The study of generalized Hurwitz–Lerch zeta function and fractional kinetic equations

ABSTRACT

In this research article, we provide an extended Hurwitz–Lerch zeta function using the extended beta function. We have also investigated the new generalized Hurwitz–Lerch zeta function’s exciting characteristics, such as integral representations, differential equations, the Mellin transform and special generating relations. In order to support the main result, we have additionally considered several notable specific cases. Furthermore, we have discussed the solution of the fractional kinetic equation using the new extended Hurwitz–Lerch zeta function.

KEYWORDS:

• Beta function
• Hurwitz–Lerch zeta function
• Mellin transform
• generating relation

MATHEMATICS SUBJECT CLASSIFICATION:

• 11B68
• 33B15
• 33C05
• 33C10
• 33C15
• 33C45
• 33E20

1. Introduction and preliminaries

In applied analysis and quantitative physics, special functions are crucial and play a significant role in applied analysis and mathematical physics. Special functions offer formulae for some differential equation solutions or some simple integral functions. One of the most significant members of the family of special functions is the classical beta function, also known as the first-order Euler integral. The study of beta functions constitutes the initial phase of creating the theory of special functions. Due to the numerous applications of these initial special functions in the fields of mathematics, physics, engineering and Lie theory, mathematicians and physicists devoted close attention to them. A fundamental special function, i.e. the classical beta function, plays an important part in many fields of science, including engineering, mathematics, physical science and statistics. Different kinds of special functions have evolved into essential tools for scientists and engineers in many fields of applied mathematics. Recently, many authors developed extensions to the Euler beta function, gamma function, Gauss hypergeometric function, confluent hypergeometric function, Hurwitz–Lerch zeta functions and many other functions (Garg et al., 2008; Khan et al., 2023, 2023; Khan et al., 2022; Lin & Srivastava, 2004; Nadeem et al., 2020). Inspired by the abovementioned work in this research article, we define a new generalization of Hurwitz–Lerch zeta function and discuss its properties. We have also discussed their one of the applications to find the solution of fractional kinetic equation using extended Hurwitz–Lerch zeta functions.

In the whole paper $\mathbb{C},{\mathbb{R}}_0^{+},{\mathbb{Z}}^{+},{\mathbb{R}}^{+}$ represent the set of complex numbers, non-negative real numbers, positive integers and positive real numbers, respectively, and $\mathrm{\Re }(\mathit{z})$ is the real part of complex number z.

An extended beta function $\mathit{B}({\theta }_1,{\theta }_2;\mathit{p})$ is defined by Chaudhry et al. (1997) as

(1.1) $\mathit{B}({\theta }_1,{\theta }_2;\mathit{p})=B_{\mathit{p}}({\theta }_1,{\theta }_2)={\int }_0^1{t}^{{\mathit{\theta }}_1-1}(1-\mathit{t}{)}^{{\mathit{\theta }}_2-1}{e}^{-\frac{\mathit{p}}{\mathit{t}(1-\mathit{t})}}\mathit{dt},$(1.1)

where $\mathrm{\Re }(\mathit{p})>0,\mathrm{\Re }({\theta }_1)>0,\mathrm{\Re }({\theta }_2)>0$. In particular, if p = 0, we get $\mathit{B}({\theta }_1,{\theta }_2,0)=\mathit{B}({\theta }_1,{\theta }_2)$.

Recently, a new extension of beta function was introduced by Khan and Husain (2022) as

(1.2) $B_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}({\theta }_1,{\theta }_2)={\int }_0^1{t}^{{\mathit{\theta }}_1-1}(1-\mathit{t}{)}^{{\mathit{\theta }}_2-1}{E}_{\mathit{p},\mathit{q}}(-\frac{x}{{\mathit{t}}^{\mathit{y}}(1-\mathit{t}{)}^{\mathit{z}}})\mathit{dt},$(1.2)

where $\mathit{x}\ge 0$, $\mathrm{\Re }({\theta }_1)>0$, $\mathrm{\Re }({\theta }_2)>0$; $\mathit{p},\mathit{q}\in {\mathbb{R}}_0^{+}$; $\mathit{y},\mathit{z}\in {\mathbb{R}}^{+}$ and $E_{\mathit{p},\mathit{q}}(.)$ is the generalized Mittag–Leffler function defined as (Wiman, 1905)

$\begin{array}{l}{E}_{\mathit{p},\mathit{q}}(\mathit{x})=\sum _{\mathit{n}=0}^{\mathrm{\infty }}\frac{x^n}{\mathrm{\Gamma }(\mathit{p}\mathit{n}+\mathit{q})}.\end{array}$

We are familiar with the Hurwitz–Lerch zeta function $\mathrm{\Phi }(\mathit{\xi },\mathit{\kappa },\mathit{\tau })$ (Erdélyi, 1953; Srivastava & Choi, 2001, 2011) defined as

(1.3) $\mathrm{\Phi }(\mathit{\xi },\mathit{\kappa },\mathit{\tau })=\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\xi }^{\mathit{m}}}{(\mathit{m}+\mathit{\tau }{)}^{\mathit{\kappa }}},$(1.3)

$\begin{array}{l}(\mathit{\tau }\in {\mathbb{Z}}^{+};\mathit{\kappa }\in \mathbb{C}\text{ when }|\mathit{\xi }|<1;\mathrm{\Re }(\mathit{\kappa })>1\text{ when }|\mathit{\xi }|=1).\end{array}$

Many generalizations of the Hurwitz–Lerch zeta functions have been given by the researchers. For example, Goyal and Laddha (1997) have introduced an extension of Hurwitz–Lerch zeta function defined as

(1.4) ${\mathrm{\Phi }}_{\mathit{\delta }}^{\ast }(\mathit{\xi },\mathit{\kappa },\mathit{\tau })=\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\mathit{\delta }{)}_{\mathit{m}}}{\mathit{m}!}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},$(1.4)

$\begin{array}{l}(\mathit{\delta }\in \mathbb{C};\mathit{\tau }\in {\mathbb{Z}}^{+};\mathit{\kappa }\in \mathbb{C}\mathit{if}|\mathit{\xi }|<1;\mathrm{\Re }(\mathit{\kappa }-\mathit{\delta })>1\mathit{if}|\mathit{\xi }|=1).\end{array}$

Lin and Srivastava (2004) also defined the Hurwitz–Lerch zeta function as

(1.5) ${\mathrm{\Phi }}_{\mathit{\zeta },\mathit{\gamma }}^{\mathit{\eta },\mathit{\omega }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau })=\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\mathit{\zeta }{)}_{\mathit{\eta m}}}{(\mathit{\gamma }{)}_{\mathit{\omega m}}}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},$(1.5)

$\begin{array}{l}\mathit{ }(\mathit{\zeta }\in \mathbb{C};\mathit{\tau },\mathit{\gamma }\in {\mathbb{Z}}^{+};\mathit{\eta },\mathit{\omega }\in {\mathbb{R}}^{+};\mathit{\eta }<\mathit{\omega }\text{ if }\mathit{\kappa },\mathit{\xi }\in \mathbb{C};\\ \mathit{\eta }=\mathit{\omega },\mathit{\kappa }\in \mathbb{C}\text{ if }|\mathit{\xi }|<1;\mathit{\eta }=\mathit{\omega },\mathrm{\Re }(\mathit{\kappa }-\mathit{\zeta }+\mathit{\gamma })>1,|\mathit{\xi }|=1).\end{array}$

Garg et al. (2008) also introduced Hurwitz–Lerch zeta function as

(1.6) ${\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau })=\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m(\mathit{\zeta }{)}_m}{(\mathit{\gamma }{)}_m(\mathit{m}+\mathit{\tau }{)}^{\kappa }}\frac{{\xi }^m}{\mathit{m}!},$(1.6)

$\begin{array}{l}(\mathit{\delta },\mathit{\zeta }\in \mathbb{C};\mathit{\gamma },\mathit{\tau }\in Z^{+};\mathit{\kappa }\in \mathbb{C}\text{ if }|\mathit{\xi }|<1;\mathrm{\Re }(\mathit{\kappa }+\mathit{\gamma }-\mathit{\delta }-\mathit{\zeta })>1\text{ if }|\mathit{\xi }|=1).\end{array}$

Parmar et al (Parmar & Raina, 2014) defined an extended Hurwitz–Lerch zeta function involving beta function (Chaudhry et al., 1997) as

(1.7) ${\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p})=\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{B_p(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},$(1.7)

$\begin{array}{l}\mathit{ }(\mathit{p}\ge 0;\mathit{\delta },\mathit{\zeta }\in \mathbb{C};\mathit{\gamma },\mathit{\tau }\in {\mathbb{Z}}^{+};\mathit{\kappa }\in \mathbb{C}\text{ if }|\mathit{\xi }|<1;\\ \mathrm{\Re }(\mathit{\kappa }+\mathit{\gamma }-\mathit{\delta }-\mathit{\zeta })>1\text{ if }|\mathit{\xi }|=1).\end{array}$

The integral representation of (1.3(1.3) $\mathrm{\Phi }(\mathit{\xi },\mathit{\kappa },\mathit{\tau })=\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\xi }^{\mathit{m}}}{(\mathit{m}+\mathit{\tau }{)}^{\mathit{\kappa }}},$(1.3) ) is given by

(1.8) $\mathrm{\Phi }(\mathit{\xi },\mathit{\kappa },\mathit{\tau })=\frac{1}{\mathrm{\Gamma }(\mathit{\kappa })}{\int }_0^{\mathrm{\infty }}\frac{{\mathit{t}}^{\mathit{\kappa }-1}{e}^{-\mathit{\tau t}}}{1-\mathit{\xi }{e}^{-\mathit{t}}}\mathit{dt}=\frac{1}{\mathrm{\Gamma }(\mathit{\kappa })}{\int }_0^{\mathrm{\infty }}\frac{{\mathit{t}}^{\mathit{\kappa }-1}{e}^{-(\mathit{\tau }-1)\mathit{t}}}{e^{-\mathit{t}}-\mathit{\xi }}\mathit{dt},$(1.8)

$\begin{array}{l}(\mathrm{\Re }(\mathit{\kappa })>0,\mathrm{\Re }(\mathit{\tau })>0\text{ if }|\mathit{\xi }|\le 1(\mathit{\xi }\ne 1);\mathrm{\Re }(\mathit{\kappa })>1\text{ if }\mathit{\xi }=1).\end{array}$

Also, the integral representations of (1.4(1.4) ${\mathrm{\Phi }}_{\mathit{\delta }}^{\ast }(\mathit{\xi },\mathit{\kappa },\mathit{\tau })=\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\mathit{\delta }{)}_{\mathit{m}}}{\mathit{m}!}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},$(1.4) ), (1.5(1.5) ${\mathrm{\Phi }}_{\mathit{\zeta },\mathit{\gamma }}^{\mathit{\eta },\mathit{\omega }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau })=\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\mathit{\zeta }{)}_{\mathit{\eta m}}}{(\mathit{\gamma }{)}_{\mathit{\omega m}}}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},$(1.5) ), (1.6(1.6) ${\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau })=\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m(\mathit{\zeta }{)}_m}{(\mathit{\gamma }{)}_m(\mathit{m}+\mathit{\tau }{)}^{\kappa }}\frac{{\xi }^m}{\mathit{m}!},$(1.6) ) and (1.7(1.7) ${\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p})=\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{B_p(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},$(1.7) ) are given by (1.9(1.9) ${\mathrm{\Phi }}_{\mathit{\delta }}^{\ast }(\mathit{\xi },\mathit{\kappa },\mathit{\tau })=\frac{1}{\mathrm{\Gamma }(\mathit{\kappa })}{\int }_0^{\mathrm{\infty }}\frac{{\mathit{t}}^{\mathit{\kappa }-1}{e}^{-\mathit{\tau t}}}{(1-\mathit{\xi }{e}^{-\mathit{t}}{)}^{\delta }}\mathit{dt}=\frac{1}{\mathrm{\Gamma }(\mathit{\kappa })}{\int }_0^{\mathrm{\infty }}\frac{{\mathit{t}}^{\mathit{\kappa }-1}{e}^{-(\mathit{\tau }-\mathit{\delta })\mathit{t}}}{(e^t-\mathit{\xi }{)}^{\delta }}\mathit{dt}$(1.9) ), (1.10(1.10) ${\mathrm{\Phi }}_{\mathit{\zeta },\mathit{\gamma }}^{(\mathit{\eta },\mathit{\omega })}(\mathit{\xi },\mathit{\kappa },\mathit{\tau })=\frac{1}{\mathrm{\Gamma }(\mathit{\kappa })}{\int }_0^{\mathrm{\infty }}{t}^{\mathit{\kappa }-1}{e}^{-\mathit{\tau t}}{ }_2{\mathrm{\Psi }}_1[\begin{array}{l}(\mathit{\zeta };\mathit{\eta }),(1;1);\\ (\mathit{\gamma };\mathit{\omega })\end{array};\mathit{\xi }{e}^{-\mathit{t}}]\mathit{dt}$(1.10) ), (1.11(1.11) ${\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta },\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau })=\frac{1}{\mathrm{\Gamma }(\mathit{\kappa })}{\int }_0^{\mathrm{\infty }}{t}^{\mathit{\kappa }-1}{e}^{-\mathit{\tau t}}{ }_2{F}_1(\mathit{\delta },\mathit{\zeta };\mathit{\gamma };\mathit{\xi }{e}^{-\mathit{t}})\mathit{dt}$(1.11) ) and (1.12(1.12) ${\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p})=\frac{1}{\mathrm{\Gamma }(\mathit{\kappa })}{\int }_0^{\mathrm{\infty }}{t}^{\mathit{\kappa }-1}{e}^{-\mathit{\tau t}}{F}_{\mathit{p}}(\mathit{\delta },\mathit{\zeta };\mathit{\gamma };\mathit{\xi }{e}^{-\mathit{t}})\mathit{dt}$(1.12) ), respectively

(1.9) ${\mathrm{\Phi }}_{\mathit{\delta }}^{\ast }(\mathit{\xi },\mathit{\kappa },\mathit{\tau })=\frac{1}{\mathrm{\Gamma }(\mathit{\kappa })}{\int }_0^{\mathrm{\infty }}\frac{{\mathit{t}}^{\mathit{\kappa }-1}{e}^{-\mathit{\tau t}}}{(1-\mathit{\xi }{e}^{-\mathit{t}}{)}^{\delta }}\mathit{dt}=\frac{1}{\mathrm{\Gamma }(\mathit{\kappa })}{\int }_0^{\mathrm{\infty }}\frac{{\mathit{t}}^{\mathit{\kappa }-1}{e}^{-(\mathit{\tau }-\mathit{\delta })\mathit{t}}}{(e^t-\mathit{\xi }{)}^{\delta }}\mathit{dt}$(1.9)

$\begin{array}{l}(\mathrm{\Re }(\mathit{\kappa })>0,\mathrm{\Re }(\mathit{\tau })>0\text{ if }|\mathit{\xi }|\le 0(\mathit{\xi }\ne 1);\mathrm{\Re }(\mathit{\kappa })>1\text{ if }\mathit{\xi }=1),\end{array}$

(1.10) ${\mathrm{\Phi }}_{\mathit{\zeta },\mathit{\gamma }}^{(\mathit{\eta },\mathit{\omega })}(\mathit{\xi },\mathit{\kappa },\mathit{\tau })=\frac{1}{\mathrm{\Gamma }(\mathit{\kappa })}{\int }_0^{\mathrm{\infty }}{t}^{\mathit{\kappa }-1}{e}^{-\mathit{\tau t}}{ }_2{\mathrm{\Psi }}_1[\begin{array}{l}(\mathit{\zeta };\mathit{\eta }),(1;1);\\ (\mathit{\gamma };\mathit{\omega })\end{array};\mathit{\xi }{e}^{-\mathit{t}}]\mathit{dt}$(1.10)

$\begin{array}{l}(\mathrm{\Re }(\mathit{\tau })>0;\mathrm{\Re }(\mathit{\kappa })>0\text{ if }|\mathit{\xi }|\le 0(\mathit{\xi }\ne 1);\mathrm{\Re }(\mathit{\kappa })>1\text{ if }\mathit{\xi }=1),\end{array}$

where ${ }_2{\mathrm{\Psi }}_1$ is the Fox–Wright function (Parmar & Raina, 2014).

(1.11) ${\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta },\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau })=\frac{1}{\mathrm{\Gamma }(\mathit{\kappa })}{\int }_0^{\mathrm{\infty }}{t}^{\mathit{\kappa }-1}{e}^{-\mathit{\tau t}}{ }_2{F}_1(\mathit{\delta },\mathit{\zeta };\mathit{\gamma };\mathit{\xi }{e}^{-\mathit{t}})\mathit{dt}$(1.11)

$\begin{array}{l}(\mathrm{\Re }(\mathit{\tau })>0;\mathrm{\Re }(\mathit{\kappa })>0\text{ if }|\mathit{\xi }|\le 0(\mathit{\xi }\ne 0);\mathrm{\Re }(\mathit{\kappa })>1\text{ if }\mathit{\xi }=1),\end{array}$

and

(1.12) ${\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p})=\frac{1}{\mathrm{\Gamma }(\mathit{\kappa })}{\int }_0^{\mathrm{\infty }}{t}^{\mathit{\kappa }-1}{e}^{-\mathit{\tau t}}{F}_{\mathit{p}}(\mathit{\delta },\mathit{\zeta };\mathit{\gamma };\mathit{\xi }{e}^{-\mathit{t}})\mathit{dt}$(1.12)

$\begin{array}{l}(\mathit{p}\ge 0;\mathrm{\Re }(\mathit{\tau })>0;\mathrm{\Re }(\mathit{\kappa })>0\text{ if }|\mathit{\xi }|\le 0(\mathit{\xi }\ne 1);\mathrm{\Re }(\mathit{\kappa })>1\text{ if }\mathit{\xi }=1),\end{array}$

where $F_p({\theta }_1,{\theta }_2;{\theta }_3;\mathit{\xi })$ is the extended hypergeometric function (Chaudhry et al., 2004) defined by

(1.13) $F_p({\theta }_1,{\theta }_2;{\theta }_3;\mathit{\xi })=\sum _{\mathit{m}=0}^{\mathrm{\infty }}({\theta }_1{)}_{\mathit{m}}\frac{{\mathit{B}}_{\mathit{p}}({\theta }_2+\mathit{m},{\theta }_3-{\theta }_2)}{\mathit{B}({\theta }_2,{\theta }_3-{\theta }_2)}\frac{(\xi {)}^m}{\mathit{m}!},$(1.13)

$\begin{array}{l}(\mathit{p}\ge 0,|\mathit{\xi }|<1;\mathrm{\Re }({\theta }_3)>\mathrm{\Re }({\theta }_2)>0).\end{array}$

Recently, Nadeem et al. (2020) introduced Hurwitz–Lerch zeta function as

(1.14) ${\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q})=\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }}$(1.14)

$\begin{array}{l}\mathit{ }(\mathit{p}\ge 0,\mathit{q}\ge 0;\mathit{\delta },\mathit{\zeta }\in \mathbb{C};\mathit{\gamma },\mathit{\tau }\in {\mathbb{Z}}^{+};\mathit{\kappa }\in \mathbb{C}\text{ if }|\mathit{\xi }|<1;\\ \mathrm{\Re }(\mathit{\kappa }+\mathit{\gamma }-\mathit{\delta }-\mathit{\zeta })>1\text{ if }|\mathit{\xi }|=1),\end{array}$

where beta and hypergeometric functions introduced by Choi et al. (2014) are defined as

(1.15) $B_{\mathit{p},\mathit{q}}({\theta }_1,{\theta }_2)={\int }_0^1{u}^{{\mathit{\theta }}_1-1}(1-\mathit{u}{)}^{{\mathit{\theta }}_2-1}\mathit{exp}(-\frac{p}{u}-\frac{q}{(1-\mathit{u})})\mathit{du}$(1.15)

$\begin{array}{l}(\mathit{p}\ge 0,\mathit{q}\ge 0;\mathrm{\Re }({\theta }_1)>0;\mathrm{\Re }({\theta }_2)>0),\end{array}$

and

(1.16) $F_{\mathit{p},\mathit{q}}({\theta }_1,{\theta }_2;{\theta }_3;\mathit{\xi })=\sum _{\mathit{m}=0}^{\mathrm{\infty }}({\theta }_1{)}_{\mathit{m}}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}({\theta }_2+\mathit{m},{\theta }_3-{\theta }_2)}{\mathit{B}({\theta }_2,{\theta }_3-{\theta }_2)}\frac{(\xi {)}^m}{\mathit{m}!}$(1.16)

$\begin{array}{l}({\theta }_1\ge 0,\mathit{p}\ge 0,\mathit{q}\ge 0,|\mathit{\xi }|<1;\mathrm{\Re }({\theta }_3)>\mathrm{\Re }({\theta }_2)>0),\end{array}$

respectively.

Motivated and inspired by the above extensions and generalizations of Hurwitz–Lerch zeta function (Nadeem et al., 2020; Parmar & Raina, 2014; Raina & Chhajed, 2004; Srivastava & Choi, 2011), we now find a more generalized form of extended Hurwitz–Lerch zeta function involving the extended beta function introduced by Khan and Husain (2022) and Khan et al. (2022), and we also study some particular cases, various integral representations and derivative formulae, the Mellin transform and the generating relations. Additionally, we use the extended Hurwitz–Lerch zeta function to find the solution of fractional kinetic equation.

2. Generalization of Hurwitz–Lerch zeta function

Here, we introduce Hurwitz–Lerch zeta function in terms of extended beta function (1.2(1.2) $B_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}({\theta }_1,{\theta }_2)={\int }_0^1{t}^{{\mathit{\theta }}_1-1}(1-\mathit{t}{)}^{{\mathit{\theta }}_2-1}{E}_{\mathit{p},\mathit{q}}(-\frac{x}{{\mathit{t}}^{\mathit{y}}(1-\mathit{t}{)}^{\mathit{z}}})\mathit{dt},$(1.2) ) as

(2.1) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})={\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q})\\ =\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},\end{array}$(2.1)

$\begin{array}{l}\mathit{ }(\mathit{p}\ge 0,\mathit{q}\ge 0;\mathit{\delta },\mathit{\zeta }\in \mathbb{C};\mathit{\gamma },\mathit{\tau }\in {\mathbb{Z}}^{+};\mathit{\kappa }\in \mathbb{C}\text{ if }|\mathit{\xi }|<1;\\ \mathrm{\Re }(\mathit{\kappa }+\mathit{\gamma }-\mathit{\delta }-\mathit{\zeta })>1\text{ if }|\mathit{\xi }|=1;\mathit{y},\mathit{z}\in {\mathbb{R}}^{+}).\end{array}$

2.1. Particular cases

Now, we discuss some well-known results obtained by putting the value of parameters in the generalized Hurwitz–Lerch zeta function (2.1(2.1) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})={\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q})\\ =\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},\end{array}$(2.1) ) in the following remarks.

Remark 2.1.

Substituting $\mathit{\delta }=1,\mathit{y}=\mathit{z}=1$ and $\mathit{p}=\mathit{q}=1$ in (2.1(2.1) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})={\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q})\\ =\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},\end{array}$(2.1) ), we get Hurwitz–Lerch zeta function introduced by Lin and Srivastava (2004):

(2.2) $\begin{array}{ll}{\mathrm{\Phi }}_{\mathit{\zeta },\mathit{\gamma }}^{1,1}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{x})& \mathit{ }={\mathrm{\Phi }}_{1,\mathit{\zeta };\mathit{\gamma }}^{\mathit{x},1,1}(\mathit{\xi },\mathit{\kappa },\mathit{\tau },1,1)\\ \mathit{ }=\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\mathit{B}}_{1,1}^{\mathit{x},1,1}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{{(\mathit{m}+\mathit{\tau })}^{\kappa }},\end{array}$(2.2)

$\begin{array}{l}\mathit{ }(\mathit{x}\ge 0;\mathit{\zeta }\in ,\mathit{\gamma },\mathit{\tau }\in +;\mathit{\kappa }\in \mathrm{if}|\mathit{\xi }|<1;\\ \mathrm{\Re }(\mathit{\kappa }+\mathit{\gamma }-1-\mathit{\zeta })>1)\mathrm{if}|\mathit{\xi }|=1).\end{array}$

Remark 2.2.

If $\mathit{\delta }=\mathit{\gamma }=1,\mathit{y}=\mathit{z}=1$ and $\mathit{p}=\mathit{q}=1$ in (2.1(2.1) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})={\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q})\\ =\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},\end{array}$(2.1) ), we get a particular case of (1.4(1.4) ${\mathrm{\Phi }}_{\mathit{\delta }}^{\ast }(\mathit{\xi },\mathit{\kappa },\mathit{\tau })=\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\mathit{\delta }{)}_{\mathit{m}}}{\mathit{m}!}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},$(1.4) ) (Goyal & Laddha, 1997) as

(2.3) $\begin{array}{l}{\mathrm{\Phi }}_{1,\mathit{\zeta },1}^{\mathit{x},1,1}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };1,1)=\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\mathit{B}}_{1,1}^{\mathit{x},1,1}(\mathit{\zeta }+\mathit{m},1-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },1-\mathit{\zeta })}\frac{{\xi }^m}{{(\mathit{m}+\mathit{\tau })}^{\kappa }},\end{array}$(2.3)

$(\mathrm{\Re }(\mathit{x})>0;\mathit{\zeta }\in ;\mathit{\tau }\in { }^{+};\mathit{\kappa }\in \mathrm{if}|\mathit{\xi }|<1;\mathrm{\Re }(\mathit{\kappa })>1\mathrm{if}|\mathit{\xi }|=1).$

Remark 2.3.

If we set x = 1 and $\mathit{\delta }=\mathit{\gamma }=1;\mathit{p}=\mathit{q}=1$ in (2.2(2.2) $\begin{array}{ll}{\mathrm{\Phi }}_{\mathit{\zeta },\mathit{\gamma }}^{1,1}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{x})& \mathit{ }={\mathrm{\Phi }}_{1,\mathit{\zeta };\mathit{\gamma }}^{\mathit{x},1,1}(\mathit{\xi },\mathit{\kappa },\mathit{\tau },1,1)\\ \mathit{ }=\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\mathit{B}}_{1,1}^{\mathit{x},1,1}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{{(\mathit{m}+\mathit{\tau })}^{\kappa }},\end{array}$(2.2) ), we get a special case of the Hurwitz–Lerch zeta function as

(2.4) $\begin{array}{ll}{\mathrm{\Phi }}_{\mathit{\delta }}^{\ast }(\mathit{\xi },\mathit{\kappa },\mathit{\tau };1)& \mathit{ }={\mathrm{\Phi }}_{1,\mathit{\zeta },1}^{1,1,1}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };1,1)\\ \mathit{ }=\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\mathit{B}}_{1,1}^{1,1,1}(\mathit{\zeta }+\mathit{m},1-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },1-\mathit{\zeta })}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},\end{array}$(2.4)

$\begin{array}{l}(\mathit{\delta }\in \mathbb{C};\mathit{\tau }\in {\mathbb{Z}}^{+};\mathit{\kappa }\in \mathbb{C}\text{ if }|\mathit{\xi }|<1;\mathrm{\Re }(\mathit{\kappa })>1\text{ if }|\mathit{\xi }|=1).\end{array}$

Remark 2.4.

The Hurwitz–Lerch zeta function (2.1(2.1) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})={\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q})\\ =\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},\end{array}$(2.1) ) has the following limiting case:

(2.5) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\zeta },\mathit{\gamma }}^{\ast }(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})=1\mathit{mu}\\ \times \underset{|\mathit{\delta }|\to \mathrm{\infty }}{lim}\left\{{\mathrm{\Phi }}_{\mathit{\zeta },\mathit{\gamma }}^{\ast }\left(\frac{\xi }{\delta },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q},\mathit{x},\mathit{y},\mathit{z}\right)\right\}=1\mathit{mu}\\ \times \sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })\mathit{m}!}\frac{{\xi }^m}{{(\mathit{m}+\mathit{\tau })}^{\kappa }},1\mathit{mu}\end{array}$(2.5)

$\begin{array}{l}(\mathit{x}\ge 0,\mathit{\zeta }\in ;\mathit{\gamma },\mathit{\tau }\in +,\mathit{\kappa }\in \mathrm{if}|\mathit{\xi }|<1;\mathrm{\Re }(\mathit{\kappa }+\mathit{\gamma }-\mathit{\zeta })>1\mathrm{if}|\mathit{\xi }|=1).\end{array}$

Remark 2.5.

Putting $\mathit{x}=0,\mathit{y}=\mathit{z}=1$ and $\mathit{p}=\mathit{q}=1$ in (2.1(2.1) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})={\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q})\\ =\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},\end{array}$(2.1) ) and (2.3(2.3) $\begin{array}{l}{\mathrm{\Phi }}_{1,\mathit{\zeta },1}^{\mathit{x},1,1}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };1,1)=\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\mathit{B}}_{1,1}^{\mathit{x},1,1}(\mathit{\zeta }+\mathit{m},1-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },1-\mathit{\zeta })}\frac{{\xi }^m}{{(\mathit{m}+\mathit{\tau })}^{\kappa }},\end{array}$(2.3) ), we get (1.6(1.6) ${\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau })=\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m(\mathit{\zeta }{)}_m}{(\mathit{\gamma }{)}_m(\mathit{m}+\mathit{\tau }{)}^{\kappa }}\frac{{\xi }^m}{\mathit{m}!},$(1.6) ) and (1.4(1.4) ${\mathrm{\Phi }}_{\mathit{\delta }}^{\ast }(\mathit{\xi },\mathit{\kappa },\mathit{\tau })=\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\mathit{\delta }{)}_{\mathit{m}}}{\mathit{m}!}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},$(1.4) ), respectively, as

(2.6) ${\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta },\mathit{\gamma }}^{0,1,1}(\mathit{\xi },\mathit{\kappa },\mathit{\tau },1,1)={\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau })=\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{(\mathit{\delta })}_m{(\mathit{\zeta })}_m}{{(\mathit{\gamma })}_m{(\mathit{m}+\mathit{\tau })}^{\kappa }}\frac{{\xi }^m}{\mathit{m}!}$(2.6)

and

(2.7) $\begin{array}{ll}{\mathrm{\Phi }}_{\mathit{\zeta }}^{\ast }(\mathit{\xi },\mathit{\kappa },\mathit{\tau };0)& \mathit{ }={\mathrm{\Phi }}_{1,\mathit{\zeta },1}^{0,1,1}(\mathit{\xi },\mathit{\kappa },\mathit{\tau },1,1)\\ \mathit{ }=\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\mathit{B}}_{1,1}^{0,1,1}(\mathit{\zeta }+\mathit{m},1-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },1-\mathit{\zeta })}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }}\\ \mathit{ }=\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\zeta {)}_m}{\mathit{m}!}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }}={\mathrm{\Phi }}_{\mathit{\zeta }}^{\ast }(\mathit{\xi },\mathit{\kappa },\mathit{\tau }).\end{array}$(2.7)

3. Integral representation and derivative formula

In this section, we introduce the various integral representations and differential formula of the extended generalized Hurwitz–Lerch zeta function (2.1(2.1) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})={\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q})\\ =\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},\end{array}$(2.1) ) as follows:

Theorem 3.1

The following integral representation of extended Hurwitz–Lerch zeta function holds true:

(3.1) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})\\ =\frac{1}{\mathrm{\Gamma }(\mathit{\kappa })}{\int }_0^{\mathrm{\infty }}{t}^{\mathit{\kappa }-1}{e}^{-\mathit{\tau t}}{F}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\delta },\mathit{\zeta },\mathit{\gamma };\mathit{\xi }{e}^{-\mathit{t}})\mathit{dt},\end{array}$(3.1)

$\begin{array}{l}\mathit{ }(\mathrm{\Re }(\mathit{x})\ge 0;\mathit{x}=0,\mathrm{\Re }(\mathit{\tau })>0;\mathrm{\Re }(\mathit{\kappa })>0\text{ if }|\mathit{\xi }|\le 0;\\ \mathrm{\Re }(\mathit{\kappa })>1\text{ if }\mathit{\xi }=1,\mathit{p},\mathit{q},\mathit{y},\mathit{z}>0,\mathrm{\Re }(\mathit{\gamma })>\mathrm{\Re }(\mathit{\zeta })>0).\end{array}$

Proof.

We have (2.1(2.1) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})={\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q})\\ =\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},\end{array}$(2.1) )

$\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})={\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta },\mathit{\gamma }}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\xi },\mathit{\kappa },\mathit{\tau },\mathit{p},\mathit{q})\\ =\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }}.\end{array}$

The identity below is obeyed by the Eulerian integral of the gamma function (see (Srivastava & Choi, 2011))

(3.2) $\frac{1}{(m+\tau {)}^{\kappa }}=\frac{1}{\mathrm{\Gamma }(\mathit{\kappa })}{\int }_0^{\mathrm{\infty }}{t}^{\mathit{\kappa }-1}{e}^{-(\mathit{m}+\mathit{\tau })\mathit{t}}\mathit{dt}$(3.2)

$\begin{array}{l}(min\{\mathrm{\Re }(\mathit{\kappa }),\mathrm{\Re }(\mathit{\tau })\}>0;\mathit{m}\in {\mathbb{N}}_0),\end{array}$

when the order of summation and integration are exchanged in (2.1(2.1) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})={\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q})\\ =\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},\end{array}$(2.1) ) by using (3.2(3.2) $\frac{1}{(m+\tau {)}^{\kappa }}=\frac{1}{\mathrm{\Gamma }(\mathit{\kappa })}{\int }_0^{\mathrm{\infty }}{t}^{\mathit{\kappa }-1}{e}^{-(\mathit{m}+\mathit{\tau })\mathit{t}}\mathit{dt}$(3.2) ), we obtain

$\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})\\ \mathit{ }=\frac{1}{\mathrm{\Gamma }(\mathit{\kappa })}{\int }_0^{\mathrm{\infty }}{t}^{\mathit{\kappa }-1}{e}^{-\mathit{\tau t}}(\sum _{\mathit{m}=0}^{\mathrm{\infty }}(\mathit{\delta }{)}_{\mathit{m}}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{(\mathit{\xi }{\mathit{e}}^{-\mathit{t}}{)}^m}{\mathit{m}!})\mathit{dt}.\end{array}$

Using (1.15(1.15) $B_{\mathit{p},\mathit{q}}({\theta }_1,{\theta }_2)={\int }_0^1{u}^{{\mathit{\theta }}_1-1}(1-\mathit{u}{)}^{{\mathit{\theta }}_2-1}\mathit{exp}(-\frac{p}{u}-\frac{q}{(1-\mathit{u})})\mathit{du}$(1.15) ) and (1.16(1.16) $F_{\mathit{p},\mathit{q}}({\theta }_1,{\theta }_2;{\theta }_3;\mathit{\xi })=\sum _{\mathit{m}=0}^{\mathrm{\infty }}({\theta }_1{)}_{\mathit{m}}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}({\theta }_2+\mathit{m},{\theta }_3-{\theta }_2)}{\mathit{B}({\theta }_2,{\theta }_3-{\theta }_2)}\frac{(\xi {)}^m}{\mathit{m}!}$(1.16) ), we obtain the desired result (3.1(3.1) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})\\ =\frac{1}{\mathrm{\Gamma }(\mathit{\kappa })}{\int }_0^{\mathrm{\infty }}{t}^{\mathit{\kappa }-1}{e}^{-\mathit{\tau t}}{F}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\delta },\mathit{\zeta },\mathit{\gamma };\mathit{\xi }{e}^{-\mathit{t}})\mathit{dt},\end{array}$(3.1) ).

Theorem 3.2

If $\mathrm{\Re }(\mathit{x})>0;\mathit{x}=0,\mathrm{\Re }(\mathit{\zeta })>\mathrm{\Re }(\mathit{\gamma })>0,\text{allowbreakmin}\{\mathrm{\Re }(\mathit{\kappa }),\mathrm{\Re }(\mathit{\tau })\}>0$, then the following integral representation holds true:

(3.3) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})\\ =\frac{1}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}{\int }_0^{\mathrm{\infty }}\frac{t^{\mathit{\zeta }-1}}{(1+\mathit{t}{)}^{\gamma }}{E}_{\mathit{p},\mathit{q}}(-\frac{x}{{\mathit{t}}^{\mathit{y}}(1-\mathit{t}{)}^z}){\mathrm{\Phi }}_{\mathit{\delta }}^{\ast }(\frac{\mathit{t\xi }}{1+\mathit{t}},\mathit{\kappa },\mathit{\tau })\mathit{dt}.\end{array}$(3.3)

Proof.

The integral representation of generalized beta function (Khan & Husain, 2022)

(3.4) $B_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}({\theta }_1,{\theta }_2)={\int }_0^{\mathrm{\infty }}\frac{t^{{\mathit{\theta }}_1-1}}{(1+\mathit{t}{)}^{{\mathit{\theta }}_1+{\theta }_2}}{E}_{\mathit{p},\mathit{q}}(-\frac{x}{{\mathit{t}}^{\mathit{y}}(1-\mathit{t}{)}^z})\mathit{dt},$(3.4)

on putting ${\theta }_1=\mathit{\zeta }+\mathit{m}$ and ${\theta }_2=\mathit{\gamma }-\mathit{\zeta }$ in (3.4(3.4) $B_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}({\theta }_1,{\theta }_2)={\int }_0^{\mathrm{\infty }}\frac{t^{{\mathit{\theta }}_1-1}}{(1+\mathit{t}{)}^{{\mathit{\theta }}_1+{\theta }_2}}{E}_{\mathit{p},\mathit{q}}(-\frac{x}{{\mathit{t}}^{\mathit{y}}(1-\mathit{t}{)}^z})\mathit{dt},$(3.4) ), we have

(3.5) $B_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })={\int }_0^{\mathrm{\infty }}\frac{t^{\mathit{\zeta }+\mathit{m}-1}}{(1+\mathit{t}{)}^{\mathit{\gamma }+\mathit{m}}}{E}_{\mathit{p},\mathit{q}}(-\frac{x}{{\mathit{t}}^{\mathit{y}}(1-\mathit{t}{)}^z})\mathit{dt},$(3.5)

from (3.5(3.5) $B_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })={\int }_0^{\mathrm{\infty }}\frac{t^{\mathit{\zeta }+\mathit{m}-1}}{(1+\mathit{t}{)}^{\mathit{\gamma }+\mathit{m}}}{E}_{\mathit{p},\mathit{q}}(-\frac{x}{{\mathit{t}}^{\mathit{y}}(1-\mathit{t}{)}^z})\mathit{dt},$(3.5) ) and (2.1(2.1) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})={\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q})\\ =\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},\end{array}$(2.1) ), we get

$\mathit{align}*\mathit{align}*$

(3.6) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})\\ =\frac{1}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}{\int }_0^{\mathrm{\infty }}\frac{t^{\mathit{\zeta }-1}}{(1+\mathit{t}{)}^{\gamma }}{E}_{\mathit{p},\mathit{q}}(-\frac{x}{{\mathit{t}}^{\mathit{y}}(1-\mathit{t}{)}^z})\\ \times \left(\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{(\xi t{)}^m}{(1+\mathit{t}{)}^m}\frac{1}{(m+\tau {)}^{\kappa }}\right)\mathit{dt}.\end{array}$(3.6)

Now, in view of (1.4(1.4) ${\mathrm{\Phi }}_{\mathit{\delta }}^{\ast }(\mathit{\xi },\mathit{\kappa },\mathit{\tau })=\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\mathit{\delta }{)}_{\mathit{m}}}{\mathit{m}!}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},$(1.4) ) in (3.6(3.6) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})\\ =\frac{1}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}{\int }_0^{\mathrm{\infty }}\frac{t^{\mathit{\zeta }-1}}{(1+\mathit{t}{)}^{\gamma }}{E}_{\mathit{p},\mathit{q}}(-\frac{x}{{\mathit{t}}^{\mathit{y}}(1-\mathit{t}{)}^z})\\ \times \left(\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{(\xi t{)}^m}{(1+\mathit{t}{)}^m}\frac{1}{(m+\tau {)}^{\kappa }}\right)\mathit{dt}.\end{array}$(3.6) ), we get the desired result (3.3(3.3) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})\\ =\frac{1}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}{\int }_0^{\mathrm{\infty }}\frac{t^{\mathit{\zeta }-1}}{(1+\mathit{t}{)}^{\gamma }}{E}_{\mathit{p},\mathit{q}}(-\frac{x}{{\mathit{t}}^{\mathit{y}}(1-\mathit{t}{)}^z}){\mathrm{\Phi }}_{\mathit{\delta }}^{\ast }(\frac{\mathit{t\xi }}{1+\mathit{t}},\mathit{\kappa },\mathit{\tau })\mathit{dt}.\end{array}$(3.3) ).

Theorem 3.3

For $\mathit{x}\ge 0,\mathrm{\Re }(\mathit{\delta })>0,\mathrm{\Re }(\mathit{\tau })>0,\text{allowbreak }\mathrm{\Re }(\mathit{\kappa })>0\text{ if }|\mathit{\xi }|\le 1;\mathrm{\Re }(\mathit{\kappa })>1\text{ if }\mathit{\xi }=1$, the following integral representation holds true

(3.7) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})\\ =\frac{1}{\mathrm{\Gamma }(\mathit{\delta })}{\int }_0^{\mathrm{\infty }}{t}^{\mathit{\delta }-1}{e}^{-\mathit{t}}{\mathrm{\Phi }}_{\mathit{\zeta },\mathit{\gamma }}^{\ast }(\mathit{\xi t},\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})\mathit{dt}\end{array}$(3.7)

where ${\mathrm{\Phi }}_{\mathit{\zeta },\mathit{\gamma }}^{\ast }(\mathit{\xi t},\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})$ is defined by (2.5(2.5) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\zeta },\mathit{\gamma }}^{\ast }(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})=1\mathit{mu}\\ \times \underset{|\mathit{\delta }|\to \mathrm{\infty }}{lim}\left\{{\mathrm{\Phi }}_{\mathit{\zeta },\mathit{\gamma }}^{\ast }\left(\frac{\xi }{\delta },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q},\mathit{x},\mathit{y},\mathit{z}\right)\right\}=1\mathit{mu}\\ \times \sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })\mathit{m}!}\frac{{\xi }^m}{{(\mathit{m}+\mathit{\tau })}^{\kappa }},1\mathit{mu}\end{array}$(2.5) ).

Proof.

The definition of the integral form of the Pochhammer symbol $(\mathit{\delta }{)}_m$ (Srivastava & Choi, 2001) is

(3.8) $(\mathit{\delta }{)}_{\mathit{m}}=\frac{1}{\mathrm{\Gamma }(\mathit{\delta })}{\int }_0^{\mathrm{\infty }}{t}^{\mathit{\delta }+\mathit{m}-1}{e}^{-\mathit{t}}\mathit{dt},$(3.8)

using (3.8(3.8) $(\mathit{\delta }{)}_{\mathit{m}}=\frac{1}{\mathrm{\Gamma }(\mathit{\delta })}{\int }_0^{\mathrm{\infty }}{t}^{\mathit{\delta }+\mathit{m}-1}{e}^{-\mathit{t}}\mathit{dt},$(3.8) ) in (2.1(2.1) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})={\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q})\\ =\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},\end{array}$(2.1) ) and interchanging the order of summation and integration under the given condition in (3.7(3.7) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})\\ =\frac{1}{\mathrm{\Gamma }(\mathit{\delta })}{\int }_0^{\mathrm{\infty }}{t}^{\mathit{\delta }-1}{e}^{-\mathit{t}}{\mathrm{\Phi }}_{\mathit{\zeta },\mathit{\gamma }}^{\ast }(\mathit{\xi t},\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})\mathit{dt}\end{array}$(3.7) ), we get

(3.9) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})\\ =\frac{1}{\mathrm{\Gamma }(\mathit{\delta })}{\int }_0^{\mathrm{\infty }}{t}^{\mathit{\delta }-1}{e}^{-\mathit{t}}\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{(\xi t{)}^m}{\mathit{m}!(\mathit{m}+\mathit{\tau }{)}^{\kappa }}\mathit{dt}.\end{array}$(3.9)

Now, with the help of (2.5(2.5) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\zeta },\mathit{\gamma }}^{\ast }(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})=1\mathit{mu}\\ \times \underset{|\mathit{\delta }|\to \mathrm{\infty }}{lim}\left\{{\mathrm{\Phi }}_{\mathit{\zeta },\mathit{\gamma }}^{\ast }\left(\frac{\xi }{\delta },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q},\mathit{x},\mathit{y},\mathit{z}\right)\right\}=1\mathit{mu}\\ \times \sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })\mathit{m}!}\frac{{\xi }^m}{{(\mathit{m}+\mathit{\tau })}^{\kappa }},1\mathit{mu}\end{array}$(2.5) ) in (3.9(3.9) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})\\ =\frac{1}{\mathrm{\Gamma }(\mathit{\delta })}{\int }_0^{\mathrm{\infty }}{t}^{\mathit{\delta }-1}{e}^{-\mathit{t}}\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{(\xi t{)}^m}{\mathit{m}!(\mathit{m}+\mathit{\tau }{)}^{\kappa }}\mathit{dt}.\end{array}$(3.9) ), we get (3.7(3.7) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})\\ =\frac{1}{\mathrm{\Gamma }(\mathit{\delta })}{\int }_0^{\mathrm{\infty }}{t}^{\mathit{\delta }-1}{e}^{-\mathit{t}}{\mathrm{\Phi }}_{\mathit{\zeta },\mathit{\gamma }}^{\ast }(\mathit{\xi t},\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})\mathit{dt}\end{array}$(3.7) ).

Theorem 3.4

For any positive integer m, the extended Hurwitz–Lerch zeta function ${\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})$ has the following differential formula:

(3.10) $\begin{array}{l}\frac{d^m}{\mathit{d}{\xi }^m}\{{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta },\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})\}\\ =\frac{(\delta {)}_m(\mathit{\zeta }{)}_m}{(\mathit{\gamma }{)}_m}{\mathrm{\Phi }}_{\mathit{\delta }+\mathit{m},\mathit{\zeta }+\mathit{m},\mathit{\gamma }+\mathit{m}}(\mathit{\xi },\mathit{\kappa },\mathit{\tau }+\mathit{m};\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z}).\end{array}$(3.10)

Proof.

The differentiation of (2.1(2.1) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})={\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q})\\ =\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},\end{array}$(2.1) ) is

(3.11) $\begin{array}{l}\frac{d}{\mathit{d\xi }}\{{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta },\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})\}\\ =\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{(\mathit{m}-1)!}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^{\mathit{m}-1}}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},\end{array}$(3.11)

by changing m by m +1 in (3.11(3.11) $\begin{array}{l}\frac{d}{\mathit{d\xi }}\{{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta },\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})\}\\ =\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{(\mathit{m}-1)!}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^{\mathit{m}-1}}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},\end{array}$(3.11) ) and applying the identity

$\begin{array}{l}\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })=\frac{\gamma }{\zeta }\mathit{B}(\mathit{\zeta }+1,\mathit{\gamma }-\mathit{\zeta }),(\mathit{\tau }{)}_{\mathit{m}+1}=\mathit{\tau }(\mathit{\tau }+1{)}_m,\end{array}$

we get

$\begin{array}{l}\frac{d}{\mathit{d\xi }}\{{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta },\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})\}\\ =\frac{\mathit{\delta \zeta }}{\gamma }{\mathrm{\Phi }}_{\mathit{\delta }+1,\mathit{\zeta }+1,\mathit{\gamma }+1}(\mathit{\xi },\mathit{\kappa },\mathit{\tau }+1,\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z}).\end{array}$

Now, the procedure of recursive application provides the desired result (3.10(3.10) $\begin{array}{l}\frac{d^m}{\mathit{d}{\xi }^m}\{{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta },\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})\}\\ =\frac{(\delta {)}_m(\mathit{\zeta }{)}_m}{(\mathit{\gamma }{)}_m}{\mathrm{\Phi }}_{\mathit{\delta }+\mathit{m},\mathit{\zeta }+\mathit{m},\mathit{\gamma }+\mathit{m}}(\mathit{\xi },\mathit{\kappa },\mathit{\tau }+\mathit{m};\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z}).\end{array}$(3.10) ).

4. Mellin transform and some generating relations

In this section, we deal with the Mellin transform and some generating relations of extended Hurwitz–Lerch zeta function.

An appropriate integrable function f(b) with index Ψ has the following definition for its Mellin transform:

(4.1) $\mathcal{M}\{\mathit{f}(\mathit{b}):\mathit{b}\to \mathrm{\Psi }\}={\int }_0^{\mathrm{\infty }}{b}^{\mathrm{\Psi }-1}\mathit{f}(\mathit{b})\mathit{db};(0<\mathit{b}<\mathrm{\infty }).$(4.1)

Theorem 4.1

The Mellin transformation of the function ${\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})$ is defined as

(4.2) $\begin{array}{l}\mathcal{M}\{{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z}):\mathit{x}\to \mathrm{\Psi }\}\\ =\frac{\pi }{\mathit{sin}(\mathrm{\Psi \pi })}\frac{\mathit{B}(\mathit{\zeta }+\mathit{y}\mathrm{\Psi },\mathit{\gamma }-\mathit{\zeta }+\mathit{z}\mathrm{\Psi })}{\mathrm{\Gamma }(1-\mathrm{\Psi }\mathit{p})\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}{\mathrm{\Phi }}_{\mathit{\delta }}^{\ast }(\mathit{\xi t},\mathit{\kappa },\mathit{\tau }),\end{array}$(4.2)

$\begin{array}{l}(\mathrm{\Re }(\mathit{x})>0,\mathrm{\Re }(\mathrm{\Psi })>0,\mathrm{\Re }(\mathit{\zeta }+\mathit{m}+\mathrm{\Psi })>0).\end{array}$

Proof.

In view of (4.1(4.1) $\mathcal{M}\{\mathit{f}(\mathit{b}):\mathit{b}\to \mathrm{\Psi }\}={\int }_0^{\mathrm{\infty }}{b}^{\mathrm{\Psi }-1}\mathit{f}(\mathit{b})\mathit{db};(0<\mathit{b}<\mathrm{\infty }).$(4.1) ), the Mellin transformation of (2.1(2.1) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})={\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q})\\ =\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},\end{array}$(2.1) ) is given by

(4.3) $\begin{array}{l}\mathcal{M}\{{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z}):\mathit{x}\to \mathrm{\Psi }\}\\ ={\int }_0^{\mathrm{\infty }}{x}^{\mathrm{\Psi }-1}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta },\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})\mathit{dx}\\ ={\int }_0^{\mathrm{\infty }}{x}^{\mathrm{\Psi }-1}\left(\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }}\right)\mathit{dx}\\ =\frac{1}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }}\times {\int }_0^{\mathrm{\infty }}{x}^{\mathrm{\Psi }-1}{B}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })\mathit{dx}\\ =\frac{1}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{(\xi t{)}^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }}{\int }_0^{\mathrm{\infty }}{x}^{\mathrm{\Psi }-1}\\ ={\int }_0^1{t}^{\mathit{\zeta }-1}(1-\mathit{t}{)}^{\mathit{\gamma }-\mathit{\zeta }-1}{E}_{\mathit{p},\mathit{q}}(-\frac{x}{t^y(1-\mathit{t}{)}^z})\mathit{dt}\mathit{dx}.\end{array}$(4.3)

Interchanging the order of integration in (4.3(4.3) $\begin{array}{l}\mathcal{M}\{{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z}):\mathit{x}\to \mathrm{\Psi }\}\\ ={\int }_0^{\mathrm{\infty }}{x}^{\mathrm{\Psi }-1}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta },\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})\mathit{dx}\\ ={\int }_0^{\mathrm{\infty }}{x}^{\mathrm{\Psi }-1}\left(\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }}\right)\mathit{dx}\\ =\frac{1}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }}\times {\int }_0^{\mathrm{\infty }}{x}^{\mathrm{\Psi }-1}{B}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })\mathit{dx}\\ =\frac{1}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{(\xi t{)}^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }}{\int }_0^{\mathrm{\infty }}{x}^{\mathrm{\Psi }-1}\\ ={\int }_0^1{t}^{\mathit{\zeta }-1}(1-\mathit{t}{)}^{\mathit{\gamma }-\mathit{\zeta }-1}{E}_{\mathit{p},\mathit{q}}(-\frac{x}{t^y(1-\mathit{t}{)}^z})\mathit{dt}\mathit{dx}.\end{array}$(4.3) ), we have

(4.4) $\begin{array}{l}\mathcal{M}\{{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z}):\mathit{x}\to \mathrm{\Psi }\}\\ =\frac{1}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{(\xi t{)}^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }}{\int }_0^1{t}^{\mathit{\zeta }-1}(1-\mathit{t}{)}^{\mathit{\gamma }-\mathit{\zeta }-1}\\ \times \left({\int }_0^{\mathrm{\infty }}{x}^{\mathrm{\Psi }-1}{E}_{\mathit{p},\mathit{q}}(-\frac{x}{t^y(1-\mathit{t}{)}^z})\mathit{dx}\right)\mathit{dt}.\end{array}$(4.4)

The second part of (4.4(4.4) $\begin{array}{l}\mathcal{M}\{{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z}):\mathit{x}\to \mathrm{\Psi }\}\\ =\frac{1}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{(\xi t{)}^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }}{\int }_0^1{t}^{\mathit{\zeta }-1}(1-\mathit{t}{)}^{\mathit{\gamma }-\mathit{\zeta }-1}\\ \times \left({\int }_0^{\mathrm{\infty }}{x}^{\mathrm{\Psi }-1}{E}_{\mathit{p},\mathit{q}}(-\frac{x}{t^y(1-\mathit{t}{)}^z})\mathit{dx}\right)\mathit{dt}.\end{array}$(4.4) )

(4.5) $\begin{array}{l}\mathit{ }{\int }_0^{\mathrm{\infty }}{x}^{\mathrm{\Psi }-1}{E}_{\mathit{p},\mathit{q}}(-\frac{x}{t^y(1-\mathit{t}{)}^z})\mathit{dx}\\ =(t^y{)}^{\mathrm{\Psi }}{((1-t{)}^z)}^{\mathrm{\Psi }}{\int }_0^{\mathrm{\infty }}{\gamma }^{\mathrm{\Psi }-1}{E}_{\mathit{p},\mathit{q}}(-\mathit{\gamma })\mathit{d\gamma }\\ =t^{\mathit{y}\mathrm{\Psi }}(1-\mathit{t}{)}^{\mathit{z}\mathrm{\Psi }}{\int }_0^{\mathrm{\infty }}{\gamma }^{\mathrm{\Psi }-1}{E}_{\mathit{p},\mathit{q}}(-\mathit{\gamma })\mathit{d\gamma },\end{array}$(4.5)

and recall

(4.6) ${\int }_0^{\mathrm{\infty }}{\gamma }^{\mathrm{\Psi }-1}{E}_{\mathit{p},\mathit{q},\mathit{\delta }}^{\mathit{\eta }}(-\mathit{\omega \gamma })\mathit{d\gamma }=\frac{\mathrm{\Gamma }(\mathrm{\Psi })\mathrm{\Gamma }(\mathit{\eta }-\mathrm{\Psi })}{\mathrm{\Gamma }(\mathit{\eta }){\mathit{\omega }}^{\mathrm{\Psi }}\mathrm{\Gamma }(\mathit{\delta }-\mathrm{\Psi p})}.$(4.6)

Putting $\mathit{\delta }=\mathit{\eta }=1$ and ω = 1 in (4.6(4.6) ${\int }_0^{\mathrm{\infty }}{\gamma }^{\mathrm{\Psi }-1}{E}_{\mathit{p},\mathit{q},\mathit{\delta }}^{\mathit{\eta }}(-\mathit{\omega \gamma })\mathit{d\gamma }=\frac{\mathrm{\Gamma }(\mathrm{\Psi })\mathrm{\Gamma }(\mathit{\eta }-\mathrm{\Psi })}{\mathrm{\Gamma }(\mathit{\eta }){\mathit{\omega }}^{\mathrm{\Psi }}\mathrm{\Gamma }(\mathit{\delta }-\mathrm{\Psi p})}.$(4.6) ), we get

(4.7) ${\int }_0^{\mathrm{\infty }}{\gamma }^{\mathrm{\Psi }-1}{E}_{\mathit{p},\mathit{q},}(-\mathit{\gamma })\mathit{d\gamma }=\frac{\mathrm{\Gamma }(\mathrm{\Psi })\mathrm{\Gamma }(1-\mathrm{\Psi })}{\mathrm{\Gamma }(1-\mathrm{\Psi }\mathit{p})},$(4.7)

therefore, by (4.5(4.5) $\begin{array}{l}\mathit{ }{\int }_0^{\mathrm{\infty }}{x}^{\mathrm{\Psi }-1}{E}_{\mathit{p},\mathit{q}}(-\frac{x}{t^y(1-\mathit{t}{)}^z})\mathit{dx}\\ =(t^y{)}^{\mathrm{\Psi }}{((1-t{)}^z)}^{\mathrm{\Psi }}{\int }_0^{\mathrm{\infty }}{\gamma }^{\mathrm{\Psi }-1}{E}_{\mathit{p},\mathit{q}}(-\mathit{\gamma })\mathit{d\gamma }\\ =t^{\mathit{y}\mathrm{\Psi }}(1-\mathit{t}{)}^{\mathit{z}\mathrm{\Psi }}{\int }_0^{\mathrm{\infty }}{\gamma }^{\mathrm{\Psi }-1}{E}_{\mathit{p},\mathit{q}}(-\mathit{\gamma })\mathit{d\gamma },\end{array}$(4.5) )

(4.8) $\begin{array}{l}\mathit{ }{\int }_0^{\mathrm{\infty }}{x}^{\mathrm{\Psi }-1}{E}_{\mathit{p},\mathit{q}}(-\frac{x}{t^y(1-\mathit{t}{)}^z})\mathit{dx}\\ =\frac{{\mathit{t}}^{\mathit{y}\mathrm{\Psi }}(1-\mathit{t}{)}^{\mathit{z}\mathrm{\Psi }}\mathrm{\Gamma }(\mathrm{\Psi })\mathrm{\Gamma }(1-\mathrm{\Psi })}{\mathrm{\Gamma }(1-\mathrm{\Psi }\mathit{p})}.\end{array}$(4.8)

Also,

(4.9) $\mathrm{\Gamma }(\mathrm{\Psi })\mathrm{\Gamma }(1-\mathrm{\Psi })=\frac{\pi }{\mathit{sin}(\mathit{\pi }\mathrm{\Psi })}.$(4.9)

Using (4.4(4.4) $\begin{array}{l}\mathcal{M}\{{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z}):\mathit{x}\to \mathrm{\Psi }\}\\ =\frac{1}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{(\xi t{)}^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }}{\int }_0^1{t}^{\mathit{\zeta }-1}(1-\mathit{t}{)}^{\mathit{\gamma }-\mathit{\zeta }-1}\\ \times \left({\int }_0^{\mathrm{\infty }}{x}^{\mathrm{\Psi }-1}{E}_{\mathit{p},\mathit{q}}(-\frac{x}{t^y(1-\mathit{t}{)}^z})\mathit{dx}\right)\mathit{dt}.\end{array}$(4.4) ), (4.8(4.8) $\begin{array}{l}\mathit{ }{\int }_0^{\mathrm{\infty }}{x}^{\mathrm{\Psi }-1}{E}_{\mathit{p},\mathit{q}}(-\frac{x}{t^y(1-\mathit{t}{)}^z})\mathit{dx}\\ =\frac{{\mathit{t}}^{\mathit{y}\mathrm{\Psi }}(1-\mathit{t}{)}^{\mathit{z}\mathrm{\Psi }}\mathrm{\Gamma }(\mathrm{\Psi })\mathrm{\Gamma }(1-\mathrm{\Psi })}{\mathrm{\Gamma }(1-\mathrm{\Psi }\mathit{p})}.\end{array}$(4.8) ), (4.9(4.9) $\mathrm{\Gamma }(\mathrm{\Psi })\mathrm{\Gamma }(1-\mathrm{\Psi })=\frac{\pi }{\mathit{sin}(\mathit{\pi }\mathrm{\Psi })}.$(4.9) ) and (1.4(1.4) ${\mathrm{\Phi }}_{\mathit{\delta }}^{\ast }(\mathit{\xi },\mathit{\kappa },\mathit{\tau })=\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\mathit{\delta }{)}_{\mathit{m}}}{\mathit{m}!}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},$(1.4) ), we get the required result (4.2(4.2) $\begin{array}{l}\mathcal{M}\{{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z}):\mathit{x}\to \mathrm{\Psi }\}\\ =\frac{\pi }{\mathit{sin}(\mathrm{\Psi \pi })}\frac{\mathit{B}(\mathit{\zeta }+\mathit{y}\mathrm{\Psi },\mathit{\gamma }-\mathit{\zeta }+\mathit{z}\mathrm{\Psi })}{\mathrm{\Gamma }(1-\mathrm{\Psi }\mathit{p})\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}{\mathrm{\Phi }}_{\mathit{\delta }}^{\ast }(\mathit{\xi t},\mathit{\kappa },\mathit{\tau }),\end{array}$(4.2) ).

Theorem 4.2

The Hurwitz–Lerch zeta function ${\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})$ has the following generating relation:

(4.10) $\begin{array}{l}\mathit{ }\sum _{\mathit{n}=0}^{\mathrm{\infty }}(\mathit{\delta }{)}_{\mathit{n}}{\mathrm{\Phi }}_{\mathit{\delta }+\mathit{n},\mathit{\zeta },\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q},\mathit{x},\mathit{y},\mathit{z})\frac{t^n}{\mathit{n}!}\\ =(1-\mathit{t}{)}^{-\mathit{\delta }}{\mathrm{\Phi }}_{\mathit{\delta }+\mathit{n},\mathit{\zeta },\mathit{\gamma }}(\frac{\xi }{1-\mathit{t}},\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q},\mathit{x},\mathit{y},\mathit{z}),\end{array}$(4.10)

$\begin{array}{l}(\mathit{x}\ge 0,\mathit{p}>0,\mathit{\delta },\mathit{\zeta },\mathit{\gamma }\in \mathbb{C}\text{ and }|\mathit{t}|<1).\end{array}$

Proof.

In LHS of (4.10(4.10) $\begin{array}{l}\mathit{ }\sum _{\mathit{n}=0}^{\mathrm{\infty }}(\mathit{\delta }{)}_{\mathit{n}}{\mathrm{\Phi }}_{\mathit{\delta }+\mathit{n},\mathit{\zeta },\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q},\mathit{x},\mathit{y},\mathit{z})\frac{t^n}{\mathit{n}!}\\ =(1-\mathit{t}{)}^{-\mathit{\delta }}{\mathrm{\Phi }}_{\mathit{\delta }+\mathit{n},\mathit{\zeta },\mathit{\gamma }}(\frac{\xi }{1-\mathit{t}},\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q},\mathit{x},\mathit{y},\mathit{z}),\end{array}$(4.10) ), using (2.1(2.1) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})={\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q})\\ =\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},\end{array}$(2.1) ), we get

(4.11) $\begin{array}{l}\begin{array}{l}\mathit{ }\sum _{\mathit{n}=0}^{\mathrm{\infty }}(\mathit{\delta }{)}_{\mathit{n}}{\mathrm{\Phi }}_{\mathit{\delta }+\mathit{n},\mathit{\zeta },\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q},\mathit{x},\mathit{y},\mathit{z})\frac{t^n}{\mathit{n}!}\\ =\sum _{\mathit{n}=0}^{\mathrm{\infty }}(\mathit{\delta }{)}_{\mathit{n}}\{\sum _{\mathit{m}=0}^{\mathrm{\infty }}(\mathit{\delta }+\mathit{n}{)}_{\mathit{m}}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{\mathit{m}!(\mathit{m}+\mathit{\tau }{)}^{\kappa }}\}\frac{t^n}{\mathit{n}!}.\end{array}\end{array}$(4.11)

With the identity $(\mathit{\delta }{)}_{\mathit{n}}(\mathit{\delta }+\mathit{n}{)}_{\mathit{m}}=(\mathit{\delta }{)}_{\mathit{m}}(\mathit{\delta }+\mathit{m}{)}_{\mathit{n}}$ and interchanging the order of summations in (4.11(4.11) $\begin{array}{l}\begin{array}{l}\mathit{ }\sum _{\mathit{n}=0}^{\mathrm{\infty }}(\mathit{\delta }{)}_{\mathit{n}}{\mathrm{\Phi }}_{\mathit{\delta }+\mathit{n},\mathit{\zeta },\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q},\mathit{x},\mathit{y},\mathit{z})\frac{t^n}{\mathit{n}!}\\ =\sum _{\mathit{n}=0}^{\mathrm{\infty }}(\mathit{\delta }{)}_{\mathit{n}}\{\sum _{\mathit{m}=0}^{\mathrm{\infty }}(\mathit{\delta }+\mathit{n}{)}_{\mathit{m}}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{\mathit{m}!(\mathit{m}+\mathit{\tau }{)}^{\kappa }}\}\frac{t^n}{\mathit{n}!}.\end{array}\end{array}$(4.11) ), we have

(4.12) $=\sum _{\mathit{m}=0}^{\mathrm{\infty }}(\mathit{\delta }{)}_{\mathit{m}}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\{\sum _{\mathit{n}=0}^{\mathrm{\infty }}(\mathit{\delta }+\mathit{m}{)}_{\mathit{n}}\frac{t^n}{\mathit{n}!}\}\frac{{\xi }^m}{\mathit{m}!(\mathit{m}+\mathit{\tau }{)}^{\kappa }}.$(4.12)

We know

(4.13) $(1-\mathit{t}{)}^{-\mathit{\delta }-\mathit{m}}=\sum _{\mathit{n}=0}^{\mathrm{\infty }}(\mathit{\delta }+\mathit{m}{)}_{\mathit{n}}\frac{t^n}{\mathit{n}!},(|\mathit{t}|<1).$(4.13)

Now, using (4.13(4.13) $(1-\mathit{t}{)}^{-\mathit{\delta }-\mathit{m}}=\sum _{\mathit{n}=0}^{\mathrm{\infty }}(\mathit{\delta }+\mathit{m}{)}_{\mathit{n}}\frac{t^n}{\mathit{n}!},(|\mathit{t}|<1).$(4.13) ) in (4.12(4.12) $=\sum _{\mathit{m}=0}^{\mathrm{\infty }}(\mathit{\delta }{)}_{\mathit{m}}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\{\sum _{\mathit{n}=0}^{\mathrm{\infty }}(\mathit{\delta }+\mathit{m}{)}_{\mathit{n}}\frac{t^n}{\mathit{n}!}\}\frac{{\xi }^m}{\mathit{m}!(\mathit{m}+\mathit{\tau }{)}^{\kappa }}.$(4.12) ), we get the desired result (4.10(4.10) $\begin{array}{l}\mathit{ }\sum _{\mathit{n}=0}^{\mathrm{\infty }}(\mathit{\delta }{)}_{\mathit{n}}{\mathrm{\Phi }}_{\mathit{\delta }+\mathit{n},\mathit{\zeta },\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q},\mathit{x},\mathit{y},\mathit{z})\frac{t^n}{\mathit{n}!}\\ =(1-\mathit{t}{)}^{-\mathit{\delta }}{\mathrm{\Phi }}_{\mathit{\delta }+\mathit{n},\mathit{\zeta },\mathit{\gamma }}(\frac{\xi }{1-\mathit{t}},\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q},\mathit{x},\mathit{y},\mathit{z}),\end{array}$(4.10) ).

Theorem 4.3

The Hurwitz–Lerch zeta function ${\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})$ holds the following generating relation:

(4.14) $\begin{array}{l}\mathit{ }\sum _{\mathit{n}=0}^{\mathrm{\infty }}\frac{(\mathit{k}{)}_{\mathit{n}}}{\mathit{n}!}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta },\mathit{\gamma }}(\mathit{\xi },\mathit{k}+\mathit{n},\mathit{\tau },\mathit{p},\mathit{q},\mathit{x},\mathit{y},\mathit{z})t^n\\ ={\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta },\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau }-\mathit{t},\mathit{p},\mathit{q},\mathit{x},\mathit{y},\mathit{z})\end{array}$(4.14)

Proof.

In RHS of (4.14(4.14) $\begin{array}{l}\mathit{ }\sum _{\mathit{n}=0}^{\mathrm{\infty }}\frac{(\mathit{k}{)}_{\mathit{n}}}{\mathit{n}!}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta },\mathit{\gamma }}(\mathit{\xi },\mathit{k}+\mathit{n},\mathit{\tau },\mathit{p},\mathit{q},\mathit{x},\mathit{y},\mathit{z})t^n\\ ={\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta },\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau }-\mathit{t},\mathit{p},\mathit{q},\mathit{x},\mathit{y},\mathit{z})\end{array}$(4.14) ), applying (2.1(2.1) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})={\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q})\\ =\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},\end{array}$(2.1) ), we get

(4.15) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta },\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau }-\mathit{t},\mathit{p},\mathit{q},\mathit{x},\mathit{y},\mathit{z})\\ =\sum _{\mathit{m}=0}^{\mathrm{\infty }}(\mathit{\delta }{)}_{\mathit{m}}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{\mathit{m}!(\mathit{m}+\mathit{\tau }-\mathit{t}{)}^k}\\ =\sum _{\mathit{m}=0}^{\mathrm{\infty }}(\mathit{\delta }{)}_{\mathit{m}}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{\mathit{m}!(\mathit{m}+\mathit{\tau }{)}^k}{(1-\frac{\mathit{t}}{\mathit{m}+\mathit{\tau }})}^{-\mathit{k}}.\end{array}$(4.15)

Using (4.13(4.13) $(1-\mathit{t}{)}^{-\mathit{\delta }-\mathit{m}}=\sum _{\mathit{n}=0}^{\mathrm{\infty }}(\mathit{\delta }+\mathit{m}{)}_{\mathit{n}}\frac{t^n}{\mathit{n}!},(|\mathit{t}|<1).$(4.13) ) in (4.15(4.15) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta },\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau }-\mathit{t},\mathit{p},\mathit{q},\mathit{x},\mathit{y},\mathit{z})\\ =\sum _{\mathit{m}=0}^{\mathrm{\infty }}(\mathit{\delta }{)}_{\mathit{m}}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{\mathit{m}!(\mathit{m}+\mathit{\tau }-\mathit{t}{)}^k}\\ =\sum _{\mathit{m}=0}^{\mathrm{\infty }}(\mathit{\delta }{)}_{\mathit{m}}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{\mathit{m}!(\mathit{m}+\mathit{\tau }{)}^k}{(1-\frac{\mathit{t}}{\mathit{m}+\mathit{\tau }})}^{-\mathit{k}}.\end{array}$(4.15) ) and then applying (2.1(2.1) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})={\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q})\\ =\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},\end{array}$(2.1) ), we get the desired result (4.14(4.14) $\begin{array}{l}\mathit{ }\sum _{\mathit{n}=0}^{\mathrm{\infty }}\frac{(\mathit{k}{)}_{\mathit{n}}}{\mathit{n}!}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta },\mathit{\gamma }}(\mathit{\xi },\mathit{k}+\mathit{n},\mathit{\tau },\mathit{p},\mathit{q},\mathit{x},\mathit{y},\mathit{z})t^n\\ ={\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta },\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau }-\mathit{t},\mathit{p},\mathit{q},\mathit{x},\mathit{y},\mathit{z})\end{array}$(4.14) ).

5. Fractional kinetic equation

The Hurwitz–Lerch zeta function (2.1(2.1) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})={\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q})\\ =\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},\end{array}$(2.1) ) is used to discuss the fractional kinetic equation (FKE). The non-stationary problem (Hohenberg & Halperin, 1977), turbulent flow (Novikov, 1994), diffusion (Mainardi, 1994) and kinetics (Gelfand & Shilov, 1964) are the various field in which the solution of FKE is used. The relationship given by Haubold and Mathai (2000) is defined as

(5.1) $\frac{\mathit{d}\mathsf{N}}{\mathit{d\ell }}=-\mathit{\sigma }({\mathsf{N}}_{\ell })+\mathit{p}({\mathsf{N}}_{\ell }),$(5.1)

where reaction rate is denoted by $\mathsf{N}=\mathsf{N}(\mathit{\ell })$, $\mathit{\sigma }({\mathsf{N}}_{\ell })=:\mathit{\sigma }$ is the destruction rate, $\mathit{p}=\mathit{p}(\mathsf{N})$ is the rate of production and ${\mathsf{N}}_{\ell }$ signifies the function defined by ${\mathsf{N}}_{\ell }({\ell }^{\ast })=\mathsf{N}(\mathit{\ell }-{\ell }^{\ast });{\ell }^{\ast }>0$.

We reach a specific case of (5.1(5.1) $\frac{\mathit{d}\mathsf{N}}{\mathit{d\ell }}=-\mathit{\sigma }({\mathsf{N}}_{\ell })+\mathit{p}({\mathsf{N}}_{\ell }),$(5.1) ) under homogeneties where the quantity ${\mathsf{N}}_{\ell }$ as (Haubold & Mathai, 2000; Kourganoff, 1973):

(5.2) $\frac{\mathit{d}{\mathsf{N}}_i}{\mathit{d\ell }}=-c_i{\mathsf{N}}_i(\mathit{\ell }),$(5.2)

where ${\mathsf{N}}_i(\mathit{\ell }=0)={\mathsf{N}}_0$ is the initial condition, representing the number of density of species i at time $\mathit{\ell }=0;c_i>0$. If the index i is not included in the integration of both sides of (5.2(5.2) $\frac{\mathit{d}{\mathsf{N}}_i}{\mathit{d\ell }}=-c_i{\mathsf{N}}_i(\mathit{\ell }),$(5.2) ), we obtain

(5.3) $\mathsf{N}(\mathit{\ell })-{\mathsf{N}}_0=-c_0{ }_0{\mathcal{D}}_{\mathit{\ell }}^{-1}-\mathsf{N}(\mathit{\ell }),$(5.3)

where the standard fractional integral operator is denoted by ${ }_0{\mathcal{D}}_{\mathit{\ell }}^{-1}$.

Also, Haubold and Mathai (2000) investigated a fractional generalization of standard kinetic (5.2(5.2) $\frac{\mathit{d}{\mathsf{N}}_i}{\mathit{d\ell }}=-c_i{\mathsf{N}}_i(\mathit{\ell }),$(5.2) ) as follows:

(5.4) $\mathsf{N}(\mathit{\ell })-{\mathsf{N}}_0=-c_0^{\mathit{\omega }}{ }_0{\mathcal{D}}_{\mathit{\ell }}^{-\mathit{\omega }}-\mathsf{N}(\mathit{\ell }),$(5.4)

where the well-known Riemann–Liouville fractional integral operator ${ }_0{\mathcal{D}}_{\mathit{\ell }}^{-\mathit{\omega }}$ is defined as

(5.5) ${ }_0{\mathcal{D}}_{\mathit{\ell }}^{-\mathit{\omega }}\mathit{f}(\mathit{\ell })=\frac{1}{\mathrm{\Gamma }(\mathit{\omega })}{\int }_0^{\mathit{\ell }}(\mathit{\ell }-\mathit{v}{)}^{\mathit{\omega }-1}\mathit{f}(\mathit{v})\mathit{dv}(\mathit{\ell }>0,\mathrm{\Re }(\mathit{\omega })>0),$(5.5)

and the solution of (5.5(5.5) ${ }_0{\mathcal{D}}_{\mathit{\ell }}^{-\mathit{\omega }}\mathit{f}(\mathit{\ell })=\frac{1}{\mathrm{\Gamma }(\mathit{\omega })}{\int }_0^{\mathit{\ell }}(\mathit{\ell }-\mathit{v}{)}^{\mathit{\omega }-1}\mathit{f}(\mathit{v})\mathit{dv}(\mathit{\ell }>0,\mathrm{\Re }(\mathit{\omega })>0),$(5.5) ) is

(5.6) $\mathsf{N}(\mathit{\ell })={\mathsf{N}}_0\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(-1{)}^m}{\mathrm{\Gamma }(\mathit{\omega m}+1)}(\mathit{c\ell }{)}^{\mathit{\omega k}}.$(5.6)

(Haubold & Mathai, 2000) obtained fractional kinetic equation as

(5.7) $\mathsf{N}(\mathit{\ell })-{\mathsf{N}}_0\mathit{f}(\mathit{\ell })=-c_0^{\mathit{\omega }}{ }_0{\mathcal{D}}_{\mathit{\ell }}^{-\mathit{\omega }}\mathsf{N}(\mathit{\ell }).(\mathrm{\Re }(\mathit{\omega })>0),$(5.7)

where the given species, $\mathsf{N}(\mathit{\ell })$ is the number of density at time $\mathit{\ell }$, ${\mathsf{N}}_0=\mathsf{N}(0)$ is the number of density of that species at time $\mathit{\ell }=0$, c is the constant and $\mathit{f}\in \mathcal{L}(0,\mathrm{\infty })$.

Applying the Laplace transform to (5.7(5.7) $\mathsf{N}(\mathit{\ell })-{\mathsf{N}}_0\mathit{f}(\mathit{\ell })=-c_0^{\mathit{\omega }}{ }_0{\mathcal{D}}_{\mathit{\ell }}^{-\mathit{\omega }}\mathsf{N}(\mathit{\ell }).(\mathrm{\Re }(\mathit{\omega })>0),$(5.7) ),

(5.8) $\begin{array}{l}\begin{array}{l}\mathit{L}\{\mathsf{N}(\mathit{\ell });\mathcal{P}\}={\mathsf{N}}_0(\sum _{\mathit{m}=0}^{\mathrm{\infty }}(-c^{\mathit{\omega }}{)}^m{\mathcal{P}}^{-\mathit{m\omega }})\mathit{F}(\mathcal{P})\\ (\mathit{m}\in {\mathbb{N}}_0,|\frac{c}{\mathcal{P}}|<1)\end{array}\end{array}$(5.8)

(5.9) $\begin{array}{l}\mathit{F}(\mathcal{P})=\mathit{L}\{\mathit{f}(\mathit{\ell });\mathcal{P}\}={\int }_0^{\mathrm{\infty }}{e}^{-\mathcal{P}\ell }\mathit{f}(\mathit{\ell })\mathit{d\ell }(\mathrm{\Re }(\mathcal{P})>0).\end{array}$(5.9)

We now move on to consider our new extended Hurwitz–Lerch zeta function (2.1(2.1) $\begin{array}{l}{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})={\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q})\\ =\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{(\delta {)}_m}{\mathit{m}!}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{{\xi }^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }},\end{array}$(2.1) ) in order to derive the solution of the generalized fractional kinetic equations.

Theorem 5.1

If d > 0, ω > 0 and $\mathit{\tau }\ne \mathit{d}$, c > 0, the solution of the equation is

(5.10) $\begin{array}{l}\mathsf{N}(\mathit{\ell })-{\mathsf{N}}_0\{{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}({\ell }^{\mathit{\omega }}\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})\}=-d_0^{\mathit{\omega }}{ }_0{\mathcal{D}}_{\mathit{\ell }}^{-\mathit{\omega }}\mathsf{N}(\mathit{\ell });\end{array}$(5.10)

is obtained by the following relation:

(5.11) $\begin{array}{l}\begin{array}{ll}\mathsf{N}(\mathit{\ell })& \mathit{ }={\mathsf{N}}_0\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{(\delta {)}_m}{\mathit{m}!}\frac{1}{(m+\tau {)}^{\kappa }}\\ \times \mathrm{\Gamma }(\mathit{m\omega }+\mathit{m}+1){\ell }^{\mathit{m\omega }+\mathit{m}}{E}_{\mathit{\omega },\mathit{m\omega }+\mathit{m}+1}(d^{\omega }{\ell }^{\omega }).\end{array}\end{array}$(5.11)

where $E_{\mathit{\varphi },\mathit{\psi }}(\mathit{\xi })$ is defined as

$\begin{array}{l}{E}_{\mathit{\varphi },\mathit{\psi }}(\mathit{\xi })=\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\xi }^m}{\mathrm{\Gamma }(\mathit{\varphi m}+\mathit{\psi })};(\mathrm{\Re }(\mathit{\varphi })>0,\mathrm{\Re }(\mathit{\psi })>0).\end{array}$

Proof.

The Riemann–Liouville fractional integral operator has the following Laplace transform:

(5.12) $\mathit{L}\{{ }_0{\mathcal{D}}_{\mathit{\ell }}^{-\mathit{\omega }}\mathit{f}(\mathit{\ell });\mathcal{P}\}={\mathcal{P}}^{-\mathit{\omega }}\mathit{F}(\mathcal{P}).$(5.12)

From (5.10(5.10) $\begin{array}{l}\mathsf{N}(\mathit{\ell })-{\mathsf{N}}_0\{{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}({\ell }^{\mathit{\omega }}\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z})\}=-d_0^{\mathit{\omega }}{ }_0{\mathcal{D}}_{\mathit{\ell }}^{-\mathit{\omega }}\mathsf{N}(\mathit{\ell });\end{array}$(5.10) ), we have

(5.13) $\begin{array}{ll}\mathit{L}\{\mathsf{N}(\mathit{\ell });\mathcal{P}\}& \mathit{ }={\mathsf{N}}_0\mathit{L}\{{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}({\ell }^{\mathit{\omega }}\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z});\mathcal{P}\}\\ -d_0^{\mathit{\omega }}\mathit{L}\{{ }_0{\mathcal{D}}_{\mathit{t}}^{-\mathit{\omega }}\mathsf{N}(\mathit{\ell });\mathcal{P}\}\\ \mathsf{N}(\mathcal{P})& \mathit{ }={\mathsf{N}}_0({\int }_0^{\mathrm{\infty }}{e}^{-\mathcal{P}\ell }\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\\ \times \frac{(\delta {)}_m}{\mathit{m}!}\frac{({\mathit{t}}^{\mathit{\omega }+1}{)}^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }}\mathit{d\ell })-d^{\mathit{\omega }}{\mathcal{P}}^{-\mathit{\omega }}\mathsf{N}(\mathcal{P})\\ \mathsf{N}(\mathcal{P})+d^{\mathit{\omega }}{\mathcal{P}}^{-\mathit{\omega }}\mathsf{N}(\mathcal{P})& \mathit{ }={\mathsf{N}}_0\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{(\delta {)}_m}{\mathit{m}!}\\ \times \frac{1}{(m+\tau {)}^{\kappa }}{\int }_0^{\mathrm{\infty }}{e}^{-\mathcal{P}\ell }{\ell }^{\mathit{\omega m}+\mathit{m}}\mathit{d\ell }\\ \mathsf{N}(\mathcal{P})+d^{\mathit{\omega }}{\mathcal{P}}^{-\mathit{\omega }}\mathsf{N}(\mathcal{P})& \mathit{ }={\mathsf{N}}_0\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{(\delta {)}_m}{\mathit{m}!}\\ \times \frac{1}{(m+\tau {)}^{\kappa }}\frac{\mathrm{\Gamma }(\mathit{m\omega }+\mathit{m}+1)}{{\mathcal{P}}^{\mathit{m\omega }+\mathit{m}+1}}\\ \mathsf{N}(\mathcal{P})& \mathit{ }={\mathsf{N}}_0\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{(\delta {)}_m}{\mathit{m}!}\frac{1}{(m+\tau {)}^{\kappa }}\\ \times \mathrm{\Gamma }(\mathit{m\omega }+\mathit{m}+1){\mathcal{P}}^{-(\mathit{m\omega }+\mathit{m}+1)}(1+d^{\mathit{\omega }}{\mathcal{P}}^{-\mathit{\omega }}{)}^{-1}\\ \mathsf{N}(\mathcal{P})& \mathit{ }={\mathsf{N}}_0\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{(\delta {)}_m}{\mathit{m}!}\frac{1}{(m+\tau {)}^{\kappa }}\\ \times \mathrm{\Gamma }(\mathit{m\omega }+\mathit{m}+1){\mathcal{P}}^{-(\mathit{m\omega }+\mathit{m}+1)}\sum _{\mathit{r}=0}^{\mathrm{\infty }}(d^{\mathit{\omega }}{\mathcal{P}}^{-\mathit{\omega }}{)}^{\mathit{r}}.\end{array}$(5.13)

Applying the inverse Laplace transform into (5.13(5.13) $\begin{array}{ll}\mathit{L}\{\mathsf{N}(\mathit{\ell });\mathcal{P}\}& \mathit{ }={\mathsf{N}}_0\mathit{L}\{{\mathrm{\Phi }}_{\mathit{\delta },\mathit{\zeta };\mathit{\gamma }}({\ell }^{\mathit{\omega }}\mathit{\xi },\mathit{\kappa },\mathit{\tau };\mathit{p},\mathit{q};\mathit{x},\mathit{y},\mathit{z});\mathcal{P}\}\\ -d_0^{\mathit{\omega }}\mathit{L}\{{ }_0{\mathcal{D}}_{\mathit{t}}^{-\mathit{\omega }}\mathsf{N}(\mathit{\ell });\mathcal{P}\}\\ \mathsf{N}(\mathcal{P})& \mathit{ }={\mathsf{N}}_0({\int }_0^{\mathrm{\infty }}{e}^{-\mathcal{P}\ell }\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\\ \times \frac{(\delta {)}_m}{\mathit{m}!}\frac{({\mathit{t}}^{\mathit{\omega }+1}{)}^m}{(\mathit{m}+\mathit{\tau }{)}^{\kappa }}\mathit{d\ell })-d^{\mathit{\omega }}{\mathcal{P}}^{-\mathit{\omega }}\mathsf{N}(\mathcal{P})\\ \mathsf{N}(\mathcal{P})+d^{\mathit{\omega }}{\mathcal{P}}^{-\mathit{\omega }}\mathsf{N}(\mathcal{P})& \mathit{ }={\mathsf{N}}_0\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{(\delta {)}_m}{\mathit{m}!}\\ \times \frac{1}{(m+\tau {)}^{\kappa }}{\int }_0^{\mathrm{\infty }}{e}^{-\mathcal{P}\ell }{\ell }^{\mathit{\omega m}+\mathit{m}}\mathit{d\ell }\\ \mathsf{N}(\mathcal{P})+d^{\mathit{\omega }}{\mathcal{P}}^{-\mathit{\omega }}\mathsf{N}(\mathcal{P})& \mathit{ }={\mathsf{N}}_0\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{(\delta {)}_m}{\mathit{m}!}\\ \times \frac{1}{(m+\tau {)}^{\kappa }}\frac{\mathrm{\Gamma }(\mathit{m\omega }+\mathit{m}+1)}{{\mathcal{P}}^{\mathit{m\omega }+\mathit{m}+1}}\\ \mathsf{N}(\mathcal{P})& \mathit{ }={\mathsf{N}}_0\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{(\delta {)}_m}{\mathit{m}!}\frac{1}{(m+\tau {)}^{\kappa }}\\ \times \mathrm{\Gamma }(\mathit{m\omega }+\mathit{m}+1){\mathcal{P}}^{-(\mathit{m\omega }+\mathit{m}+1)}(1+d^{\mathit{\omega }}{\mathcal{P}}^{-\mathit{\omega }}{)}^{-1}\\ \mathsf{N}(\mathcal{P})& \mathit{ }={\mathsf{N}}_0\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{(\delta {)}_m}{\mathit{m}!}\frac{1}{(m+\tau {)}^{\kappa }}\\ \times \mathrm{\Gamma }(\mathit{m\omega }+\mathit{m}+1){\mathcal{P}}^{-(\mathit{m\omega }+\mathit{m}+1)}\sum _{\mathit{r}=0}^{\mathrm{\infty }}(d^{\mathit{\omega }}{\mathcal{P}}^{-\mathit{\omega }}{)}^{\mathit{r}}.\end{array}$(5.13) ), we obtain

$\begin{array}{l}\begin{array}{l}{L}^{-1}\{\mathsf{N}(\mathcal{P})\}={\mathsf{N}}_0\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{(\delta {)}_m}{\mathit{m}!}\frac{1}{(m+\tau {)}^{\kappa }}\\ \mathit{ }\times \mathrm{\Gamma }(\mathit{m\omega }+\mathit{m}+1)L^{-1}\left\{\sum _{\mathit{r}=0}^{\mathrm{\infty }}{d}^{\mathit{\omega r}}{\mathcal{P}}^{-\mathit{\omega r}-\mathit{m\omega }-\mathit{m}-1}\right\}\\ \mathsf{N}(\mathit{\ell })={\mathsf{N}}_0\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{(\delta {)}_m}{\mathit{m}!}\frac{1}{(m+\tau {)}^{\kappa }}\\ \mathit{ }\times \mathrm{\Gamma }(\mathit{m\omega }+\mathit{m}+1)\left\{\sum _{\mathit{r}=0}^{\mathrm{\infty }}{d}^{\mathit{\omega r}}\frac{{\ell }^{\mathit{\omega r}+\mathit{m\omega }+\mathit{m}}}{\mathrm{\Gamma }(\mathit{\omega r}+\mathit{m\omega }+\mathit{m}+1)}\right\}\\ \mathsf{N}(\mathit{\ell })={\mathsf{N}}_0\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{(\delta {)}_m}{\mathit{m}!}\frac{1}{(m+\tau {)}^{\kappa }}\\ \mathit{ }\times \mathrm{\Gamma }(\mathit{m\omega }+\mathit{m}+1){\ell }^{\mathit{m\omega }+\mathit{m}}\left\{\sum _{\mathit{r}=0}^{\mathrm{\infty }}\frac{{\mathit{d}}^{\mathit{\omega r}}{\ell }^{\mathit{\omega r}}}{\mathrm{\Gamma }(\mathit{\omega r}+\mathit{m\omega }+\mathit{m}+1)}\right\}\end{array}\end{array}$

$\begin{array}{l}\begin{array}{l}\mathsf{N}(\mathit{\ell })={\mathsf{N}}_0\sum _{\mathit{m}=0}^{\mathrm{\infty }}\frac{{\mathit{B}}_{\mathit{p},\mathit{q}}^{\mathit{x},\mathit{y},\mathit{z}}(\mathit{\zeta }+\mathit{m},\mathit{\gamma }-\mathit{\zeta })}{\mathit{B}(\mathit{\zeta },\mathit{\gamma }-\mathit{\zeta })}\frac{(\delta {)}_m}{\mathit{m}!}\frac{1}{(m+\tau {)}^{\kappa }}\\ \mathit{ }\times \mathrm{\Gamma }(\mathit{m\omega }+\mathit{m}+1){\ell }^{\mathit{m\omega }+\mathit{m}}{E}_{\mathit{\omega },\mathit{m\omega }+\mathit{m}+1}(d^{\omega }{\ell }^{\omega }),\end{array}\end{array}$

which is the required result.

6. Conclusion

The results obtained in this manuscript are connected with the Hurwitz–Lerch zeta function that will be used to solve a variety of problems of fractional kinetic equations, integral transforms, integral representation and generating relations. The fraction kinetic equations in various forms have been broadly and usefully employed when describing and solving several important problems in physics and astrophysics. Recently, fractional kinetic equations associated with some special functions have proven themselves to be useful tools for applications in many fields of research. Because of the effectiveness and great importance of the fractional kinetic equation in certain astrophysical problems, the authors develop a fractional kinetic equation along with the Hurwitz–Lerch zeta function.

It is worth stressing that the generalized Hurwitz–Lerch zeta function obtained and the results computed are amenable to further generalization and investigation. We have attempted to exploit the close connection of the generalized Hurwitz–Lerch zeta function with several important special functions and compute certain integral representations, limiting form and differential formulae, the Mellin transforms and generating relation. Therefore, the investigated results in this paper would at once give many results involving special functions occurring in the problem of astrophysics, mathematical physics, and engineering.

Disclosure statement

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