APMAdvances in Pure Mathematics2160-0368Scientific Research Publishing10.4236/apm.2016.63013APM-63987ArticlesPhysics&Mathematics Modified Double Zeta Function and Its Properties rifM. Khan1*Department of Mathematics, Jodhpur Institute of Engineering &amp; Technology, Jodhpur, India* E-mail:khanarif76@gmail.com2602201606031591674 November 2015accepted 24 February 29 February 2016© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

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.

Modified Zeta Function Riemann-Liouville Operator Tremblay Fractional Operators Hypergeometric Function
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

and

The generalized double zeta function of Bin-Saad  is defined by

where;,

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.

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 

where,; and

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

where;

,.

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:

where;

, (31)

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

If then

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;, ,

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 Equation (52) reduces to

Theorem 4.4. If, and all conditions mentioned in theorem 4.1 holds, then

where;.

Proof. From (16) we have

Now employing series representation of at R.H.S in above equation by using (13)

Clearly

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].

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  , we obtain

and substituting, we have

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

again

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).

6. 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  . 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.

Cite this paper

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

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