Keywords: Dirichlet average, Generalized MillerRoss Function, fractional derivative and Fractional calculus operators
Turkish Journal of Analysis and Number Theory, 2015 3 (1),
pp 3032.
DOI: 10.12691/tjant317
Received December 09, 2014; Revised January 14, 2015; Accepted March 01, 2015
Copyright © 2015 Science and Education Publishing. All Rights Reserved.
1. Introduction
Carlson [15]^{[1]} has defined Dirichlet average of functions which represents certain type of integral average with respect to Dirichlet measure. He showed that various important special functions can be derived as Dirichlet averages for the ordinary simple functions like etc. He has also pointed out ^{[3]} that the hidden symmetry of all special functions which provided their various transformations can be obtained by averaging etc. Thus he established a unique process towards the unification of special functions by averaging a limited number of ordinary functions. Almost all known special functions and their well known properties have been derived by this process.
Recently, Gupta and Agarwal ^{[9, 10]} found that averaging process is not altogether new but directly connected with the old theory of fractional derivative. Carlson overlooked this connection whereas he has applied fractional derivative in so many cases during his entire work. Deora and Banerji ^{[6]} have found the double Dirichlet average of e^{x} by using fractional derivatives and they have also found the Triple Dirichlet Average of x^{t} by using fractional derivatives ^{[7]}.
In the present paper the Dirichlet average of Generalized MillerRoss Function has been obtained.
2. Definitions
Some definitions which are necessary in the preparation of this paper.
2.1. Standard Simplex in :Denote the standard simplex in , by [^{[1]}, p. 62].
2.2. Dirichlet MeasureLet and let be the standard simplex in The complex measure is defined by E ^{[1]}.
known as Dirichlet measure.
Here
open right half plane and k is the Cartesian power of .
2.3. Dirichlet Average[[1], p. 75]Let be the convex set in , let and let be a convex combination of . Let be a measureable function on and let be a Dirichlet measure on the standard simplex in .Define
 (2.3) 
F is the Dirichlet measure of with variables and parameters .
Here
If , define
2.4. Generalized MillerRoss FunctionThis function is introduced by the author as follows:
 (2.4) 
Here, upper parameters and lower parameters and are pochammer symbols. The function (3.6.1.1) is defined when none of the denominator parameters is a negative integer or zero. If any parameter is negative then the function (2.4) terminates into a polynomial in x. By using ratio test, it is evident that function (2.4) is convergent for all x, when , it is convergent for when , divergent when . In some cases the series is convergent for . Let us consider take,
when, the series is absolutely convergent for if , convergent for if and divergent for if 1 which is a special case of Wright function.
2.5. Fractional Derivative [[8], p. 181]The theory of fractional derivative with respect to an arbitrary function has been used by Erdelyi ^{[8]}. The general definition for the fractional derivative of order found in the literature on the “RiemannLiouville integral” is
 (2.5) 
where and is the form of , where is analytic at
3. Equivalence
In this section we shall prove the equivalence of single Dirichlet average of Generalized MillerRoss Function with the fractional derivative i.e.
 (3.1) 
Proof:
Putting we have
On changing the order of integration and summation, we have
Or
Hence by the definition of fractional derivative, we get
This completes the analysis.
4. Particular Cases
If and no upper and lower parameter in (3.1) then
 (4.1) 
This confluent hyper geometric function ^{[11]}
Then
(ii) If and from (4.1), then
where be the generalization of MittagLeffler function ^{[12]}.
5. Applications
Dirichlet average is average given by Dirichlet. The Dirichlet average of elementary function like power function, exponential function etc is given by many notable mathematician, Actually, We have convert the elementary function into the summation form after that taking Dirichlet average of those function, using fractional integral and get new results. These results will be used in future by mathematician and scientist. Thus we have find a connection Dirichlet average of a function and fractional integral.
Acknowledgements
The authors express their thanks to the anonymous learned referee(s) and the editor for their constructive comments, valuable suggestions, which resulted in the subsequent improvement of this research article. The authors are grateful to all editorial board members and reviewers of esteemed journal i.e. Turkish Journal of Applied Analysis and Number Theory (TJANT). All the authors conceived of the study and participated in its design and coordination. All authors drafted the manuscript, participated in the sequence alignment. All the authors read and approved the final version of manuscript. The authors declare that there is no conflict of interests regarding the publication of this research article.
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