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)

On Generating Functions of Quadruple Hypergeometric Function

Maged G. Bin-Saad*, Jihad Ahmed Younis

Department of Mathematics, Aden University, Aden, Khormaksar, P.O.Box 6014, Yemen

Abstract: In this work, by using Laplace integral representation of quadruple function defined in [1], we introduce new generating functions involving some triple hyper-geometric functions and itself. Some particular cases and consequences of our main results are also considered.

Keywords: Laplace transforms, triple hyper-geometric functions, quadruple hypergeometric functions, generating functions

Received November 09, 2018; Revised January 18, 2019; Accepted January 27, 2019

Citation: Maged G. Bin-Saad, Jihad Ahmed Younis, On Generating Functions of Quadruple Hypergeometric Function . Turkish Journal of Analysis and Number Theory, 7 (1). 5-10. January 2019. doi:10.12691/tjant-7-1-2

1. Introduction

Recently, Bin-Saad et al. [1] introduced five new quadruple hyper-geometric functions whose names are to investigate their five Laplace integral representations which include the confluent hyper-geometric function , , a Humbert functions and in their kernels. Very recently Bin-Saad and Younis [2] established new integral representations of Euler type for some hyper-geometric functions of four variables, whose kernels include the quadruple hyper-geometric functions , of which is defined as follows

[picture] | (1.1)

where denotes the Pochhammer symbol given by and .

The following is the Laplace integral representation of the function (see [1]):

[picture] | (1.2)

In diverse areas in applied mathematics and mathematical physics, generating functions play an important role in the investigation of various useful properties of the sequences which they generate. These are used to find certain properties and formulas for numbers and polynomials in a wide range of research subjects such as modern combinatory. One can refer the extensive work of Srivastava and Manocha [3] for a systematic introduction to, and several interesting and useful applications of the various methods of obtaining linear, bilinear, bilateral or mixed multilateral generating functions for a fairly wide variety of sequences of hyper-geometric functions and polynomials in one, two or more variables, among much abundant literature. In fact, a remarkable large number of generating functions involving a variety of hyper-geometric functions have been developed by many authors (for example [4,5,6,7] and the related references therein). Here, we use the integral representation of the hyper-geometric function of four variables o obtain new generating functions involving Exton’s functions ,,,of three variables, the Lauricella functions of three variables and the quadruple functions itself. Some special cases of the main results here are also considered.

2. Generating Functions

For our purpose, we begin by recalling Exton's functions of three variables ,, and defined by

[picture] | (2.1)

[picture] | (2.2)

[picture] | (2.3)

and

[picture] | (2.4)

respectively (see [8]). Lauricella hyper-geometric functions of three variables are as below (see [9])

[picture] | (2.5)

[picture] | (2.6)

Now, we begin the following theorem:

[picture] | (2.7)

[picture] | (2.8)

[picture] | (2.9)

[picture] | (2.10)

[picture] | (2.11)

[picture] | (2.12)

[picture] | (2.13)

[picture] | (2.14)

[picture] | (2.15)

[picture] | (2.16)

Proof. To prove the above relations, we need the following formulae (cf. [3,10,11,12]):

[picture] | (2.17)

[picture] | (2.18)

[picture] | (2.19)

[picture] | (2.20)

[picture] | (2.21)

[picture] | (2.22)

[picture] | (2.23)

[picture] | (2.24)

For the convenience, we denote the left hand side of (2.7) with , using (1.2)

by using (2.23), we have

The function which appears in above equation can be replaced by its series form and then interchanging the order of the summation and integral sign which is permissible here, we get

Now, use of (2.17), (2.18) and (2.19), in above equation and then simplified with series manipulation completes the proof of relation (2.7). From the relations (2.17) to (2. 24), one can easily obtain the other generating functions.

3. Special Cases

It is easy to observe that the main results (2.7) to (2.16) gave a number generating functions and transformations for the hhyper-geometric series of four variables . In the present section, we will mention only some special cases. setting in (2.7) to (2.16), we obtain the following relations:

[picture] | (3.1)

[picture] | (3.2)

[picture] | (3.3)

[picture] | (3.4)

[picture] | (3.5)

[picture] | (3.6)

[picture] | (3.7)

[picture] | (3.8)

[picture] | (3.9)

[picture] | (3.10)

Equations (3.2), (3.7) and (3.9) with , yield Exton's results [8]. Equations (3.4), (3.5) and (3.10) with , yields the known results [8]. If we put in (3.8), we get

[picture] | (3.11)

Now, if in (2.7) and (2.9), we take , we shall obtain the following generating functions :

[picture] | (3.12)

[picture] | (3.13)

A special case of (3.13) when yields the well-known results (see [8]).

4. Conclusion and Observation

Based on Laplace integral representation of the quadruple function defined in [1], we introduce new generating functions for the function involving triple hyper-geometric functions. Some particular cases and consequences of our main results are also considered. We conclude this investigation by remarking that the schema suggested in the derivation of the results in this work can be applied to find other new generating functions of other four variables hyper-geometric series and study their particular cases.

References

[1]   | Maged G. Bin-Saad, A.Younis, Jihad, and R. Aktaş, Integral representations for certain quadruple hyper-geometric series", Far East J. Math. Sci., 103, 21-44.2018.
[2]   | Maged G. Bin-Saad and A.Younis, Jihad, "Some Integral Representations for Certain Quadruple Hyper-geometric Functions", MATLAB J., 1, 61-68. 2018.
[3]   | H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Ellis Horwood Lt1., Chichester, 1984.
[4]   | P. Agarwal and C. L. Koul, "On generating functions", J. Rajasthan Acad. Phy., Sci. 2(3), 173-180. 2003.
[5]   | J. Choi and P. Agarwal, "Certain generating functions involving Appell series", Far East J. Math. Sci., 84, 25-32.2014.
[6]   | H. M. Srivastava, "Certain generating functions of several variables", Z. Angew. Math. Mech. 57, 339-340. 1977.
[7]   | H. M. Srivastava, P. Agarwal and S. Jain, "Generating functions for the generalized Gauss hyper-geometric functions", Appl. Math. Comput., 247, 348-352.2014.
[8]   | H. Exton, "Hyper-geometric functions of three variables", J. Indian Acad. Math. 4, 113-119. 1982.
[9]   | Lauricella, G., "Sull funzioni ipergeometric a più variabili.:, Rend. Cric. Mat. Palermo 7, 111-158. 1893.
[10]   | A. Erdélyi, W. Magnus, F. Oberhettinger and F. G., Tricomi, Higher Transcendental Functions. Vol. I, McGraw-Hill Book Company, New York, Toronto and London, 1953.
[11]   | E. D. Rainville, Special Functions, Macmillan Company, New York, 1960, Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.
[12]   | H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press, Bristone, London, New York and Toronto, 1985.