_{1}

^{*}

The present paper aims at introducing and investigating a new class of generalized double zeta function i.e. modified double zeta function which involves the Riemann, Hurwitz, Hurwitz-Lerch, Barnes double zeta function and Bin-Saad generalized double zeta function as particular cases. The results are obtained by suitably applying Riemann-Liouville type and Tremblay fractional integral and differential operators. We derive the expansion formula for the proposed function with some of its properties via fractional operators and discuss the link with known results.

The Hurwitz-Lerch zeta function [

The further generalization of Hurwitz-Lerch zeta function

where

In [

and

The generalized double zeta function of Bin-Saad [

where

The alternate representation is

where

The generalized hypergeometric function in classical form has been defined [

where

Bin-Saad [

where F_{1} is the Appell’s function of two variables [

We further recall the following well known expansion formula of Hurwitz-Lerch zeta function [

where

is Hurwitz zeta function which is generalization of the Riemann zeta function given as

Due to great potential and significant role of special functions especially hypergeometric functions in various problems occurring in mathematical physics, engineering [

In section 2 we introduce and discuss some properties of the modified double zeta function. Section 3 devoted to discuss the Trembley [

In a sequel of result (5) here we introduce a modified double zeta function as follows

where

We can readily obtain following relationship

Integration and differentiation of fractional order are traditionally defined by the left side Riemann fractional integral operator

Further for

A generalization of Riemann-Liouville fractional derivatives

(throughout this paper we apply all operators with respect to x variable).

Lemma 2.1. (Mathai and Haubold [

Lemma 2.2. (Srivastava and Tomovski [

Lemma 2.3. If

Proof. On using definition (16), we get

Simplify and using definition (16) again, yields the proof of (29).

Now we define the integral operator as follows:

where

, (31)

The fractional calculus operator

where

We can easily obtain the following result of

where

The operator

Theorem 4.1 If

If

Proof. L.H.S of (38) after using (21) gives

Using definition (16) suitably changing the order of summation and integration, we have

By virtue of (27)

Finally by using definition (16), yields result (38).

Further to prove (39), we use (16) and (23)

Using (38) we get

Finally using lemma 2.3 yields R.H.S of (39).

To prove (40), we have

Using Equation (28), yields proof of (40).

Theorem 4.2 If

Proof. We have

On using (35) we get

After little simplification and using definition (16), yields the results (46).

Remark 4.1. For

Remark 4.2. On putting y = 0 in Equation (49), we get

Remark 4.3. Further if we set b = c in Equation (46), it reduces to known identity due to Trembley [

Theorem 4.3. If

then

Proof. Expressing modified zeta function in L.H.S as series and changing the order of integration and summation, gives

employing (37), yields

which completes the proof.

Corollary 4.1. On putting

Theorem 4.4. If

where

Proof. From (16) we have

Now employing series representation of

Clearly

After little simplification

This completes the proof of (55).

Remark 4.4. For b = c equation (55) yields the result [Bin-Saad [

Theorem 5.1. With all conditions on parameters as stated in Equations (27) and (30), the following properties holds true

Proof. From (21) and (30), we have

Therefore

Interchanging the order of integration and using Dirichlet formula [

and substituting

Making use of (21) leads

this leads the proof of L.H.S of (58).

again

Using the Dirichlet formula [

Substituting

making use of (59) readily leads to the proof of R.H.S of (58).

Recently fractional operator’s theory was recognized to be a good tool for modeling complex problems, kinetic equations, fractional reaction, diffusion equations, etc. In this work we introduce and study the new class of generalized zeta function through Riemann Liouville type and Tremblay fractional integral and differential operators. In section 4, interesting images of modified double zeta function have been obtained and useful link between generalized and modified zeta function has been established through Trembley fractional operator. Series expansion of the new class of generalized zeta function is a significant contribution in the direction along that developed in [

Arif M.Khan, (2016) Modified Double Zeta Function and Its Properties. Advances in Pure Mathematics,06,159-167. doi: 10.4236/apm.2016.63013