Received 4 November 2015; accepted 24 February 2016; published 29 February 2016
1. Introduction and Preliminaries
The Hurwitz-Lerch zeta function  is defined by
is an analytic function in both variables y and z in suitable region.
The further generalization of Hurwitz-Lerch zeta function is defined by 
where denotes the Pochhammer’s symbol, ,
In   Bin-Saad and Al-Gonah introduced two hypergeometric type generating functions of generalized zeta function as follows
The generalized double zeta function of Bin-Saad  is defined by
The alternate representation is
where is the generalized zeta function defined by (2).
The generalized hypergeometric function in classical form has been defined  as
where; denominator parameters are neither zero nor negative integers.
Bin-Saad  discussed following relationships.
where F1 is the Appell’s function of two variables  defined as
We further recall the following well known expansion formula of Hurwitz-Lerch zeta function 
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   , the author has motivated to further investigate the topic. Several generalizations of hypergeometric functions have been made by many authors   . Recently Rao  defined Wright type generalized hypergeometric function via fractional calculus. Many authors investigated the fractional calculus approach in study of generalized hypergeometric type function   . The subject fractional calculus has gained much attention amongst researchers due to its vast potential of demonstrated mathematical models in various fields of science and engineering such as diffusion, oscillation, dynamical process in porous structures, propagation of waves, diffusive transport, fluid flow, etc. The present paper aims at introducing and investigating a new kind of hypergeometric type function that is modified double zeta function via fractional calculus. The layout of the paper is as follows
In section 2 we introduce and discuss some properties of the modified double zeta function. Section 3 devoted to discuss the Trembley  well poised fractional calculus operator together with its properties. In section 4, we establish some interesting results of modified double zeta function through fractional operators and also derive its summation formula. In section 5, we develop some properties of fractional operators. Many Lemmas and particular cases have been discussed to relate known results.
2. Modified Double Zeta Function
In a sequel of result (5) here we introduce a modified double zeta function as follows
We can readily obtain following relationship
Integration and differentiation of fractional order are traditionally defined by the left side Riemann fractional integral operator and right hand operator and the corresponding R-L fractional derivative operators and  , which are given as follows
Further for the left sided and right sided Riemann fractional differential operators are defined as
A generalization of Riemann-Liouville fractional derivatives  is given by
(throughout this paper we apply all operators with respect to x variable).
is the space of Lebesgue measurable real or complex valued functions such that
Lemma 2.1. (Mathai and Haubold  ) If; then
Lemma 2.2. (Srivastava and Tomovski  ) If, , then
Lemma 2.3. If, , , w > 0 then
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:
3. The Well Poised Fractional Calculus Operator
The fractional calculus operator that was introduced by Tremblay  is given as
where and due to  we have
We can easily obtain the following result of
where is Gauss hypergeometric function.
The operator has lot more interesting properties and applications. Tremblay introduced this operator to deal with special function more efficiently.
4. The Main Results
Theorem 4.1 If;;, then for following results holds true
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, , then
Proof. We have
On using (35) we get
After little simplification and using definition (16), yields the results (46).
Remark 4.1. For, z = 1 Equation (46) yields.
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;, ,
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 Equation (52) reduces to
Theorem 4.4. If, and all conditions mentioned in theorem 4.1 holds, then
Proof. From (16) we have
Now employing series representation of at R.H.S in above equation by using (13)
After little simplification
This completes the proof of (55).
Remark 4.4. For b = c equation (55) yields the result [Bin-Saad  : p. 273, Equation (2.18), theorem 2.1].
5. Some Properties of the Operator
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
Interchanging the order of integration and using Dirichlet formula  , we obtain
and substituting, we have
Making use of (21) leads
this leads the proof of L.H.S of (58).
Using the Dirichlet formula  and interchanging the order of integration we get
Substituting, we get
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  . In section 5, interesting properties of operator have been derived. Many lemmas, corollaries and remarks are obtained to link results with earlier known work. Composition results of Trembley fractional operators and modified zeta function are very useful due to general nature proposed function which may lead several functions and open vast scope of further research in the operator’s field.
 Bin-Saad, M.G. (2007) Sums and Partial Sums of Double Power Series Associated with the Generalized Zeta Function and Their N Fractional Calculus. Mathematical Journal of Okayama University, 49, 37-52.
 Prajapati, J.C., Saxena, R.K., Jana, R.K. and Shukla, A.K. (2013) Some Results on Mittag Leffler Function Operator. Journal of Inequalities and Applications, 2013, 33.
 Rao, S.B., Prajapati, J.C., Patel, A.D. and Shukla A.K. (2014) Some Properties of Wright Type Hypergeometric Functionvia Fractional Calculus. Advances in Difference Equation, 2014, 119.
 Srivastva, H.M. and Tomovski, Z. (2009) Fractional Calculus with an integral Operator Containing a Generalized Mittag Leffler in the Kernel. Applied Mathematics and Computation, 211, 198-210.