On Generalized Fractional Derivatives Involving Generalized k-Mittag Leffler Function

TL;DR

This paper introduces generalized fractional derivatives involving the k-Mittag-Leffler function and derives their transform formulas, providing a broad framework with various special cases for advanced mathematical analysis.

Contribution

It presents new generalized fractional derivative formulas involving the k-Mittag-Leffler function and establishes their transform representations, expanding the theoretical framework.

Findings

Derived generalized fractional derivative formulas with k-Mittag-Leffler function

Established image formulas using Beta, Laplace, and Whittaker transforms

Discussed special cases of the main results

Abstract

In this paper, certain generalized fractional derivative formulae are introduced involving the k-Mittag-Leffler function. Then their image formulae (using Beta transform, Laplace transform and Whittaker transform) are also established. The results obtained here are quite general in nature. The special cases of our findings are also discussed.

Figures

Tables

Table 1. Table 1: Numerical Values of the Equation ( 2.12 )

$x$	$\sigma =0.1$	$\sigma =0.2$	$\sigma =0.3$	$\sigma =0.4$
0.00	0.00	0.00	0.00	0.00
0.50	3.30	3.09	2.48	1.44
1.00	4.57	4.20	3.30	1.88
1.50	5.53	5.02	3.90	2.19
2.00	6.33	5.70	4.38	2.44
2.50	7.03	6.28	4.80	2.66
3.00	7.66	6.81	5.18	2.85
3.50	8.24	7.29	5.51	3.02
4.00	8.77	7.73	5.82	3.18
4.50	9.27	8.14	6.11	3.33
5.00	9.74	8.52	6.38	3.46
5.50	10.19	8.89	6.64	3.59
6.00	10.61	9.24	6.88	3.71
6.50	11.02	9.57	7.11	3.83
7.00	11.41	9.88	7.32	3.94
7.50	11.79	10.19	7.54	4.04
8.00	12.15	10.48	7.74	4.14
8.50	12.50	10.77	7.93	4.24
9.00	12.84	11.04	8.12	4.33
9.50	13.17	11.31	8.30	4.42
10.00	13.49	11.56	8.48	4.51

Table 2. Table 2: Numerical Values of the Equation ( 2.19 )

$x$	$\sigma =0.1$	$\sigma =0.2$	$\sigma =0.3$	$\sigma =0.4$
0	0	0	0	0
0.50	3.47	3.83	4.22	4.67
1.00	4.81	5.19	5.61	6.08
1.50	5.82	6.20	6.63	7.09
2.00	6.66	7.04	7.46	7.91
2.50	7.40	7.77	8.17	8.61
3.00	8.06	8.42	8.80	9.23
3.50	8.66	9.01	9.38	9.79
4.00	9.22	9.55	9.91	10.30
4.50	9.75	10.06	10.40	10.77
5.00	10.24	10.54	10.86	11.21
5.50	10.71	10.99	11.29	11.62
6.00	11.16	11.42	11.70	12.01
6.50	11.59	11.83	12.09	12.38
7.00	12.00	12.22	12.46	12.73
7.50	12.39	12.59	12.82	13.07
8.00	12.78	12.96	13.16	13.40
8.50	13.15	13.31	13.49	13.71
9.00	13.50	13.65	13.81	14.01
9.50	13.85	13.97	14.12	14.30
10.00	14.19	14.29	14.42	14.58

Equations

$$\begin{array}{rl}& (\vartheta {)}_{n,k}:=\{\begin{array}{rl}& \frac{{\Gamma }_k(\vartheta +nk)}{{\Gamma }_k(\vartheta )}(k\in \mathbb{R};\vartheta \in \mathbb{C}\setminus \{0\})\\ & \vartheta (\vartheta +k)...(\vartheta +(n-1)k)(n\in \mathbb{N};\vartheta \in \mathbb{C}),\end{array}\end{array}$$

$$\begin{array}{r}{\Gamma }_k(\vartheta )=k^{\frac{\vartheta }{k}-1}\Gamma \left(\frac{\vartheta }{k}\right),\end{array}$$

$$\begin{array}{rl}& {\Gamma }_s(\vartheta )={\left(\frac{s}{k}\right)}^{\frac{\vartheta }{s}-1}{\Gamma }_k\left(\frac{k\vartheta }{s}\right),\end{array}$$

$${\Gamma }_k(\vartheta )=k^{\frac{\vartheta }{k}-1}\Gamma \left(\frac{\vartheta }{k}\right).$$

$$(\vartheta {)}_{nq,s}={\left(\frac{s}{k}\right)}^{nq}{\left(\frac{k\vartheta }{s}\right)}_{nq},$$

$$(\vartheta {)}_{nq,k}=(k{)}^{nq}{\left(\frac{\vartheta }{k}\right)}_{nq}.$$

$$E_{k,\xi ,\zeta }^{\vartheta ,q}(z)=\sum _{n=0}^{\infty }\frac{(\vartheta {)}_{nq,k}{z}^n}{{\Gamma }_k(n\xi +\zeta )n!},$$

$$E_{k,\xi ,\zeta }^{\vartheta ,1}(z)=\sum _{n=0}^{\infty }\frac{(\vartheta {)}_{n,k}{z}^n}{{\Gamma }_k(n\xi +\zeta )n!}=E_{k,\xi ,\zeta }^{\vartheta }(z).$$

$$E_{1,\xi ,\zeta }^{\vartheta ,q}(z)=\sum _{n=0}^{\infty }\frac{(\vartheta {)}_{nq}{z}^n}{\Gamma (n\xi +\zeta )n!}=E_{\xi ,\zeta }^{\vartheta ,q}(z).$$

$$E_{1,\xi ,\zeta }^{\vartheta ,1}(z)=\sum _{n=0}^{\infty }\frac{(\vartheta {)}_n{z}^n}{\Gamma (n\xi +\zeta )n!}=E_{\xi ,\zeta }^{\vartheta }(z).$$

$$E_{1,\xi ,\zeta }^{1,1}(z)=\sum _{n=0}^{\infty }\frac{z^n}{\Gamma (n\xi +\zeta )}=E_{\xi ,\zeta }(z).$$

$$E_{1,\xi ,1}^{1,1}(z)=\sum _{n=0}^{\infty }\frac{z^n}{\Gamma (n\xi +1)}=E_{\xi }(z).$$

$$\displaystyle\begin{aligned} &{}_{p}\Psi_{q}[z]={}_{p}\Psi_{q}\left[\begin{array}[]{cc}(a_{1},\xi_{1}),...,(a_{p},\xi_{p});\\ (b_{1},\zeta_{1}),...,(b_{q},\zeta_{q});\end{array}z\right]\\ &\hskip 11.38109pt={}_{p}\Psi_{q}\left[\begin{array}[]{cc}(a_{i},\xi_{i})_{1,p};\\ (b_{j},\zeta_{j})_{1,q};\end{array}z\right]=\sum^{\infty}_{n=0}\frac{\prod^{p}_{i=1}\Gamma(a_{i}+\xi_{i}n)}{\prod^{q}_{j=1}\Gamma(b_{j}+\zeta_{j}n)}\frac{z^{n}}{n!},\end{aligned}$$

$$\begin{array}{rl}& 1+\sum _{j=1}^q{\zeta }_j-\sum _{i=1}^p{\xi }_i\ge 0.\end{array}$$

$$(I_{a+}^{\xi }f)(x)=\frac{1}{\Gamma (\xi )}{\int }_a^x(x-\tau {)}^{\xi -1}f(\tau )d\tau ;x>a$$

$$(I_{b-}^{\xi }f)(x)=\frac{1}{\Gamma (\xi )}{\int }_x^b(\tau -x{)}^{\xi -1}f(\tau )d\tau ;x<b,$$

$$\begin{array}{rl}& \left({}^{\eta }{I}_{a+}^{\sigma }f\right)(x)=\frac{(\eta {)}^{1-\sigma }}{\Gamma (\sigma )}{\int }_a^x\frac{t^{\eta -1}f(t)}{(x^{\eta }-t^{\eta }{)}^{1-\sigma }}dt,\end{array}$$

$$\begin{array}{rl}& \left({}^{\eta }{I}_{b-}^{\sigma }f\right)(x)=\frac{(\eta {)}^{1-\sigma }}{\Gamma (\sigma )}{\int }_x^b\frac{t^{\eta -1}f(t)}{(t^{\eta }-x^{\eta }{)}^{1-\sigma }}dt.\end{array}$$

$$\begin{array}{rl}& \left({}^{\eta }{I}_{0+}^{\sigma }f\right)(x)=\frac{(\eta {)}^{1-\sigma }}{\Gamma (\sigma )}{\int }_0^x\frac{t^{\eta -1}f(t)}{(x^{\eta }-t^{\eta }{)}^{1-\sigma }}dt,\end{array}$$

$$\begin{array}{rl}& \left({}^{\eta }{I}_{0-}^{\sigma }f\right)(x)=\frac{(\eta {)}^{1-\sigma }}{\Gamma (\sigma )}{\int }_x^0\frac{t^{\eta -1}f(t)}{(t^{\eta }-x^{\eta }{)}^{1-\sigma }}dt.\end{array}$$

$$\begin{array}{rl}& \left({}^{\eta }{D}_{a+}^{\sigma }f(t)\right)(x)={\left(x^{1-\eta }\frac{d}{dx}\right)}^n\left({}^{\eta }{I}_{a+}^{n-\sigma }f(t)\right)(x)\\ & =\frac{{\eta }^{\sigma -n+1}}{\Gamma (n-\xi )}{\left(x^{1-\eta }\frac{d}{dx}\right)}^n{\int }_a^x\frac{t^{\eta -1}f(t)}{(x^{\eta }-t^{\eta }{)}^{\sigma -n+1}}dtforx>a\end{array}$$

$$\begin{array}{rl}& \left({}^{\eta }{D}_{b-}^{\sigma }f(t)\right)(x)={\left(-x^{1-\eta }\frac{d}{dx}\right)}^n\left({}^{\eta }{I}_{b-}^{n-\sigma }f(t)\right)(x)\\ & =\frac{{\eta }^{\sigma -n+1}}{\Gamma (n-\xi )}{\left(-x^{1-\eta }\frac{d}{dx}\right)}^n{\int }_x^b\frac{t^{\eta -1}f(t)}{(t^{\eta }-x^{\eta }{)}^{\sigma -n+1}}dtforx<b,\end{array}$$

$$\begin{array}{rl}& \frac{d^n}{d{x}^n}(x^{\sigma })=\frac{\Gamma (\sigma +1)}{\Gamma (\sigma +1-n)}{x}^{\sigma -n},\Re (\sigma )>0.\end{array}$$

$$\begin{array}{rl}& \left({}^{\eta }{D}_{0+}^{\sigma }f(t)\right)(x)={\left(x^{1-\eta }\frac{d}{dx}\right)}^n\left({}^{\eta }{I}_{0+}^{n-\sigma }f(t)\right)(x)=\frac{{\eta }^{\sigma -n+1}}{\Gamma (n-\xi )}{\left(x^{1-\eta }\frac{d}{dx}\right)}^n{\int }_0^x\frac{t^{\eta -1}f(t)}{(x^{\eta }-t^{\eta }{)}^{\sigma -n+1}}dt,\end{array}$$

$$\begin{array}{rl}& \left({}^{\eta }{D}_{0-}^{\sigma }f(t)\right)(x)={\left(-x^{1-\eta }\frac{d}{dx}\right)}^n\left({}^{\eta }{I}_{0-}^{n-\sigma }f(t)\right)(x)=\frac{{\eta }^{\sigma -n+1}}{\Gamma (n-\xi )}{\left(-x^{1-\eta }\frac{d}{dx}\right)}^n{\int }_x^0\frac{t^{\eta -1}f(t)}{(t^{\eta }-x^{\eta }{)}^{\sigma -n+1}}dt.\end{array}$$

$$\displaystyle\begin{aligned} &\left({}^{\eta}D^{\sigma}_{0+}t^{\mu}E^{\vartheta,q}_{k,\xi,\zeta}(t^{\nu})\right)(x)=x^{\mu-\sigma\eta}\frac{{\eta}^{\sigma}}{\Gamma\left(\frac{\vartheta}{k}\right)}k^{1-\frac{\zeta}{k}}{}_{2}\Psi_{2}\left[\begin{array}[]{cc}\left(\frac{\vartheta}{k},{q}\right),\left(\frac{\mu}{\eta}+1,\frac{\nu}{\eta}\right)\\ \left(\frac{\zeta}{k},\frac{\xi}{k}\right),\left(\frac{\mu}{\eta}-\sigma+1,\frac{\nu}{\eta}\right)\end{array}\left|k^{(q-\frac{\xi}{k})}x^{\nu}\right.\right].\end{aligned}$$

$$\begin{array}{rl}& \mathcal{D}=\sum _{r=0}^{\infty }\frac{(\vartheta {)}_{rq,k}}{{\Gamma }_k(r\xi +\zeta )}\frac{1}{r!}\left({}^{\eta }{D}_{0+}^{\sigma }{t}^{r\nu +\mu }\right),\end{array}$$

$$\begin{array}{rl}& \mathcal{D}=\sum _{r=0}^{\infty }\frac{(\vartheta {)}_{rq,k}}{{\Gamma }_k(r\xi +\zeta )}\frac{1}{r!}\frac{{\eta }^{\sigma }}{\Gamma (1-\sigma )}\left(x^{1-\eta }\frac{d}{dx}\right){\int }_0^x\frac{t^{\eta -1}{t}^{r\nu +\mu }}{(x^{\eta }-t^{\eta }{)}^{\sigma }}dt.\end{array}$$

$$\begin{array}{rl}& \mathcal{D}=\sum _{r=0}^{\infty }\frac{{\eta }^{\sigma }}{\Gamma (1-\sigma )}{k}^{1-\frac{\zeta }{k}}\frac{1}{r!}\frac{\Gamma (\frac{\vartheta }{k}+rq)}{\Gamma (\frac{\vartheta }{k})\Gamma (\frac{\zeta }{k}+\frac{\xi }{k}r)}{k}^{r(q-\frac{\xi }{k})}\frac{1}{\eta }\left(x^{1-\eta }\frac{d}{dx}\right)\\ & \times \left(x^{\eta +r\nu +\mu -\sigma \eta }{\int }_0^1{z}^{\frac{r\nu +\mu }{\eta }}(1-z{)}^{-\sigma }dz\right).\end{array}$$

$$\begin{array}{rl}& \mathcal{D}=\sum _{r=0}^{\infty }\frac{{\eta }^{\sigma }}{\Gamma (1-\sigma )}{k}^{1-\frac{\zeta }{k}}\frac{1}{r!}\frac{\Gamma (\frac{\vartheta }{k}+rq)}{\Gamma (\frac{\vartheta }{k})\Gamma (\frac{\zeta }{k}+\frac{\xi }{k}r)}{k}^{r(q-\frac{\xi }{k})}\times \left(x^{r\nu +\mu -\sigma \eta }\frac{\Gamma \left(\frac{\mu }{\eta }+1+\frac{\nu }{\eta }r\right)\Gamma (1-\sigma )}{\Gamma \left(\frac{\mu }{\eta }+1-\sigma +\frac{\nu }{\eta }r\right)}\right),\end{array}$$

Full text

††journal: ….

1 Introduction

Diaz and Pariguan [1] introduced the $k$-Pochhammer symbol and $k$-gamma function defined as follows:

[TABLE]

and the relation with the classical Euler’s gamma function as:

[TABLE]

where $\vartheta\in\mathbb{C},k\in\mathbb{R}$ and $n\in\mathbb{N}.$

When $k=1$, (1.1) reduces to the classical Pochhammer symbol and Euler’s gamma function respectively.

Also let $\vartheta\in\mathbb{C},k,s\in\mathbb{R}$, then the following identity holds

[TABLE]

in particular,

[TABLE]

Further, let $\vartheta\in\mathbb{C},k,s\in\mathbb{R}$ and $\vartheta\in\mathbb{C}$, then the following identity holds

[TABLE]

in particular,

[TABLE]

For more details of k-Pochhammer symbol, k-special function and fractional Fourier transform one can refer to the papers by Romero et. al.[6, 7].

Let $k\in\mathbb{R},\xi,\zeta,\vartheta\in\mathbb{C};\Re(\xi)>0,\Re(\zeta)>0,\Re{(\vartheta)}>0$ and $q\in\mathbb{R}^{+}$, then the generalized k-Mittag-Leffler function, denoted by $E^{\vartheta,q}_{k,\xi,\zeta}(z)$, is defined as

[TABLE]

where $(\vartheta)_{nq,k}$ denotes the k-Pochhammer symbol given by equation (1.6) and $\Gamma_{k}(\vartheta)$ is the k-gamma function given by the equation (1.4) as (also see[9]).

Particular cases of $E^{\vartheta,q}_{k,\xi,\zeta}(z):$

For $q=1$, equation(1.7) yields k-Mittag-Leffler function defined as:

[TABLE]

For $k=1$, equation(1.7) yields Mittag-Leffler function, defined as (Shukla and Prajapati [10])

[TABLE]

For $q=1$ and $k=1$, equation(1.7) gives Mittag-Leffler function, defined as

[TABLE]

For $q=1,k=1$ and $\vartheta=1$, equation(1.7) gives Mittag-Leffler function (Wiman [11]), defined as

[TABLE]

For $q=1,k=1,\vartheta=1$ and $\zeta=1$, equation(1.7) gives Mittag-Leffler function is defined as

[TABLE]

The Fox-Wright function ${}_{p}\Psi_{q}[z]$ defined as

[TABLE]

where the coefficients $\xi_{1},...,\xi_{p}$, $\zeta_{1},...,\zeta_{q}$ $\in\mathbb{R}^{+}$ such that

[TABLE]

2 Fractional integration

In this section, we will establish some fractional integral formulas for the generalized k-Mittag-Leffler function. To do this, we need to recall the following pair of fractional integral operators.

The Riemann-Liouville fractional integrals $I^{\xi}_{a+}f$ and $I^{\xi}_{b-}f$ of order $\xi\in\mathbb{C},\Re(\xi)>0$, are defined by [2, 3, 4, 5, 8],

[TABLE]

and

[TABLE]

respectively. Here $\tau(.)$ is the Gamma function. These integrals are called the left-sided and right-sided fractional integrals, respectively. When $\xi=n\in\mathbb{N}$, the integrals (2.1) and (2.2) coincide with the n-fold integrals [2].

Lemma 1**.**

Let $\Omega=[a,b](-\infty<a<b<\infty)$ be a finite interval on the real axis $\mathbb{R}$. The generalized fractional integral ${}^{\eta}I_{a+}^{\sigma}f$ of order $\sigma\in\mathbb{C}$ for $x>a$ and $\Re(\sigma)>0$ is defined as

[TABLE]

similarly generalized fractional integral ${}^{\eta}I_{b-}^{\sigma}f$ of order $\sigma\in\mathbb{C}$ for $x<b$ and $\Re(\sigma)>0$ is defined as

[TABLE]

If we choose $a=b=0$ the above Lemma 1 reduces to

Lemma 2**.**

The generalized fractional integral ${}^{\eta}I_{0+}^{\sigma}f$ of order $\sigma\in\mathbb{C}$ for $x>0$ and $\Re(\sigma)>0$ is defined as

[TABLE]

similarly generalized fractional integral ${}^{\eta}I_{0-}^{\sigma}f$ of order $\sigma\in\mathbb{C}$ for $x<0$ and $\Re(\sigma)>0$ is defined as

[TABLE]

Lemma 3**.**

Riemann-type fractional derivatives ${}^{\eta}D^{\sigma}_{a+}f$ and ${}^{\eta}D^{\sigma}_{b-}f$ of order $\sigma\in\mathbb{C},\Re(\sigma)>0$ are defined as

[TABLE]

and

[TABLE]

where $n=[\Re(\sigma)]+1.$

[TABLE]

If we choose $a=b=0$ the above Lemma 3 reduces to

Lemma 4**.**

The generalized fractional integral ${}^{\eta}I_{0+}^{\sigma}f$ of order $\sigma\in\mathbb{C}$ for $x>0$ and $\Re(\sigma)>0$ is defined as

[TABLE]

similarly generalized fractional integral ${}^{\eta}I_{0-}^{\sigma}f$ of order $\sigma\in\mathbb{C}$ for $x<0$ and $\Re(\sigma)>0$ is defined as

[TABLE]

The main results are given in the following theorem.

Theorem 1**.**

Let $x>0,\sigma,\eta,\mu,\xi,\zeta,\vartheta\in\mathbb{C},k\in\mathbb{R}$, $\Re(\sigma)>0,\Re(\zeta)>0,\Re(\vartheta)>0$ and $q\in\mathbb{R}^{+}$, such that $\eta>0$, then

[TABLE]

Proof.

For convenience, we denote the left-hand side of the result (2.12) by $\mathscr{D}$. Using (1.7), and then changing the order of integration and summation, then

[TABLE]

applying the result (2.10), the above equation (2.13) reduced to

[TABLE]

Put $t^{\eta}=x^{\eta}z$ in equation (2.14) and by proper substitution we have

[TABLE]

[TABLE]

after simplification, the above equation (2.16) reduces to

[TABLE]

By using equation (2.17) and simplification, we have

[TABLE]

∎

Theorem 2**.**

Let $x>0,\sigma,\mu,\eta,\xi,\zeta,\vartheta\in\mathbb{C},k\in\mathbb{R}$, $\Re(\sigma)>0,\Re(\zeta)>0,\Re(\vartheta)>0$ and $q\in\mathbb{R}^{+}$, such that $\eta>0$, then

[TABLE]

Proof.

For convenience, we denote the left-hand side of the result (2.19) by $\mathscr{D}$. Using (1.7), and then changing the order of integration and summation, then

[TABLE]

applying the result (2.11), the above equation (2.20) reduced to

[TABLE]

Put $t^{\eta}=x^{\eta}z$ in equation (2.21) and by proper substitution we have

[TABLE]

[TABLE]

after simplification, the above equation (2.23) reduces to

[TABLE]

By using equation (2.24) and simplification, we have

[TABLE]

∎

2.1 Numerical results and graphical interpretation

In this section we plot the graphs and obtained the numerical value of our findings in equation (2.12) and (2.19). For this purpose, we select the values of the parameters involving in these results as $\mu=0.5;\nu=0.8;\eta=0.3;\xi=0.5;\zeta=0.2;\vartheta=0.5;q=0.4;k=0.5$ and $\sigma=0.1:0.1:0.4$ for Figure LABEL:ig-1. In Figure 2 the values of the figure are taken as $\eta=1:2:7;\sigma=0.02$

Theorem 3**.**

Let $x>0,\sigma,\mu,\eta,\xi,\zeta,\vartheta\in\mathbb{C},k\in\mathbb{R}$, $\Re(\sigma)>0,\Re(\zeta)>0,\Re(\vartheta)>0,\Re(\mu)>0,\Re(l)>0,\Re(m)>0>$ and $q\in\mathbb{R}^{+}$, such that $\eta>0$, then

[TABLE]

Proof.

For convenience, we denote the left-hand side of the result (2.26) by $\mathscr{B}$. Using the definition of Beta transform, the LHS of (2.26) becomes:

[TABLE]

further using (1.7) and then changing the order of integration and summation,which is valid under the conditions of Theorem 1, then

[TABLE]

applying the result (2.10), after simplification Eq.(2.28) reduced to

[TABLE]

applying the definition of Beta transform, Eq.(2.29) reduced to

[TABLE]

[TABLE]

∎

Theorem 4**.**

Let $x>0,\sigma,\mu,\eta,\xi,\zeta,\vartheta\in\mathbb{C},k\in\mathbb{R}$, $\Re(\sigma)>0,\Re(\zeta)>0,\Re(\vartheta)>0,\Re(\mu)>0,\Re(l)>0,\Re(m)>0>$ and $q\in\mathbb{R}^{+}$, such that $\eta>0$, then

[TABLE]

Proof.

For convenience, we denote the left-hand side of the result (2.32) by $\mathscr{B}$. Using the definition of Beta transform, the LHS of (2.32) becomes:

[TABLE]

further using (1.7) and then changing the order of integration and summation,which is valid under the conditions of Theorem 2, then

[TABLE]

applying the result (2.11), after simplification Eq.(2.34) reduced to

[TABLE]

applying the definition of Beta transform, Eq.(2.35) reduced to

[TABLE]

[TABLE]

∎

Theorem 5**.**

Let $x>0,\sigma,\mu,\eta,\xi,\zeta,\vartheta\in\mathbb{C},k\in\mathbb{R}$, $\Re(\sigma)>0,\Re(\zeta)>0,\Re(\vartheta)>0,\Re(\mu)>0,\Re(l)>0,\Re(m)>0>$ and $q\in\mathbb{R}^{+}$, such that $\eta>0$, then

[TABLE]

Proof.

For convenience, we denote the left-hand side of the result (2.38) by $\mathscr{L}$. Using the definition of Laplace transform, the LHS of (2.38) becomes:

[TABLE]

further using (1.7) and then changing the order of integration and summation,which is valid under the conditions of Theorem 1, then applying the result (2.10), after simplification Eq.(2.39) reduced to

[TABLE]

Eq.(2.40) reduced to

[TABLE]

[TABLE]

∎

Theorem 6**.**

Let $x>0,\sigma,\mu,\eta,\xi,\zeta,\vartheta\in\mathbb{C},k\in\mathbb{R}$, $\Re(\sigma)>0,\Re(\zeta)>0,\Re(\vartheta)>0,\Re(\mu)>0,\Re(l)>0,\Re(m)>0>$ and $q\in\mathbb{R}^{+}$, such that $\eta>0$, then

[TABLE]

Proof.

For convenience, we denote the left-hand side of the result (2.43) by $\mathscr{L}$. Using the definition of Laplace transform, the LHS of (2.38) becomes:

[TABLE]

further using (1.7) and then changing the order of integration and summation,which is valid under the conditions of Theorem 2, then applying the result (2.11), after simplification Eq.(2.44) reduced to

[TABLE]

Eq.(2.45) reduced to

[TABLE]

[TABLE]

∎