Dirichlet Average of Generalized Miller-Ross Function and Fractional Derivative

Mohd. Farman Ali, Manoj Sharma, Lakshmi Narayan Mishra, Vishnu N. Mishra

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Dirichlet Average of Generalized Miller-Ross Function and Fractional Derivative

Mohd. Farman Ali1, Manoj Sharma2, Lakshmi Narayan Mishra3, 4, Vishnu N. Mishra5,

1School of Mathematics and Allied Sciences, Jiwaji University, Gwalior

2Department of Mathematics RJIT, BSF Academy, Tekanpur, Gwalior

3Department of Mathematics, National Institute of Technology, Silchar - 788 010, District - Cachar (Assam), India

4L. 1627 Awadh Puri Colony Beniganj, Phase – III, Opposite – Industrial Training Institute (I.T.I.), Faizabad - 224 001 (Uttar Pradesh), India

5Applied Mathematics and Humanities Department, Sardar Vallabhbhai National Institute of Technology, Ichchhanath Mahadev Dumas Road, Surat – 395 007 (Gujarat), India


The object of the present paper is to establish the results of single Dirichlet average of Generalized Miller-Ross Function, using Riemann-Liouville Fractional Integral. The Generalized Miller-Ross Function can be measured as a Dirichlet average and connected with fractional calculus. In this paper the solution comes in compact form of single Dirichlet average of Generalized Miller-Ross Function. The special cases of our results are same as earlier obtained by Saxena et al. [12], for single Dirichlet average of Generalized Miller-Ross Function.

Cite this article:

  • Ali, Mohd. Farman, et al. "Dirichlet Average of Generalized Miller-Ross Function and Fractional Derivative." Turkish Journal of Analysis and Number Theory 3.1 (2015): 30-32.
  • Ali, M. F. , Sharma, M. , Mishra, L. N. , & Mishra, V. N. (2015). Dirichlet Average of Generalized Miller-Ross Function and Fractional Derivative. Turkish Journal of Analysis and Number Theory, 3(1), 30-32.
  • Ali, Mohd. Farman, Manoj Sharma, Lakshmi Narayan Mishra, and Vishnu N. Mishra. "Dirichlet Average of Generalized Miller-Ross Function and Fractional Derivative." Turkish Journal of Analysis and Number Theory 3, no. 1 (2015): 30-32.

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1. Introduction

Carlson [1-5][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 ex by using fractional derivatives and they have also found the Triple Dirichlet Average of xt by using fractional derivatives [7].

In the present paper the Dirichlet average of Generalized Miller-Ross 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 Measure

Let and let be the standard simplex in The complex measure is defined by E [1].

known as Dirichlet measure.


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


F is the Dirichlet measure of with variables and parameters .


If , define

2.4. Generalized Miller-Ross Function

This function is introduced by the author as follows:


Here, upper parameters and lower parameters and are pochammer symbols. The function ( 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 “Riemann-Liouville integral” is


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 Miller-Ross Function with the fractional derivative i.e.



Putting we have

On changing the order of integration and summation, we have


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


This confluent hyper geometric function [11]


(ii) If and from (4.1), then

where be the generalization of Mittag-Leffler 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.


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.


[1]  Carlson, B.C., Special Function of Applied Mathematics, Academic Press, New York, 1977.
In article      
[2]  Carlson, B.C., Appell’s function F4 as a double average, SIAM J.Math. Anal. 6 (1975), 960-965.
In article      CrossRef
[3]  Carlson, B.C., Hidden symmetries of special functions, SIAM Rev. 12 (1970), 332-345.
In article      CrossRef
[4]  Carlson, B.C., Dirichlet averages of x t log x, SIAM J.Math. Anal. 18(2) (1987), 550-565.
In article      CrossRef
[5]  Carlson, B.C., A connection between elementary functions and higher transcendental functions, SIAM J. Appl. Math. 17 (1969), 116-140.
In article      CrossRef
[6]  Deora, Y. and Banerji, P.K., Double Dirichlet average of ex using fractional derivatives, J. Fractional Calculus 3 (1993), 81-86.
In article      
[7]  Deora, Y. and Banerji, P.K., Double Dirichlet average and fractional derivatives, Rev.Tec.Ing.Univ. Zulia 16 (2) (1993), 157-161.
In article      
[8]  Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi , F.G., Tables of Integral Transforms, Vol. 2 McGraw-Hill, New York, 1954.
In article      PubMed
[9]  Gupta,S.C. and Agrawal, B.M., Dirichlet average and fractional derivatives, J. Indian Acad.Math. 12(1) (1990), 103-115.
In article      
[10]  Gupta,S.C. and Agrawal, Double Dirichlet average of ex using fractional derivatives, Ganita Sandesh 5 (1) (1991),47-52.
In article      
[11]  Mathai, A.M. and Saxena,R.K., The H-Function with Applications in Stastistics and other Disciplines, Wiley Halsted, New York, 1978.
In article      
[12]  Saxena,R.K., Mathai,A.M and Haubold, H.J., Unified fractional kinetic equation and a fractional diffusion equation, J. Astrophysics and Space Science 209 (2004) , 299-310.
In article      CrossRef
[13]  Sharma, Manoj and Jain, Renu, Dirichlet Average and Fractional Derivatie, J. Indian Acad. Math. Vol. 12, No. 1(1990).
In article      
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