Further extension of the generalized Hurwitz-Lerch Zeta function of two variables

TL;DR

This paper introduces a new generalization of the two-variable Hurwitz-Lerch Zeta function, exploring its properties, integral representations, and connections with hypergeometric functions, along with special cases.

Contribution

It provides a novel generalization of the two-variable Hurwitz-Lerch Zeta function and investigates its properties and special cases.

Findings

Derived integral representations of the generalized function.

Established summation formulas and connections with hypergeometric functions.

Explored important special cases of the generalized function.

Abstract

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

Equations

$$F\left({\beta }_1,\cdots ,{\beta }_p;{\delta }_1,\cdots ,{\delta }_q;z\right)$$

$$=\sum _{n=0}^{\infty }\frac{{\prod }_{i=1}^p{\left({\beta }_i\right)}_n{z}^n}{{\prod }_{j=1}^q{\left({\beta }_j\right)}_{n}n!},$$

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

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

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

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

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

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

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

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

$${\Phi }_{\alpha ,\beta ;\gamma ,\lambda ,\mu ;\nu }\left(z,t,s,a\right):=\sum _{k,l=0}^{\infty }\frac{(\alpha {)}_k(\beta {)}_k(\lambda {)}_l(\mu {)}_l}{(\gamma {)}_k(\nu {)}_{l}k!l!}\frac{z^k{t}^l}{(k+l+a{)}^s},$$

$$\gamma ,\nu ,a\ne \left\{0,-1,-2,\cdots ,\right\},s\in \mathbb{C};$$

$$\Re \left(s+\gamma +\nu -\alpha -\beta -\mu -\lambda \right)>0\text{when}\left|z\right|=1\text{and}\left|t\right|=1.$$

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

$$(\mu ,\eta ,{\eta }^{\prime }\in \mathbb{C};a,\nu \in \mathbb{C}\setminus {\mathbb{Z}}_0^{-};s,z,t\in \mathbb{C}\text{when}\left|z\right|<1\text{and}\left|t\right|<1;$$

$$\text{and}\Re \left(s+\nu -\mu -\eta -{\eta }^{\prime }\right)>0\text{when}\left|z\right|=1\text{and}\left|t\right|=1).$$

$$\begin{array}[]{ccccc}&\Phi_{\mu,\eta,\eta^{\prime},\delta,\delta^{\prime};\nu,\xi,\xi^{\prime}}\left(z,t,s,a\right):=\sum_{k,l=0}^{\infty}\frac{(\mu)_{k+l}(\eta)_{k}(\eta^{\prime})_{l}(\delta)_{k}(\delta^{\prime})_{l}}{(\nu)_{k+l}(\xi)_{k}(\xi^{\prime})_{l}k!l!}\frac{z^{k}t^{l}}{(k+l+a)^{s}},\\ &\left(\mu,\eta,\eta^{\prime},\delta,\delta^{\prime}\in\mathbb{C};a,\nu,\xi,\xi^{\prime}\in\mathbb{C}\setminus\mathbb{Z}_{0}^{-};\,s,z,t\in\mathbb{C}\quad\text{when}\quad\left|z\right|<1\quad\text{and}\quad\left|t\right|<1;\right.\\ &\left.\text{and}\quad\Re\left(s+\nu+\xi+\xi^{\prime}-\mu-\eta-\eta^{\prime}-\delta-\delta^{\prime}\right)>0\quad\text{when}\left|z\right|=1\text{and}\left|t\right|=1\right).\end{array}$$

$${\Phi }_{\eta ,{\eta }^{\prime }}\left(z,t,s,a\right):=\sum _{k,l=0}^{\infty }\frac{(\eta {)}_k({\eta }^{\prime }{)}_l}{k!l!}\frac{z^k{t}^l}{(k+l+a{)}^s},$$

$$(\eta ,{\eta }^{\prime }\in \mathbb{C};a\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-};s\in \mathbb{C}\text{when}\left|z\right|<1\text{and}\left|t\right|<1;$$

$$\text{and}\Re \left(s-\eta -{\eta }^{\prime }\right)>0\text{when}\left|z\right|=1\text{and}\left|t\right|=1).$$

$${\Phi }_{\mu ,\eta ,\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right):=\underset{{\eta }^{\prime }\to \infty }{\lim }\left\{{\Phi }_{\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t/{\eta }^{\prime },s,a\right)\right\}$$

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

$$(\mu ,\eta ,\delta ,{\delta }^{\prime }\in \mathbb{C};a,\nu ,\xi ,{\xi }^{\prime }\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-};s,z,t\in \mathbb{C}\text{when}\left|z\right|<1\text{and}\left|t\right|<1;$$

$$\text{and}\Re \left(s+\nu +\xi +{\xi }^{\prime }-\mu -\eta -\delta -{\delta }^{\prime }\right)>0\text{when}\left|z\right|=1\text{and}\left|t\right|=1).$$

$${\Phi }_{\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right):=\underset{\mu \to \infty }{\lim }\left\{{\Phi }_{\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z/\mu ,t/\mu ,s,a\right)\right\}$$

$$=\sum _{k,l=0}^{\infty }\frac{(\eta {)}_k({\eta }^{\prime }{)}_l(\delta {)}_k({\delta }^{\prime }{)}_l}{(\nu {)}_{k+l}(\xi {)}_k({\xi }^{\prime }{)}_{l}k!l!}\frac{z^k{t}^l}{(k+l+a{)}^s},$$

$$(\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime }\in \mathbb{C};a,\nu ,\xi ,{\xi }^{\prime }\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-};s,z,t\in \mathbb{C}\text{when}\left|z\right|<1\text{and}\left|t\right|<1;$$

$$\text{and}\Re \left(s+\nu +\xi +{\xi }^{\prime }-\eta -\delta -{\delta }^{\prime }\right)>0\text{when}\left|z\right|=1\text{and}\left|t\right|=1).$$

$${\Phi }_{\eta ,\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right):=\underset{min(|\mu |,|{\eta }^{\prime }|)\to \infty }{\lim }\left\{{\Phi }_{\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(\frac{z}{\mu },\frac{t}{(\mu {\eta }^{\prime })},s,a\right)\right\}$$

$$=\sum _{k,l=0}^{\infty }\frac{(\eta {)}_k(\delta {)}_k({\delta }^{\prime }{)}_l}{(\nu {)}_{k+l}(\xi {)}_k({\xi }^{\prime }{)}_{l}k!l!}\frac{z^k{t}^l}{(k+l+a{)}^s},$$

$$(\eta ,\delta ,{\delta }^{\prime }\in \mathbb{C};a,\nu ,\xi ,{\xi }^{\prime }\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-};s,z,t\in \mathbb{C}\text{when}\left|z\right|<1\text{and}\left|t\right|<1;$$

$$\text{and}\Re \left(s+\nu +\xi +{\xi }^{\prime }-\eta -\delta -{\delta }^{\prime }\right)>0\text{when}\left|z\right|=1\text{and}\left|t\right|=1).$$

$${\Phi }_{\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right)$$

$$=\frac{1}{\Gamma (s)}{\int }_0^{\infty }{x}^{s-1}{e}^{-ax}{M}_4\left(\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime };z{e}^{-x},t{e}^{-x}\right)dx,$$

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

$$\Re (s)>1,\text{when}z=1,t=1).$$

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

$${\Phi }_{\mu ,\eta ,{\eta }^{\prime },\delta ,{\delta }^{\prime };\nu ,\xi ,{\xi }^{\prime }}\left(z,t,s,a\right)$$

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

Full text

*Corresponding author

1. Introduction

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

[TABLE]

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

The Appell hypergeometric function $F_{1}$ of two variables [2] is defined by

[TABLE]

The confluent forms of Humbert functions are [2]:

[TABLE]

[TABLE]

and

[TABLE]

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

[TABLE]

If we set $\mu=\nu,\delta=\xi,\delta^{\prime}=\xi^{\prime}$ in (1.6) then

[TABLE]

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

[TABLE]

[TABLE]

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, 11, 12, 13, 10].

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

[TABLE]

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

[TABLE]

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

[TABLE]

Special cases:

Case 1. If $\delta=\xi,\delta^{\prime}=\xi^{\prime}$, then (LABEL:FEGHLZ) reduces to (3) of [15] which is given in (1.10).

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

[TABLE]

The limiting cases of (LABEL:FEGHLZ) are as follows:

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

[TABLE]

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

[TABLE]

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

[TABLE]

2. Integral Representations

Theorem 2.1**.**

The following integral representation of (LABEL:FEGHLZ) holds true:

[TABLE]

Proof.

Using the following Eulerian integral

[TABLE]

in (LABEL:FEGHLZ), we get

[TABLE]

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

[TABLE]

In view of (1.6), we arrived the desired result. ∎

Similarly, if we use (2.2) in the limiting cases (1.13), (1.14) and (1.15) then we obtain the following corollaries:

Corollary 2.1**.**

The following integral representations for $\Phi_{\mu,\eta,\delta,\delta^{\prime};\nu,\xi,\xi^{\prime}}\left(z,t,s,a\right),\Phi_{\eta,\eta^{\prime},\delta,\delta^{\prime};\nu,\xi,\xi^{\prime}}\left(z,t,s,a\right)$ and $\Phi_{\eta,\delta,\delta^{\prime};\nu,\xi,\xi^{\prime}}\left(z,t,s,a\right)$ in (1.13), (1.14) and (1.15) holds true when $\delta=\xi,\delta^{\prime}=\xi^{\prime}$ :

[TABLE]

which is (14) of [15].

[TABLE]

which is (15) of [15] and

[TABLE]

$\left(\min\left\{\Re(s),\Re(a)\right\}>0\text{when}\left|z\right|\leq 1(z\neq 1),\left|t\right|\leq 1(t\neq 1),\right.\\$* $\left.\quad\quad\quad\quad\quad\quad\quad\Re(s)>1,\text{when}~{}z=1,t=1\right)$, which is (16) of [15].*

Corollary 2.2**.**

In view of (1.7), we have

[TABLE]

$\left(\min\Re(s)>0,\Re(a)>0\quad\text{when}\left|z\right|\leq 1(z\neq 1),\left|t\right|\leq 1(t\neq 1),\right.$

$\left.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\Re(s)>1,\text{when}~{}z=1,t=1\right).$**

Remark 2.1**.**

If we take $t=0$ in (2.2), then it gives (19) of [15] and by setting $t=0,\eta=1$ then (2.2) reduces to (20) of [15]

Theorem 2.2**.**

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

[TABLE]

and

[TABLE]

Proof.

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

[TABLE]

gives

[TABLE]

[TABLE]

[TABLE]

Now substituting (2.12) in (LABEL:FEGHLZ), we get

[TABLE]

interchanging integration and summation gives

[TABLE]

In view of (LABEL:FEGHLZ) and (1.9) we arrived the desired result.

Now, we prove the second integral. From (2.3), $\Phi_{\mu,\eta,\eta^{\prime},\delta,\delta^{\prime};\nu,\xi,\xi^{\prime}}\left(z,t,s,a\right)$ can be written as

[TABLE]

Now using (2.12), we get

[TABLE]

∎

Corollary 2.3**.**

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

[TABLE]

Theorem 2.3**.**

The following summation formula hold true.

[TABLE]

Proof.

Using (LABEL:FEGHLZ), we have

[TABLE]

In view of definition (LABEL:FEGHLZ), we reach the required result. ∎

3. A connection with generalized hypergeometric function

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

Theorem 3.1**.**

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

[TABLE]

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

Proof.

Using (LABEL:FEGHLZ) and the identity [16, page 56, Equation (1)]

[TABLE]

which implies that

[TABLE]

Now,

[TABLE]

we get,

[TABLE]

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

Corollary 3.1**.**

If we set $\delta=\xi$ in Theorem 3.1, then we get (28) of [14] as

[TABLE]

Corollary 3.2**.**

If we set $\nu=\eta,\delta=\xi$ and $\delta^{\prime}=\xi^{\prime}$ in Theorem 3.1, then we get (29) of [14] as

[TABLE]

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 [15, 14] which help to derive some interesting special cases and are mentioned in Remark 2.1 and Corollaries 2.1, 2.2, 2.3, 3.1, and 3.2.

Declaration:

This article is a corrected version of the preprint https://arxiv.org/abs/1706.03516.