A note on the Laplace transforms of certain generalized fractional integral operators

TL;DR

This paper derives formulas for the Laplace transforms of two recently introduced generalized fractional integral operators, extending known results and providing insights into their properties.

Contribution

It introduces new formulas for Laplace transforms of generalized fractional integrals, broadening the understanding of these operators.

Findings

Derived Laplace transform formulas for two generalized fractional integrals

Generalized several known results in fractional calculus

Provided remarks on the implications of these formulas

Abstract

In this paper, we derive certain formulas giving the Laplace transforms of two generalized fractional integral operators introduced recently in [Fract. Calc. Appl. Anal. 20 (2) (2017), 422--446]. The main results provide generalizations to various known results. Some useful remarks related to the results presented in this paper are also mentioned.

Equations

$${\left(a\right)}_k:=\frac{\Gamma \left(a+k\right)}{\Gamma \left(a\right)}=\{\begin{array}{ll}1& \left(k=0;\text{leavevmode}a\in \mathbb{C}\setminus \{0\}\right),\\ a\left(a+1\right)\cdots \left(a+k-1\right)& \left(k\in \mathbb{N};\text{leavevmode}a\in \mathbb{C}\right).\end{array}$$

$${}_p{F}_q\left[\begin{array}{c}(a_p)\\ (b_q)\end{array};z\right]:=\sum _{k=0}^{\infty }\frac{((a_p){)}_k}{((b_p){)}_k}\frac{z^k}{k!}\text{leavevmode}(|z|<1).$$

$$\left(\mathcal{I}\phi \right)(x)$$

$$:=\frac{\nu x^{-\delta -\nu (\mu +h)}}{\Gamma (\mu )}{\int }_0^x(x^{\nu }-s^{\nu }{)}^{\mu -1}{}_{r+2}{F}_{r+1}\left[\begin{array}{c}a,b,\\ \mu ,\end{array}\text{leavevmode}\begin{array}{c}(f_r+m_r)\\ (f_r)\end{array};1-\frac{s^{\nu }}{x^{\nu }}\right]\phi (s)s^{\nu h+\nu -1}\mathrm{d}s,$$

$$\left(\mathcal{J}\phi \right)(x)$$

$$:=\frac{\nu x^{\nu h+\nu -1}}{\Gamma (\mu )}{\int }_x^{\infty }(s^{\nu }-x^{\nu }{)}^{\mu -1}{}_{r+2}{F}_{r+1}\left[\begin{array}{c}a,b,\\ \mu ,\end{array}\text{leavevmode}\begin{array}{c}(f_r+m_r)\\ (f_r)\end{array};1-\frac{x^{\nu }}{s^{\nu }}\right]\phi (s)s^{-\delta -\nu (\mu +h)}\mathrm{d}s.$$

$$\mathcal{L}[x^{\lambda }\mathcal{I}\phi ](s)\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{and}\text{leavevmode}\text{leavevmode}\text{leavevmode}\mathcal{L}[x^{\lambda }\mathcal{J}\phi ](s),$$

$$\mathcal{L}[\phi ](s):={\int }_0^{\infty }{\mathrm{e}}^{-st}\phi (t)\mathrm{d}t.$$

$$\begin{array}{rl}{\mathfrak{c}}_1(t)& :=1+h+\frac{t}{\nu },\\ {\mathfrak{c}}_2(t)& :={\mathfrak{c}}_1(\delta -1)-\frac{t}{\nu }.\end{array}\text{leavevmode}\text{leavevmode}\}$$

$${\mathfrak{p}}_k:=\mu -a-b-k.$$

$$A_k=\sum _{j=k}^m\left\{\genfrac{}{}{0ex}{}{j}{k}\right\}{\sigma }_{m-j},\text{leavevmode}\text{leavevmode}{A}_0=(f_1{)}_{m_1}\cdots (f_r{)}_{m_r},\text{leavevmode}\text{leavevmode}{A}_m=1,$$

$$(f_1+x{)}_{m_1}\cdots (f_r+x{)}_{m_r}=\sum _{j=0}^m{\sigma }_{m-j}{x}^j.$$

$$\mathcal{I}{x}^{\lambda }=x^{\lambda -\delta }\sum _{k=0}^m\frac{A_k}{A_0}\frac{(a{)}_k(b{)}_k\Gamma ({\mathfrak{c}}_1(\lambda ))\Gamma ({\mathfrak{c}}_1(\lambda )+{\mathfrak{p}}_k)}{\Gamma ({\mathfrak{c}}_1(\lambda )+\mu -a)\Gamma ({\mathfrak{c}}_1(\lambda )+\mu -b)},$$

$$\mathcal{J}{x}^{\lambda }=x^{\lambda -\delta }\sum _{k=0}^m\frac{A_k}{A_0}\frac{(a{)}_k(b{)}_k\Gamma ({\mathfrak{c}}_2(\lambda ))\Gamma ({\mathfrak{c}}_2(\lambda )+{\mathfrak{p}}_k)}{\Gamma ({\mathfrak{c}}_2(\lambda )+\mu -a)\Gamma ({\mathfrak{c}}_2(\lambda )+\mu -b)},$$

$$H_{p,q}^{m,n}\left[z|\begin{array}{c}(a_i,A_i{)}_{1,p}\\ (b_j,B_j{)}_{1,q}\end{array}\right]$$

$$=\frac{1}{2\pi \mathrm{i}}{\int }_L\frac{\prod _{k=1}^m\Gamma (B_{k}s+b_k)\prod _{i=1}^n\Gamma (1-a_i-A_{i}s)}{\prod _{k=m+1}^q\Gamma (1-b_k-B_{k}s)\prod _{i=n+1}^p\Gamma (A_{i}s+a_i)}{z}^{-s}\mathrm{d}s,$$

$$a^{\ast }$$

$$\Delta$$

$${\delta }^{\ast }$$

$${\mu }^{\ast }$$

$$a_1^{\ast }$$

$$G_{p,q}^{m,n}\left[z|\begin{array}{c}(a_i,1{)}_{1,p}\\ (b_j,1{)}_{1,q}\end{array}\right]$$

$$=\frac{1}{2\pi \mathrm{i}}{\int }_L\frac{\prod _{k=1}^m\Gamma (s+b_k)\prod _{i=1}^n\Gamma (1-a_i-s)}{\prod _{k=m+1}^q\Gamma (1-b_k-s)\prod _{i=n+1}^p\Gamma (s+a_i)}{z}^{-s}\mathrm{d}s,$$

$$H_{p,q+1}^{1,p}\left[z|\begin{array}{c}(1-a_i,{\alpha }_i{)}_{1,p}\\ (0,1),(1-b_j,{\beta }_j{)}_{1,q}\end{array}\right]={}_p{\Psi }_q\left[\begin{array}{c}(a_j,{\alpha }_j{)}_{1,p}\\ (b_j,{\beta }_j{)}_{1,q}\end{array}|-z\right],$$

$${}_p{\Psi }_q\left[\begin{array}{c}(a_j,{\alpha }_j{)}_{1,p}\\ (b_j,{\beta }_j{)}_{1,q}\end{array}|z\right]:=\sum _{k=0}^{\infty }\frac{\prod _{j=1}^p\Gamma (a_j+{\alpha }_{j}k)}{\prod _{j=1}^q\Gamma (b_j+{\beta }_{j}k)}\frac{z^k}{k!}.$$

$${\Delta }^{\prime }:=\sum _{j=1}^q{\beta }_j-\sum _{\ell =1}^p{\alpha }_{\ell }>-1,$$

$$\Vert \phi {\Vert }_1:={\int }_A|\phi (t)|\mathrm{d}t.$$

$$\Re ({\mathfrak{c}}_2(\lambda ))>-\min [0,\Re ({\mathfrak{p}}_m)].$$

$$\mathcal{L}[x^{\lambda }\mathcal{I}\phi ](s)={\int }_0^{\infty }{K}_{\mathcal{I}}(s,x)\phi (x)\mathrm{d}x,$$

$$K_{\mathcal{I}}(s,x)\equiv \mathcal{J}{x}^{\lambda }{\mathrm{e}}^{-sx}$$

$$\cdot H_{2,3}^{3,0}\left[sx|\begin{array}{c}({\mathfrak{c}}_2(\lambda )+\mu -a,1/\nu ),({\mathfrak{c}}_2(\lambda )+\mu -b,1/\nu )\\ (0,1),({\mathfrak{c}}_2(\lambda ),1/\nu ),({\mathfrak{c}}_2(\lambda )+{\mathfrak{p}}_k,1/\nu )\end{array}\right].$$

$$\mathcal{L}[x^{\lambda }\mathcal{I}\phi ](x)$$

$$\displaystyle\hskip 28.45274pt\cdot{}_{r+2}F_{r+1}\left[\begin{matrix}a,b,\\ \mu,\end{matrix}\leavevmode\nobreak\ \begin{matrix}(f_{r}+m_{r})\\ (f_{r})\end{matrix};1-\frac{t^{\nu}}{x^{\nu}}\right]\varphi(t)t^{\nu h+\nu-1}\mathrm{d}t\Bigg{\}}\mathrm{d}x$$

$$\displaystyle=\int_{0}^{\infty}\varphi(t)\Bigg{\{}\frac{\nu t^{\nu h+\nu-1}}{\Gamma(\mu)}\int_{t}^{\infty}(x^{\nu}-t^{\nu})^{\mu-1}$$

$$\displaystyle\hskip 28.45274pt\cdot{}_{r+2}F_{r+1}\left[\begin{matrix}a,b,\\ \mu,\end{matrix}\leavevmode\nobreak\ \begin{matrix}(f_{r}+m_{r})\\ (f_{r})\end{matrix};1-\frac{t^{\nu}}{x^{\nu}}\right]\mathrm{e}^{-sx}x^{\lambda-\delta-\nu(\mu+h)}\mathrm{d}x\Bigg{\}}\mathrm{d}t$$

$$={\int }_0^{\infty }\phi (t)(\mathcal{J}{x}^{\lambda }{\mathrm{e}}^{-sx})(t)\mathrm{d}x={\int }_0^{\infty }\phi (x)K_{\mathcal{I}}(s,x)\mathrm{d}x,$$

$$I$$

Full text

**Keywords: Fox-Wright function, Fractional integral operator, Laplace transform, $H$-function.

**

Mathematics Subject Classification (2020): 26A33; 33C60.

*1**Department of Mathematics, School of Mathematics and Statistics,

Donghua University, Shanghai 201620,

People’s Republic of China.*

E-mail: mathwinnie@live.com, mathwinnie@dhu.edu.cn

*2**Department of Mathematics, School of Mathematics and Statistics,

Donghua University, Shanghai 201620,

People’s Republic of China.*

E-mail: Shen031127@outlook.com

*3**M.P. University of Agriculture and Technology, Udaipur (Rajasthan), India

Present address: 10/11, Ganpati Vihar, Opposite Sector 5,

Udaipur-313002, Rajasthan, India.*

E-mail: rkraina1944@gmail.com; rkraina_7@hotmail.com

1 Introduction

The study of the Laplace transforms of various types of fractional integral and derivative operators is a basic and fundamental area of study in Fractional Calculus which has extensive applications in solving fractional integral and differential equations (see [2], [4, Chapter 5], [8] and [11, Chapter 4]). Since 2017, the first and third authors of this paper have published a series of papers ([5, 6, 7]) introducing a pair of fractional integral operators whose kernels involve a very special class of generalized hypergeometric function ${}_{r+2}F_{r+1}$ (see Definition 1 below). The Laplace transforms of these operators have so far not been investigated in detail, as suggested in [8, pp. 79–80]. The purpose of this paper is therefore to study the Laplace transformation of such generalized forms of fractional calculus operators.

For any $m\in\mathbb{Z}$, let $\mathbb{Z}_{\geq m}:=\{n\in\mathbb{Z};n\geq m\}$, and also let $\mathbb{R}_{>0}:=\{x\in\mathbb{R};x>0\}=(0,\infty)$. As usual, the Pochhammer symbol $(a)_{k}$ is defined by

[TABLE]

We shall use the convention of writing the finite sequence parameters $a_{1}$, $\cdots$, $a_{p}$ by $(a_{p})$ and the product of $p$ Prochhammer symbols by $((a_{p}))_{k}\equiv(a_{1})_{k}\cdots(a_{p})_{k}$, where an empty product $p=0$ is treated as unity. The generalized hypergeometric function ${}_{p}F_{q}$ is then defined by the series

[TABLE]

For its conditions of convergence and analytic continuation via Mellin-Barnes type integral, we refer the reader to [4, pp. 30–31] and [9].

Definition 1** ([5, p. 423]).**

Let $x,h,\nu\in\mathbb{R}_{>0}$, $\delta,a,b,f_{1},\cdots,f_{r}\in\mathbb{C}$ and $m_{1},\cdots,m_{r}\in\mathbb{Z}_{\geq 0}$. Also, let $\Re(\mu)>0$ and $\varphi(s)$ be a suitable complex-valued function defined on $\mathbb{R}_{>0}$. Then, the fractional integral of a function $\varphi(x)$ of the first kind is defined by

[TABLE]

and the fractional integral of the second kind of a function $\varphi\left(x\right)$ is defined by

[TABLE]

Operators (1) and (1) include many important known fractional integral operators as special cases, such as the Riemann-Liouville operators, the Erdélyi-Kober operators and the Saigo operators (see [7]).

In Section 2, we shall present some useful results regarding (1) and (1) and provide definitions of some special functions to be used later. In Section 3, we prove our main theorems, namely, the formulas for

[TABLE]

where $\lambda\in\mathbb{C}$ and $\mathcal{L}$ denotes the Laplace transform of the function $\varphi(t)$ defined by

[TABLE]

2 Preliminaries

Following the earlier work in [5, 6] and [7], we use for convenience sake the notations $\mathfrak{c}_{1}(t)$ and $\mathfrak{c}_{2}(t)$ given by

[TABLE]

Also, for $k\in\mathbb{Z}_{\geq 0}$, we define

[TABLE]

Throughout the present paper, the sequence $A_{k}$ $\left(0\leq k\leq m:=m_{1}+\cdots+m_{r}\right)$ is always defined by

[TABLE]

where ${j\brace k}$ denotes the Stirling numbers of the second kind and $\sigma_{j}$ $(0\leq j\leq m)$ are generated by the relation

[TABLE]

Lemma 2.1** ([5, p. 426]).**

Let $x,h,\nu\in\mathbb{R}_{>0}$, $\delta,\lambda,a,b,f_{1},\cdots,f_{r}\in\mathbb{C}$ and $m_{1},\cdots,m_{r}\in\mathbb{Z}_{\geq 0}$. Also, let $\mathfrak{c}_{1}(t)$, $\mathfrak{c}_{2}(t)$ and $\mathfrak{p}_{k}$ be defined, respectively, by (3) and (4). Then there holds the following formulas:

[TABLE]

provided that $\Re(\mu)>0$ and $\Re(\mathfrak{c}_{1}(\lambda))>-\min[0,\Re(\mathfrak{p}_{m})]$, and

[TABLE]

if $\Re(\mu)>0$ and $\Re(\mathfrak{c}_{2}(\lambda))>-\min[0,\Re(\mathfrak{p}_{m})],$ where $A_{k}$ $(0\leq k\leq m)$ is defined by (5).

Let $m,n,p,q\in\mathbb{Z}_{\geq 0}$, $0\leq m\leq q$, $0\leq n\leq p$ and let $a_{i},b_{j}\in\mathbb{C},A_{i},B_{j}\geq 0$, where $1\leq i\leq p,1\leq j\leq q$. The Fox $H$-function is defined by (see [3, p. 1] and [4, p. 58]):

[TABLE]

where $L$ is a suitable contour that separates the poles of $\Gamma\left(B_{k}s+b_{k}\right)$ from the poles of $\Gamma\left(1-a_{i}-A_{i}s\right)$. To further clarify the definition, we take $L=L_{\mathrm{i}c\infty}$, which is a contour starting at the point $c-\mathrm{i}\infty$ and terminating at the point $c+\mathrm{i}\infty$, where $c\in\mathbb{R}$. The properties of the $H$-function depend on the following indexes:

[TABLE]

When $A_{i}=B_{j}=1$ $(i=1,\cdots,p;j=1,\cdots,q)$, the $H$-function (2) reduces to Meijer’s $G$-function

[TABLE]

where $L$ is the same contour taken for the $H$-function defined in (2). We also have [4, p. 67, Eq. (1.12.68)]:

[TABLE]

where ${}_{p}\Psi_{q}$ denotes the Fox-Wright function defined by [4, p. 56, Eq. (1.11.14)]

[TABLE]

If

[TABLE]

then the series in (14) is absolutely convergent for all $z\in\mathbb{C}$ (see [4, p. 56]).

3 Main results

Let $L^{1}(A)$ be the space of all Lebesgue measurable, complex-valued functions $\varphi:A\rightarrow\mathbb{C}$ with finite norm

[TABLE]

Theorem 3.1**.**

Let the conditions in Definition 1 be satisfied and let

[TABLE]

Also, let $x^{\lambda-\delta}\varphi(x)\in L^{1}(\mathbb{R}_{>0})$. Then, for $\Re(s)>0$, we have

[TABLE]

where

[TABLE]

Proof.

By Fubini’s theorem, it is easy to see that

[TABLE]

provided that

[TABLE]

Note that

[TABLE]

Since $x^{\lambda-\delta}\varphi(x)\in L^{1}(\mathbb{R}_{>0})$ and $\Re(s)>0$, it is sufficient to guarantee the convergence of the integral

[TABLE]

Recall that

[TABLE]

as $z\rightarrow 0^{+}$, where $\psi_{p}:=\sum_{\ell=1}^{p}b_{\ell}-\sum_{\ell=1}^{p+1}a_{\ell}$ (see [1]). So we have

[TABLE]

as $u\rightarrow+\infty$, where $\mathfrak{p}_{m}$ is defined by (4). In addition,

[TABLE]

as $u\rightarrow 1^{+}$. The condition (16) is therefore obtained by ensuring the convergence of the integral

[TABLE]

We now evaluate $K_{\mathcal{I}}(s,x)\equiv\mathcal{J}x^{\lambda}\mathrm{e}^{-sx}$ given by (3.1). Let us express $\mathrm{e}^{-sx}$ as its Mellin-Barnes integral

[TABLE]

where $c>0$ and $|\arg(s)|<\pi/2$. Under the condition (16), we use (6) to obtain

[TABLE]

From the definition (2) of the $H$-function, we have

[TABLE]

The corresponding indexes (8)–(12) concerning the above $H$-function $H_{2,3}^{3,0}$ satisfies

[TABLE]

Using (22) and (23), we obtain (3.1) and the result (17) of Theorem 3.1 follows from (3.1) and (19). This completes the proof. ∎

Remark 3.2*.*

Since the kernel $K_{\mathcal{I}}(s,x)$ is a finite sum of $H$-functions, a direct analysis of its behaviour near zero and infinity would therefore be interesting to expedite. Under the conditions given in (24), we can use Corollary 1.10.1 of [3] to obtain

[TABLE]

and thus from (3.1) we have

[TABLE]

On the other hand, by using [4, p. 61, Eq. (1.12.23)], we obtain

[TABLE]

where $\rho_{k}^{*}:=\min[0,\nu\Re(\mathfrak{c}_{2}(\lambda)),\nu\Re(\mathfrak{c}_{2}(\lambda))+\nu\Re(\mathfrak{p}_{m})]$. Hence, it follows from (3.1) that

[TABLE]

In view of the condition (16), we observe that the expression $\nu\min[0,\Re(\mathfrak{c}_{2}(\lambda)),\Re(\mathfrak{c}_{2}(\lambda))+\Re(\mathfrak{p}_{m})]$ in (26) equals to zero, and consequently, we infer that

[TABLE]

Thus, the assertions (25) and (27) suggest the imposition of the condition $x^{\lambda-\delta}\varphi(x)\in L^{1}(\mathbb{R}_{>0})$ as stated in the hypothesis of Theorem 3.1.

Theorem 3.3**.**

Let the conditions in Definition 1 be satisfied, and let

[TABLE]

Also, let $x^{\lambda-\delta}\varphi(x)\in L^{1}(\mathbb{R}_{>0})$. Then, for $\Re(s)>0$, we have

[TABLE]

where

[TABLE]

Proof.

By Fubini’s theorem, we have

[TABLE]

provided that

[TABLE]

Note that

[TABLE]

Since $x^{\lambda-\delta}\varphi(x)\in L^{1}(\mathbb{R}_{>0})$ and $\Re(s)>0$, it is sufficient to guarantee the convergence of the integral

[TABLE]

In view of (20), we know that $J^{\prime}$ is finite if the condition (28) is satisfied.

Next, we evaluate $K_{\mathcal{J}}(s,x)\equiv\mathcal{I}x^{\lambda}\mathrm{e}^{-sx}$ involved in (31). Using the integral representation (21), we obtain

[TABLE]

In view of (2) and the reduction formula (13), we find that

[TABLE]

The relation (15) concerning the ${}_{2}\Psi_{2}$-function satisfies $\Delta^{\prime}=0$ and the indexes (8) and (9) satisfies

[TABLE]

Now combining (32) and (33), we get immediately (30) and the desired result (29) of Theorem 3.3 follows from (30) and (31). This proof completes the proof. ∎

Remark 3.4*.*

As in Remark 3.2, we give a direct analysis of the behaviour of the kernel $K_{\mathcal{J}}(s,x)$ involved in the integral operator (29). Using the formula [3, p. 11, Eq. (1.5.13)], we have

[TABLE]

where $\rho_{k}:=-\nu\min[\Re(\mathfrak{c}_{1}(\lambda)),\Re(\mathfrak{c}_{1}(\lambda))+\Re(\mathfrak{p}_{m})]$ and $|\arg(s)|<\pi/2$. Thus,

[TABLE]

On the other hand, we have

[TABLE]

and therefore

[TABLE]

The relations (3.4) and (35) therefore suggest the condition that $x^{\lambda-\delta}\varphi(x)\in L^{1}(\mathbb{R}_{>0})$, as given in the hypothesis of Theorem 3.3. It is worth mentioning here that the same asymptotic behaviour can also be obtained by using the theory of the Fox-Wright function; see Paris and Kaminski [10, p. 57, Case (i)].

Finally, we conclude this section by pointing out that Theorems 3.1 and 3.3 provide generalizations to the results of Srivastava et al. [12, p. 6, Theorem 3].