Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

Dirichlet Average of Generalized Miller-Ross Function and Fractional Derivative

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

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

Abstract: 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.

Keywords: Dirichlet average, Generalized Miller-Ross Function, fractional derivative and Fractional calculus operators

Received December 19, 2014; Revised January 14, 2015; Accepted March 01, 2015

Citation: Mohd. Farman Ali, Manoj Sharma, Lakshmi Narayan Mishra, Vishnu N. Mishra, Dirichlet Average of Generalized Miller-Ross Function and Fractional Derivative. Turkish Journal of Analysis and Number Theory, 3 (1). 30-32. January 2015. doi:10.12691/tjant-3-1-7

1. Introduction

Carlson [1-5] 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.

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

[picture] | (2.3)

F is the Dirichlet measure of with variables and parameters .

Here

If , define

2.4. Generalized Miller-Ross Function

This function is introduced by the author as follows:

[picture] | (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 “Riemann-Liouville integral” is

[picture] | (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 Miller-Ross Function with the fractional derivative i.e.

[picture] | (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

[picture] | (4.1)

This confluent hyper geometric function [11]

Then

(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.

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.

References

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