Further Extension of the Generalized Hurwitz-Lerch Zeta Function of Two Variables

Abstract

The main aim of this paper is to provide a new generalization of Hurwitz-Lerch Zeta function of two variables. We also investigate several interesting properties such as integral representations, summation formula, and a connection with the generalized hypergeometric function. To strengthen the main results we also consider some important special cases.

1. Introduction

The generalized hypergeometric function $F(-)$ [1] defined by

$$\begin{array}{cc}\hfill F\left({\beta }_1,\cdots ,{\beta }_p;{\delta }_1,\cdots ,{\delta }_q;z\right)& {=}_p{F}_q\left({\beta }_1,\cdots ,{\beta }_p;{\delta }_1,\cdots ,{\delta }_q;z\right)\hfill \\ & =\sum _{n=0}^{\infty }\frac{{\prod }_{i=1}^p{\left({\beta }_i\right)}_n{z}^n}{{\prod }_{j=1}^q{\left({\beta }_j\right)}_{n}n!},\hfill \end{array}$$

(1)

where $p,q\in {\mathbb{Z}}^{+};b_j\ne 0,-1,-2,\cdots$.

The Appell hypergeometric function $F_1$ of two variables [2] is defined by

$$\begin{array}{c}\hfill F_1[a,b,b^{\prime };c;z,t]=\sum _{k,l=0}^{\infty }\frac{{\left(a\right)}_{k+l}{\left(b\right)}_k{\left(b^{\prime }\right)}_l}{{\left(c\right)}_{k+l}}\frac{z^k}{k!}\frac{t^l}{l!},\\ \hfill =\sum _{k=0}^{\infty }\frac{{\left(a\right)}_k{\left(b\right)}_k}{{\left(c\right)}_k}{}_2{F}_1\left[\begin{array}{r}a+k,b^{\prime };\\ \hfill c+k;\end{array}t\right]\frac{z^k}{k!},\\ \hfill \left(max\left\{\Re \left(z\right),\Re \left(t\right)\right\}\le 1\mathrm{and}\Re \left(a\right)>0\right).\end{array}$$

(2)

The confluent forms of Humbert functions are [2]:

$${\mathsf{\Phi }}_1[a,b;c;z,t]=\sum _{k,l=0}^{\infty }\frac{{\left(a\right)}_{k+l}{\left(b\right)}_k}{{\left(c\right)}_{k+l}}\frac{z^k}{k!}\frac{t^l}{l!},\left(\left|z\right|<1,\left|t\right|<\infty \right),$$

(3)

$${\mathsf{\Phi }}_2[b,b^{\prime };c;z,t]=\sum _{k,l=0}^{\infty }\frac{{\left(b\right)}_k{\left(b^{\prime }\right)}_l}{{\left(c\right)}_{k+l}}\frac{z^k}{k!}\frac{t^l}{l!},\left(\left|z\right|<\infty ,\left|t\right|<\infty \right),$$

(4)

and

$${\mathsf{\Phi }}_3[b;c;z,t]=\sum _{k,l=0}^{\infty }\frac{{\left(b\right)}_k}{{\left(c\right)}_{k+l}}\frac{z^k}{k!}\frac{t^l}{l!},\left(\left|z\right|<\infty ,\left|t\right|<\infty \right).$$

(5)

The Appell’s type generalized functions $M_i$ by considering product of two ${}_3{F}_2$ functions is given in [3]. From these expansions, we recall one of the generalized Appell’s type functions of two variables $M_4$ and is defined by

$$M_4\left(\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime };x,y\right)=\sum _{k=0}^{\infty }\sum _{l=0}^{\infty }\frac{{\left(\mu \right)}_{k+l}{\left(\eta \right)}_k{\left({\eta }^{\prime }\right)}_l{\left(\delta \right)}_k{\left({\delta }^{\prime }\right)}_l}{{\left(\nu \right)}_{k+l}{\left(\xi \right)}_k{\left({\xi }^{\prime }\right)}_l}\frac{x^k}{k!}\frac{y^l}{l!}.$$

(6)

If we set $\mu =\nu ,\delta =\xi ,{\delta }^{\prime }={\xi }^{\prime }$ in (6) then

$$M_4[\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\mu ,\delta ,{\delta }^{\prime };x,y]={\left(1-x\right)}^{-\eta }{\left(1-y\right)}^{-{\eta }^{\prime }}.$$

(7)

The Hurwitz-Lerch Zeta function $\mathsf{\Phi }(z,s,a)$ is defined by (see [4,5]):

$$\mathsf{\Phi }(z,s,a)=\sum _{k=0}^{\infty }\frac{z^k}{{(k+a)}^s},$$

(8)

$$\left(a\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-};s\in \mathbb{C}\right)\mathrm{when}\left|z\right|<1;\Re \left(s\right)>1\mathrm{when}\left|z\right|=1.$$

For more details about the properties and particular cases found in [1,4,5]. Various type of generalizations, extensions, and properties of the Hurwitz-Lerch Zeta function can be found in [6,7,8,9,10,11,12,13].

Recently, Pathan and Daman [14] give another generalization of the form

$$\begin{array}{c}{\mathsf{\Phi }}_{\alpha ,\beta ;\gamma ,\lambda ,\mu ;\nu }\left(z,t,s,a\right):=\sum _{k,l=0}^{\infty }\frac{{\left(\alpha \right)}_k{\left(\beta \right)}_k{\left(\lambda \right)}_l{\left(\mu \right)}_l}{{\left(\gamma \right)}_k{\left(\nu \right)}_{l}k!l!}\frac{z^k{t}^l}{{(k+l+a)}^s},\hfill \\ \gamma ,\nu ,a\ne \left\{0,-1,-2,\cdots ,\right\},s\in \mathbb{C};\hfill \\ \Re \left(s+\gamma +\nu -\alpha -\beta -\mu -\lambda \right)>0\mathrm{when}\left|z\right|=1\mathrm{and}\left|t\right|=1.\hfill \end{array}$$

(9)

Very recently, Choi and Parmar [15] introduced two variable generalization by

$$\begin{array}{c}{\mathsf{\Phi }}_{\mu ,\eta ,{\eta }^{\prime };\nu }\left(z,t,s,a\right):=\sum _{k,l=0}^{\infty }\frac{{\left(\mu \right)}_{k+l}{\left(\eta \right)}_k{\left({\eta }^{\prime }\right)}_l}{{\left(\nu \right)}_{k+l}k!l!}\frac{z^k{t}^l}{{(k+l+a)}^s},\hfill \\ \left(\mu ,\eta ,{\eta }^{\prime }\in \mathbb{C};a,\nu \in \mathbb{C}\setminus {\mathbb{Z}}_0^{-};s,z,t\in \mathbb{C}\mathrm{when}\left|z\right|<1\mathrm{and}\left|t\right|<1;\right.\hfill \\ \left.\mathrm{and}\Re \left(s+\nu -\mu -\eta -{\eta }^{\prime }\right)>0\mathrm{when}\left|z\right|=1\mathrm{and}\left|t\right|=1\right).\hfill \end{array}$$

(10)

In this paper, we further extended the Hurwitz-Lerch Zeta function of two variables and is defined by

$$\begin{array}{c}{\mathsf{\Phi }}_{\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right):={\sum }_{k,l=0}^{\infty }\frac{{\left(\mu \right)}_{k+l}{\left(\eta \right)}_k{\left({\eta }^{\prime }\right)}_l{\left(\delta \right)}_k{\left({\delta }^{\prime }\right)}_l}{{\left(\nu \right)}_{k+l}{\left(\xi \right)}_k{\left({\xi }^{\prime }\right)}_{l}k!l!}\frac{z^k{t}^l}{{(k+l+a)}^s},\\ \left(\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime }\in \mathbb{C};a,\nu ,\xi ,{\xi }^{\prime }\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-};s,z,t\in \mathbb{C}\mathrm{when}\left|z\right|<1\mathrm{and}\left|t\right|<1;\right.\\ \left.\mathrm{and}\Re \left(s+\nu +\xi +{\xi }^{\prime }-\mu -\eta -{\eta }^{\prime }-\delta -{\delta }^{\prime }\right)>0\mathrm{when}\left|z\right|=1\mathrm{and}\left|t\right|=1\right).\end{array}$$

(11)

Special cases:

Case 1. If $\delta =\xi ,{\delta }^{\prime }={\xi }^{\prime }$, then (11) reduces to (3) of [15] which is given in (10).

Case 2. If $\mu =\nu$ and $\delta =\xi ,{\delta }^{\prime }={\xi }^{\prime }$ in (11), then we get the generalized Hurwitz-Lerch Zeta function of [14]:

$$\begin{array}{l}{\mathsf{\Phi }}_{\eta ,{\eta }^{\prime }}\left(z,t,s,a\right):={\displaystyle \sum _{k,l=0}^{\infty }}\frac{{\left(\eta \right)}_k{\left({\eta }^{\prime }\right)}_l}{k!l!}\frac{z^k{t}^l}{{(k+l+a)}^s},\\ \left(\eta ,{\eta }^{\prime }\in \mathbb{C};a\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-};s\in \mathbb{C}\mathrm{when}\left|z\right|<1\mathrm{and}\left|t\right|<1;\right.\\ \left.\mathrm{and}\Re \left(s-\eta -{\eta }^{\prime }\right)>0\mathrm{when}\left|z\right|=1\mathrm{and}\left|t\right|=1\right).\end{array}$$

(12)

The limiting cases of (11) are as follows:

Case 3. If ${\eta }^{\prime }\to \infty$ then we have

$$\begin{array}{l}{\mathsf{\Phi }}_{\mu ,\eta ,\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right):={\displaystyle \underset{{\eta }^{\prime }\to \infty }{lim}}\left\{{\mathsf{\Phi }}_{\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t/{\eta }^{\prime },s,a\right)\right\}\\ ={\displaystyle \sum _{k,l=0}^{\infty }}\frac{{\left(\mu \right)}_{k+l}{\left(\eta \right)}_k{\left(\delta \right)}_k{\left({\delta }^{\prime }\right)}_l}{{\left(\nu \right)}_{k+l}{\left(\xi \right)}_k{\left({\xi }^{\prime }\right)}_{l}k!l!}\frac{z^k{t}^l}{{(k+l+a)}^s},\\ \left(\mu ,\eta ,\delta ,{\delta }^{\prime }\in \mathbb{C};a,\nu ,\xi ,{\xi }^{\prime }\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-};s,z,t\in \mathbb{C}\mathrm{when}\left|z\right|<1\mathrm{and}\left|t\right|<1;\right.\\ \left.\mathrm{and}\Re \left(s+\nu +\xi +{\xi }^{\prime }-\mu -\eta -\delta -{\delta }^{\prime }\right)>0\mathrm{when}\left|z\right|=1\mathrm{and}\left|t\right|=1\right).\end{array}$$

(13)

Case 4. If $\mu \to \infty$ then we have

$$\begin{array}{l}{\mathsf{\Phi }}_{\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right):={\displaystyle \underset{\mu \to \infty }{lim}}\left\{{\mathsf{\Phi }}_{\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z/\mu ,t/\mu ,s,a\right)\right\}\\ ={\displaystyle \sum _{k,l=0}^{\infty }}\frac{{\left(\eta \right)}_k{\left({\eta }^{\prime }\right)}_l{\left(\delta \right)}_k{\left({\delta }^{\prime }\right)}_l}{{\left(\nu \right)}_{k+l}{\left(\xi \right)}_k{\left({\xi }^{\prime }\right)}_{l}k!l!}\frac{z^k{t}^l}{{(k+l+a)}^s},\\ \left(\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime }\in \mathbb{C};a,\nu ,\xi ,{\xi }^{\prime }\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-};s,z,t\in \mathbb{C}\mathrm{when}\left|z\right|<1\mathrm{and}\left|t\right|<1;\right.\\ \left.\mathrm{and}\Re \left(s+\nu +\xi +{\xi }^{\prime }-\eta -\delta -{\delta }^{\prime }\right)>0\mathrm{when}\left|z\right|=1\mathrm{and}\left|t\right|=1\right).\end{array}$$

(14)

Case 5. If $min(\left|\mu \right|,\left|{\eta }^{\prime }\right|)\to \infty$ then we have

$$\begin{array}{l}{\mathsf{\Phi }}_{\eta ,\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right):={\displaystyle \underset{min(\left|\mu \right|,\left|{\eta }^{\prime }\right|)\to \infty }{lim}}\left\{{\mathsf{\Phi }}_{\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(\frac{z}{\mu },\frac{t}{\left(\mu {\eta }^{\prime }\right)},s,a\right)\right\}\hfill \\ ={\displaystyle \sum _{k,l=0}^{\infty }}\frac{{\left(\eta \right)}_k{\left(\delta \right)}_k{\left({\delta }^{\prime }\right)}_l}{{\left(\nu \right)}_{k+l}{\left(\xi \right)}_k{\left({\xi }^{\prime }\right)}_{l}k!l!}\frac{z^k{t}^l}{{(k+l+a)}^s},\hfill \\ \left(\eta ,\delta ,{\delta }^{\prime }\in \mathbb{C};a,\nu ,\xi ,{\xi }^{\prime }\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-};s,z,t\in \mathbb{C}\mathrm{when}\left|z\right|<1\mathrm{and}\left|t\right|<1;\right.\\ \left.\mathrm{and}\Re \left(s+\nu +\xi +{\xi }^{\prime }-\eta -\delta -{\delta }^{\prime }\right)>0\mathrm{when}\left|z\right|=1\mathrm{and}\left|t\right|=1\right).\end{array}$$

(15)

2. Integral Representations

Theorem 1.

The following integral representation of (11) holds true:

$$\begin{array}{c}{\mathsf{\Phi }}_{\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right)\hfill \\ =\frac{1}{\mathsf{\Gamma }\left(s\right)}{\int }_0^{\infty }{x}^{s-1}{e}^{-ax}{M}_4\left(\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime };z{e}^{-x},t{e}^{-x}\right)dx,\hfill \\ \left(min\left\{\Re \left(s\right),\Re \left(a\right)\right\}>0\mathit{when}\left|z\right|\le 1(z\ne 1),\left|t\right|\le 1(t\ne 1),\right.\hfill \\ \left.\Re \left(s\right)>1,\mathit{when}z=1,t=1\right).\hfill \end{array}$$

(16)

Proof. 

Using the following Eulerian integral

$$\frac{1}{{\left(k+l+a\right)}^s}:=\frac{1}{\mathsf{\Gamma }\left(s\right)}{\int }_0^{\infty }{t}^{s-1}{e}^{-(k+l+a)t}dt\left(min\Re \left(s\right),\Re \left(a\right)>0,k,l\in {\mathbb{N}}_0\right)$$

(17)

in (11), we get

$$\begin{array}{cc}& {\mathsf{\Phi }}_{\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right)\hfill \\ =& \sum _{k,l=0}^{\infty }\frac{{\left(\mu \right)}_{k+l}{\left(\eta \right)}_k{\left({\eta }^{\prime }\right)}_l{\left(\delta \right)}_k{\left({\delta }^{\prime }\right)}_l}{{\left(\nu \right)}_{k+l}{\left(\xi \right)}_k{\left({\xi }^{\prime }\right)}_l}\frac{z^k{t}^l}{k!l!}\frac{1}{\mathsf{\Gamma }\left(s\right)}{\int }_0^{\infty }{x}^{s-1}{e}^{-(k+l+a)x}dx.\hfill \end{array}$$

Interchanging the order of integration and summation, which is verified by uniform convergence of the involved series under the given conditions, we have

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}_{\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right)\hfill \\ =& \frac{1}{\mathsf{\Gamma }\left(s\right)}{\int }_0^{\infty }{x}^{s-1}{e}^{-ax}\sum _{k,l=0}^{\infty }\frac{{\left(\mu \right)}_{k+l}{\left(\eta \right)}_k{\left({\eta }^{\prime }\right)}_l{\left(\delta \right)}_k{\left({\delta }^{\prime }\right)}_l}{{\left(\nu \right)}_{k+l}{\left(\xi \right)}_k{\left({\xi }^{\prime }\right)}_l}\frac{z^k{t}^l}{k!l!}{\left(e^{-t}\right)}^k{\left(e^{-t}\right)}^{l}dx.\hfill \end{array}$$

(18)

In view of (6), we arrived the desired result. □

Similarly, if we use (17) in the limiting cases (13), (14) and (15) then we obtain the following corollaries:

Corollary 1.

The following integral representations for ${\mathsf{\Phi }}_{\mu ,\eta ,\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right),{\mathsf{\Phi }}_{\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right)$ and ${\mathsf{\Phi }}_{\eta ,\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right)$ in (13), (14) and (15) holds true when $\delta =\xi ,{\delta }^{\prime }={\xi }^{\prime }$:

$${\mathsf{\Phi }}_{\mu ,\eta ;\nu }\left(z,t,s,a\right)=\frac{1}{\mathsf{\Gamma }\left(s\right)}{\int }_0^{\infty }{x}^{s-1}{e}^{-ax}{\mathsf{\Phi }}_1\left(\mu ,\eta ;\nu ;z{e}^{-x},t{e}^{-x}\right)dx,$$

(19)

which is (14) of [15].

$${\mathsf{\Phi }}_{\eta ,{\eta }^{\prime };\nu }\left(z,t,s,a\right)=\frac{1}{\mathsf{\Gamma }\left(s\right)}{\int }_0^{\infty }{x}^{s-1}{e}^{-ax}{\mathsf{\Phi }}_2\left(\eta ,{\eta }^{\prime };\nu ;z{e}^{-x},t{e}^{-x}\right)dx,$$

(20)

which is (15) of [15] and

$${\mathsf{\Phi }}_{\eta ;\nu }\left(z,t,s,a\right)=\frac{1}{\mathsf{\Gamma }\left(s\right)}{\int }_0^{\infty }{x}^{s-1}{e}^{-ax}{\mathsf{\Phi }}_3\left(\eta ;\nu ;z{e}^{-x},t{e}^{-x}\right)dx.$$

(21)

$\left(min\left\{\Re \left(s\right),\Re \left(a\right)\right\}>0\mathit{when}\left|z\right|\le 1(z\ne 1),\left|t\right|\le 1(t\ne 1),\right.$$\left.\Re \left(s\right)>1,\mathit{when}z=1,t=1\right)$, which is (16) of [15].

Corollary 2.

In view of (7), we have

$$\begin{array}{c}{\mathsf{\Phi }}_{\mu ,\eta ,{\eta }^{\prime };\mu }\left(z,t,s,a\right)\hfill \\ =\frac{1}{\mathsf{\Gamma }\left(s\right)}{\int }_0^{\infty }\frac{x^{s-1}{e}^{-ax}}{{\left(1-z{e}^{-x}\right)}^{\eta }{\left(1-t{e}^{-x}\right)}^{-{\eta }^{\prime }}}dx,\hfill \end{array}$$

(22)

$\left(min\Re \left(s\right)>0,\Re \left(a\right)>0\mathit{when}\left|z\right|\le 1(z\ne 1),\left|t\right|\le 1(t\ne 1),\right.$$\left.\Re \left(s\right)>1,\mathit{when}z=1,t=1\right).$

Remark 1.

If we take $t=0$ in (22), then it gives (19) of [15] and by setting $t=0,\eta =1$ then (22) reduces to (20) of [15]

Theorem 2.

Each of the following integrals for ${\mathsf{\Phi }}_{\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right)$ holds true

$$\begin{array}{c}{\mathsf{\Phi }}_{\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right)\\ =\frac{\mathsf{\Gamma }\left(\nu \right)}{\mathsf{\Gamma }\left(\mu \right)\mathsf{\Gamma }(\nu -\mu )}{\int }_0^{\infty }\frac{y^{\mu -1}}{{(1+y)}^{\nu }}{\mathsf{\Phi }}_{\eta ,{\eta }^{\prime },\xi ,\delta ;{\delta }^{\prime },{\xi }^{\prime }}\left(\frac{zy}{1+y},\frac{ty}{1+y},s,a\right),\end{array}$$

(23)

and

$$\begin{array}{l}{\mathsf{\Phi }}_{\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right)\\ =\frac{\mathsf{\Gamma }\left(\nu \right)}{\mathsf{\Gamma }\left(s\right)\mathsf{\Gamma }\left(\mu \right)\mathsf{\Gamma }(\nu -\mu )}{\int }_0^{\infty }{\int }_0^{\infty }\frac{x^{s-1}{e}^{-ax}{y}^{\mu -1}}{{(1+y)}^{\nu }}\\ \times {\displaystyle \sum _{k=0}^{\infty }}\frac{{\left(\eta \right)}_k{\left(\delta \right)}_k}{k!{\left(\xi \right)}_k}{\left(\frac{zy{e}^{-x}}{1+y}\right)}^k{\displaystyle \sum _{l=0}^{\infty }}\frac{{\left({\eta }^{\prime }\right)}_l{\left({\delta }^{\prime }\right)}_k}{l!{\left({\xi }^{\prime }\right)}_k}{\left(\frac{ty{e}^{-x}}{1+y}\right)}^{l}dxdy.\end{array}$$

(24)

Proof. 

Setting $a=\mu +k+l,b=\nu +k+l$ in the Eulerian beta function formula,

$$B\left(a,b-a\right)=\frac{\mathsf{\Gamma }\left(a\right)\mathsf{\Gamma }(b-a)}{\mathsf{\Gamma }\left(b\right)}={\int }_0^{\infty }\frac{y^{a-1}}{{(1+y)}^b}dy,\Re \left(b\right)>\Re \left(a\right)>0,$$

(25)

gives

$$\frac{\mathsf{\Gamma }(\mu +k+l)\mathsf{\Gamma }(\nu -\mu )}{\mathsf{\Gamma }(\nu +k+l)}={\int }_0^{\infty }\frac{y^{\mu +k+l-1}}{{(1+y)}^{\nu +k+l}}dy,$$

(26)

$$\begin{array}{c}\hfill \Rightarrow \frac{{\left(\mu \right)}_{k+l}\mathsf{\Gamma }\left(\mu \right)\mathsf{\Gamma }(\nu -\mu )}{{\left(\nu \right)}_{k+l}\mathsf{\Gamma }\left(\nu \right)}={\int }_0^{\infty }\frac{y^{\mu +k+l-1}}{{(1+y)}^{\nu +k+l}}dy,\\ \hfill \Re \left(\nu \right)>\Re \left(\mu \right)>0,k,l\in \mathbb{N}.\end{array}$$

$$\Rightarrow \frac{{\left(\mu \right)}_{k+l}}{{\left(\nu \right)}_{k+l}}=\frac{\mathsf{\Gamma }\left(\nu \right)}{\mathsf{\Gamma }\left(\mu \right)\mathsf{\Gamma }(\nu -\mu )}{\int }_0^{\infty }\frac{y^{\mu +k+l-1}}{{(1+y)}^{\nu +k+l}}dy.$$

(27)

Now substituting (27) in (11), we get

$$\begin{array}{c}{\mathsf{\Phi }}_{\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right)\\ =\sum _{k=0}^{\infty }\frac{\mathsf{\Gamma }\left(\nu \right)}{\mathsf{\Gamma }\left(\mu \right)\mathsf{\Gamma }(\nu -\mu )}{\int }_0^{\infty }\frac{y^{\mu +k+l-1}}{{(1+y)}^{\nu +k+l}}\frac{{\left(\eta \right)}_k{\left({\eta }^{\prime }\right)}_l{\left(\delta \right)}_k{\left({\delta }^{\prime }\right)}_l}{{\left(\xi \right)}_k{\left({\xi }^{\prime }\right)}_{l}k!l!}\frac{z^k{t}^l}{{(k+l+a)}^s}dy\end{array}$$

(28)

interchanging integration and summation gives

$$\begin{array}{c}{\mathsf{\Phi }}_{\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right)\\ =\frac{\mathsf{\Gamma }\left(\nu \right)}{\mathsf{\Gamma }\left(\mu \right)\mathsf{\Gamma }(\nu -\mu )}{\int }_0^{\infty }\frac{y^{\mu -1}}{{(1+y)}^{\nu }}\sum _{k=0}^{\infty }\frac{{\left(\eta \right)}_k{\left({\eta }^{\prime }\right)}_l{\left(\delta \right)}_k{\left({\delta }^{\prime }\right)}_l}{{\left(\xi \right)}_k{\left({\xi }^{\prime }\right)}_{l}k!l!}{\left(\frac{zy}{1+y}\right)}^k{\left(\frac{ty}{1+y}\right)}^l\frac{1}{{\left(k+l+a\right)}^s}dy.\end{array}$$

(29)

In view of (11) and (9) we arrived the desired result.

Now, we prove the second integral. From (18), ${\mathsf{\Phi }}_{\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right)$ can be written as

$$\begin{array}{cc}& {\mathsf{\Phi }}_{\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right)\hfill \\ =& \frac{1}{\mathsf{\Gamma }\left(s\right)}{\int }_0^{\infty }{x}^{s-1}{e}^{-ax}\sum _{k,l=0}^{\infty }\frac{{\left(\mu \right)}_{k+l}{\left(\eta \right)}_k{\left({\eta }^{\prime }\right)}_l{\left(\delta \right)}_k{\left({\delta }^{\prime }\right)}_l}{{\left(\nu \right)}_{k+l}{\left(\xi \right)}_k{\left({\xi }^{\prime }\right)}_l}\frac{{\left(z{e}^{-x}\right)}^k}{k!}\frac{{\left(t{e}^{-x}\right)}^l}{l!}dx,\hfill \end{array}$$

Now using (27), we get

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_{\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right)& =\frac{1}{\mathsf{\Gamma }\left(s\right)}{\int }_0^{\infty }{x}^{s-1}{e}^{-ax}\sum _{k,l=0}^{\infty }\frac{\mathsf{\Gamma }\left(\nu \right)}{\mathsf{\Gamma }\left(\mu \right)\mathsf{\Gamma }(\nu -\mu )}{\int }_0^{\infty }\frac{y^{\mu +k+l-1}}{{(1+y)}^{\nu +k+l}}dy\hfill \\ & \times \frac{{\left(\eta \right)}_k{\left({\eta }^{\prime }\right)}_l{\left(\delta \right)}_k{\left({\delta }^{\prime }\right)}_l}{{\left(\xi \right)}_k{\left({\xi }^{\prime }\right)}_l}\frac{{\left(z{e}^{-x}\right)}^k}{k!}\frac{{\left(t{e}^{-x}\right)}^l}{l!}dx,\hfill \\ & =\frac{\mathsf{\Gamma }\left(\nu \right)}{\mathsf{\Gamma }\left(s\right)\mathsf{\Gamma }\left(\mu \right)\mathsf{\Gamma }(\nu -\mu )}{\int }_0^{\infty }{\int }_0^{\infty }\frac{x^{s-1}{e}^{-ax}{y}^{\mu -1}}{{(1+y)}^{\nu }}\hfill \\ & \times \sum _{k=0}^{\infty }\frac{{\left(\eta \right)}_k{\left(\delta \right)}_k}{{\left(\xi \right)}_{k}k!}{\left(\frac{zy{e}^{-x}}{1+y}\right)}^k\sum _{l=0}^{\infty }\frac{{\left({\eta }^{\prime }\right)}_l{\left({\delta }^{\prime }\right)}_l}{{\left({\xi }^{\prime }\right)}_{l}l!}{\left(\frac{ty{e}^{-x}}{1+y}\right)}^{l}dxdy.\hfill \end{array}$$

 □

Corollary 3.

If $\delta ={\delta }^{\prime }=1$ and $\xi ={\xi }^{\prime }=1$, then we get the result (22) of [15] as

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_{\mu ,\eta ,{\eta }^{\prime };\nu }\left(z,t,s,a\right)& =\frac{\mathsf{\Gamma }\left(\nu \right)}{\mathsf{\Gamma }\left(s\right)\mathsf{\Gamma }\left(\mu \right)\mathsf{\Gamma }(\nu -\mu )}\hfill \\ & \times {\int }_0^{\infty }{\int }_0^{\infty }\frac{x^{s-1}{e}^{-ax}{y}^{\mu -1}}{{(1+y)}^{\nu }}{\left(1-\frac{zy{e}^{-x}}{1+y}\right)}^{-\eta }{\left(1-\frac{ty{e}^{-x}}{1+y}\right)}^{-{\eta }^{\prime }}dxdy,\hfill \\ & \left(\Re \left(\nu \right)>\Re \left(\mu \right)>0;min\left\{\Re \left(s\right),\Re \left(a\right)\right\}>0\right).\hfill \end{array}$$

Theorem 3.

The following summation formula hold true.

$$\sum _{r=0}^{\infty }\frac{{\left(s\right)}_r}{r!}{\mathsf{\Phi }}_{\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s+r,a\right)x^r={\mathsf{\Phi }}_{\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a-x\right),\left(\left|x\right|<\left|a\right|;s\ne 1\right).$$

(30)

Proof. 

Using (11), we have

$$\begin{array}{cc}& {\mathsf{\Phi }}_{\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a-x\right)\hfill \\ & =\sum _{k,l=0}^{\infty }\frac{{\left(\mu \right)}_{k+l}{\left(\eta \right)}_k{\left({\eta }^{\prime }\right)}_l{\left(\delta \right)}_k{\left({\delta }^{\prime }\right)}_l}{{\left(\nu \right)}_{k+l}{\left(\xi \right)}_k{\left(\xi \right)}_l}\frac{z^k{t}^l}{k!l!{(k+l+a-x)}^s},\hfill \\ & =\sum _{k,l=0}^{\infty }\frac{{\left(\mu \right)}_{k+l}{\left(\eta \right)}_k{\left({\eta }^{\prime }\right)}_l{\left(\delta \right)}_k{\left({\delta }^{\prime }\right)}_l}{{\left(\nu \right)}_{k+l}{\left(\xi \right)}_k{\left(\xi \right)}_l}\frac{z^k{t}^l}{k!l!{(k+l+a)}^s}{\left(1-\frac{x}{k+l+a}\right)}^{-s},\hfill \\ \hfill \mathrm{using}\mathrm{binomial}\mathrm{series},\mathrm{we}\mathrm{get}& \\ & =\sum _{r=0}^{\infty }\frac{{\left(s\right)}_r}{s!}\left\{\sum _{k,l=0}^{\infty }\frac{{\left(\mu \right)}_{k+l}{\left(\eta \right)}_k{\left({\eta }^{\prime }\right)}_l{\left(\delta \right)}_k{\left({\delta }^{\prime }\right)}_l}{{\left(\nu \right)}_{k+l}{\left(\xi \right)}_k{\left(\xi \right)}_l}\frac{z^k{t}^l}{k!l!{(k+l+a)}^{s+r}}\right\}{x}^r.\hfill \end{array}$$

In view of definition (11), we reach the required result. □

3. A Connection with Generalized Hypergeometric Function

In this section, we establish the connection between (11) and generalized hypergeometric function.

Theorem 4.

For $a\ne \left\{-1,-2,\cdots \right\}$ and $z\ne 0$, the following explicit series representation holds true

$$\begin{array}{cc}{\mathsf{\Phi }}_{\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right)\hfill & ={\sum }_{k=0}^{\infty }\frac{{\left(\mu \right)}_k{\left(\eta \right)}_k{\left(\delta \right)}_k{z}^k}{{\left(\nu \right)}_k{\left(\xi \right)}_k{(a+k)}^{s}k!}\hfill \\ & \times F\left({\eta }^{\prime },{\delta }^{\prime },1-\xi -k,-k;1-\eta -k,1-\delta -k,{\xi }^{\prime };\frac{t}{z}\right),\hfill \end{array}$$

(31)

where $F(-)$ is the generalized hypergeometric function defined in (1).

Proof. 

Using (11) and the identity ([16] page 56, Equation (1))

$$\sum _{k=0}^{\infty }\sum _{l=0}^{\infty }A\left(l,k\right)=\sum _{k=0}^{\infty }\sum _{l=0}^{k}A\left(l,k-l\right),$$

which implies that

$${\mathsf{\Phi }}_{\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right)=\sum _{k=0}^{\infty }\sum _{l=0}^k\frac{{\left(\mu \right)}_k{\left(\eta \right)}_{k-l}{\left({\eta }^{\prime }\right)}_l{\left(\delta \right)}_{k-l}{\left({\delta }^{\prime }\right)}_l}{{\left(\nu \right)}_k{\left(\xi \right)}_{k-l}{\left({\xi }^{\prime }\right)}_l(k-l)!l!}\frac{z^{k-l}{t}^l}{{(k+a)}^s}.$$

Now,

$$\begin{array}{c}\hfill \left(k-l\right)!=\frac{{(-1)}^{l}k!}{{(-k)}_l},0\le l\le k\\ \hfill {\left(\eta \right)}_{k-l}=\frac{{(-1)}^l{\left(\eta \right)}_k}{{(1-\eta -k)}_l},0\le l\le k,\end{array}$$

we get,

$$\begin{array}{c}{\mathsf{\Phi }}_{\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right)\hfill \\ =\sum _{k=0}^{\infty }\sum _{l=0}^k\frac{{\left(\mu \right)}_k{\left(\eta \right)}_k{\left({\eta }^{\prime }\right)}_l{\left(\delta \right)}_k{\left({\delta }^{\prime }\right)}_l{(1-\xi -k)}_l{(-k)}_l{z}^{k-l}{t}^l}{{\left(\nu \right)}_k{(1-\eta -k)}_l{(1-\delta -k)}_l{\left(\xi \right)}_k{\left({\xi }^{\prime }\right)}_l\left(k\right)!l!}\frac{1}{{(k+a)}^s}.\hfill \end{array}$$

Lastly, summing the l-series, we get the required result. □

Corollary 4.

If we set $\delta =\xi$ in Theorem 4, then we get (28) of [14] as

$${\mathsf{\Phi }}_{\mu ,\eta ,{\eta }^{\prime },\xi ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right)=\sum _{k=0}^{\infty }\frac{{\left(\mu \right)}_k{\left(\eta \right)}_k{z}^k}{{\left(\nu \right)}_k{(k+a)}^{s}k!}F\left({\eta }^{\prime },{\delta }^{\prime },-k,1-\xi -k;{\xi }^{\prime },1-\eta -k,1-\xi -k;\frac{t}{z}.\right)$$

(32)

Corollary 5.

If we set $\nu =\eta ,\delta =\xi$ and ${\delta }^{\prime }={\xi }^{\prime }$ in Theorem 4, then we get (29) of [14] as

$${\mathsf{\Phi }}_{\mu ,\eta ,{\eta }^{\prime },\xi ,{\xi }^{\prime };\eta ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right)=\sum _{k=0}^{\infty }\frac{{\left(\mu \right)}_k{z}^k}{{(a+k)}^{s}k!}F\left({\eta }^{\prime },-k;1-\eta -k;\frac{t}{z}\right).$$

(33)

4. Concluding Remarks

An extension of a generalized Hurwitz-Lerch Zeta function is defined and some of its properties are studied in this paper. An integral representation is established and a relation with Appell’s type function is given. Finally, a connection with the hypergeometric function is also given. The results derived here are more general in nature by comparing the results of the papers [14,15] which help to derive some interesting special cases and are mentioned in Remark 1 and Corollaries 1–5.