Modified Double Zeta Function and Its Properties

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.


Introduction and Preliminaries
The Hurwitz-Lerch zeta function [1] is defined by The generalized double zeta function of Bin-Saad [5] is defined by where where * µ ∅ is the generalized zeta function defined by (2).The generalized hypergeometric function in classical form has been defined [6] as , , , ; , , , ; !

(
) ( ) where F 1 is the Appell's function of two variables [7] defined as We further recall the following well known expansion formula of Hurwitz-Lerch zeta function [1] 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 [8] [9], the author has motivated to further investigate the topic.Several generalizations of hypergeometric functions have been made by many authors [10] [11].Recently Rao [12] defined Wright type generalized hypergeometric function via fractional calculus.Many authors investigated the fractional calculus approach in study of generalized hypergeometric type function [13] [14].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 [15] 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.

Modified Double Zeta Function
In a sequel of result (5) here we introduce a modified double zeta function as follows where  .We can readily obtain following relationship the left sided and right sided Riemann fractional differential operators are defined as A generalization of Riemann-Liouville fractional derivatives ( )( ) (throughout this paper we apply all operators with respect to x variable).

The Well Poised Fractional Calculus Operator
The fractional calculus operator z O α β that was introduced by Tremblay [15] is given as where We can easily obtain the following result of z O α β { } where 2 1 F is Gauss hypergeometric function.The operator z O α β has lot more interesting properties and applications.Tremblay introduced this operator to deal with special function more efficiently.
Proof.We have < and all conditions mentioned in theorem 4.1 holds, then where 1, 2, 3, This completes the proof of (55).
Proof.From ( 21) and (30), we have , ; , d d Γ Interchanging the order of integration and using Dirichlet formula [17], we obtain Making use of ( this leads the proof of L.H.S of (58).again Using the Dirichlet formula [17] and interchanging the order of integration we get making use of (59) readily leads to the proof of R.H.S of (58).

Conclusion
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 [5].In section 5, interesting properties of operator

∅
is an analytic function in both variables y and z in suitable region.The further generalization of Hurwitz-Lerch zeta function In [3] [4] Bin-Saad and Al-Gonah introduced two hypergeometric type generating functions of generalized zeta function as follows of fractional order are traditionally defined by the left side Riemann fractional integral operator I f µ α + and right hand operator I f µ β − and the corresponding R-L fractional derivative operawhich are given as follows

Remark 4 . 4 .Theorem 5 . 1 .
For b = c equation (55) yields the result [Bin-Saad [5]: p. 273, Equation (2.18), theorem 2.1].With all conditions on parameters as stated in Equations (27) and (30), the following properties holds true . 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.