 21.

Assume that condition of Theorem 7 is fulfilled then using generalized Wright-type Function (4), the Formula (73) reduces to the following form.

$$\begin{array}{cc}\hfill & {\int }_0^{\infty }{y}^{\varphi -1}{(y+b+\sqrt{y^2+2by})}^{-\chi }{S}_c^d\left(\frac{v}{(y+b+\sqrt{y^2+2by})}\right)\hfill \\ \hfill & \times F_{-\varsigma }^{\omega ,\epsilon }\left(\frac{-v}{(y+b+\sqrt{y^2+2by})}\right)dy\hfill \\ \hfill & =b^{\varphi -\chi }{2}^{1-\varphi }\frac{\mathsf{\Gamma }\left(2\varphi \right)\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\frac{v}{b}\right)}^k\hfill \\ \hfill & \times {}_3{\psi }_4\left[\begin{array}{c}(\omega ,1),(\chi +k+1,1),(\chi +\varphi +k,1);\\ (\epsilon ,1),(0,-\varsigma ),(\chi +k,1),(1+\chi +\varphi +k,1);\end{array}\frac{-v}{b}\right].\hfill \end{array}$$

(83)

Corollary 22.

Assume that condition of Theorem 8 is fulfilled then using generalized Wright-type Function (4), the Formula (77) reduces to the following form.

$$\begin{array}{cc}\hfill & {\int }_0^{\infty }{y}^{\varphi -1}{(y+b+\sqrt{y^2+2by})}^{-\chi }{S}_c^d\left(\frac{vy}{(y+b+\sqrt{y^2+2by})}\right)\hfill \\ \hfill & \times F_{-\varsigma }^{\omega ,\epsilon }\left(\frac{-vy}{(y+b+\sqrt{y^2+2by})}\right)dy\hfill \\ \hfill & =b^{\varphi -\chi }{2}^{1-\varphi }\frac{\mathsf{\Gamma }(\chi -\varphi )\mathsf{\Gamma }\left(\epsilon \right)}{\mathsf{\Gamma }\left(\omega \right)}\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\frac{v}{2}\right)}^k\hfill \\ \hfill & \times {}_3{\psi }_4\left[\begin{array}{c}(\omega ,1),(\chi +k+1,1),(2\varphi +2k,2);\\ (\epsilon ,1),(0,-\varsigma ),(\chi +k,1),(1+\chi +\varphi +2k,2);\end{array}\frac{-v}{2}\right].\hfill \end{array}$$

(84)

Corollary 23.

Assume that condition of Theorem 7 is fulfilled then using generalized Wright-type Function (5), the Formula (73) reduces to the following form.

$$\begin{array}{cc}\hfill & {\int }_0^{\infty }{y}^{\varphi -1}{(y+b+\sqrt{y^2+2by})}^{-\chi }{S}_c^d\left(\frac{v}{(y+b+\sqrt{y^2+2by})}\right)\hfill \\ \hfill & \times R_{\varsigma ,\tau }\left(\frac{-v}{(y+b+\sqrt{y^2+2by})}\right)dy\hfill \\ \hfill & =b^{\varphi -\chi }{2}^{1-\varphi }\mathsf{\Gamma }\left(2\varphi \right)\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\frac{v}{b}\right)}^k\hfill \\ \hfill & \times {}_2{\psi }_3\left[\begin{array}{c}(\chi +k+1,1),(\chi +\varphi +k,1);\\ (\tau ,\varsigma ),(\chi +k,1),(1+\chi +\varphi +k,1);\end{array}\frac{-v}{b}\right].\hfill \end{array}$$

(85)

Corollary 24.

Assume that condition of Theorem 8 is fulfilled then using normalized Wright Function (5), the Formula (77) reduces to the following form.

$$\begin{array}{cc}\hfill & {\int }_0^{\infty }{y}^{\varphi -1}{(y+b+\sqrt{y^2+2by})}^{-\chi }{S}_c^d\left(\frac{vy}{(y+b+\sqrt{y^2+2by})}\right)\hfill \\ \hfill & \times R_{\varsigma ,\tau }\left(\frac{vy}{(y+b+\sqrt{y^2+2by})}\right)dy\hfill \\ \hfill & =b^{\varphi -\chi }{2}^{1-\varphi }\mathsf{\Gamma }(\chi -\varphi )\sum _{k=0}^{\left[\frac{d}{c}\right]}\frac{{(-c)}_{dk}}{k!}{G}_{c,k}{\left(\frac{v}{2}\right)}^k\hfill \\ \hfill & \times {}_2{\psi }_3\left[\begin{array}{c}(\chi +k+1,1),(2\varphi +2k,2);\\ (\tau ,\varsigma ),(\chi +k,1),(1+\chi +\varphi +2k,2);\end{array}\frac{v}{2}\right].\hfill \end{array}$$

(86)

6. Conclusions

The present study is based on a well-known technique to explore certain general formulae involving special functions by skilfully utilizing the different integral and differential operators. In this article, we established new integral and differential formulas involving the product of Srivastava’s polynomial and extended Wright function. The main consequences are presented in terms of Fox–Wright hypergeometric function. Unified integral representations of some of its special cases are also derived. The extended Wright function is related to Mittag–Leffler function (9), Meijer G-function (10) and Fox H-function (11), therefore all obtained results can be expressed in the form of these functions as well. Moreover, the general class of polynomials gives many known classical orthogonal polynomials as special cases for given suitable values for the coefficient $G_{c,k}$. The Hermite, Laguerre, Jacobi, and Konhauser polynomials are only a few examples. In continuation of this study, one can obtain the integral representation of more generalized special functions that have applicability in physics and engineering sciences.