Turkish Journal of Analysis and Number Theory
ISSN (Print): 2333-1100 ISSN (Online): 2333-1232 Website: https://www.sciepub.com/journal/tjant
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Turkish Journal of Analysis and Number Theory. 2020, 8(6), 97-106
DOI: 10.12691/tjant-8-6-1
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Certain Generating Functions Involving Some Hypergeometric Series of Four Variables by Means of Operational Representations

Jihad A. Younis1, Maged G. Bin-Saad1 and Kottakkaran S. Nisar2,

1Department of Mathematics, Aden University, Aden, Yemen

2Department of Mathematics, College of Arts and Sciences, Wadi Aldawaser, Prince Sattam bin Abdulaziz University, Saudi Arabia

Pub. Date: November 15, 2020

Cite this paper:
Jihad A. Younis, Maged G. Bin-Saad and Kottakkaran S. Nisar. Certain Generating Functions Involving Some Hypergeometric Series of Four Variables by Means of Operational Representations. Turkish Journal of Analysis and Number Theory. 2020; 8(6):97-106. doi: 10.12691/tjant-8-6-1


The main aim of this present paper is to present certain generating functions of some hypergeometric functions in four variables by using the integral and symbolic representations for these quadruple functions. A few interesting special cases have also been considered.

Laplace integrals symbolic representations quadruple hypergeometric functions generating functions

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[1]  Srivastava, M. H and Manocha, H. A., Treatise on Generating Functions. Halsted Press, Bristone, London, New York and Toronto, 1985.
[2]  Agarwal, P., Chand, M and Purohit, S. D., A note on generating functions involving the generalized Gauss hypergeometric function. National Acad. Sci. Lett., 3, 457-459, 2014.
[3]  Bin-Saad, M. G and Younis, J. A., Certain generating functions of some quadruple hypergeometric series. Eurasian Bulletin Math., 2, 56-62, 2019.
[4]  Desale, B. S and Qashash, G. A., Generating functions of special triple hypergeometric functions. International Mathematical Forum, 9, 1677-1693, 2014.
[5]  Liu, H and Wang, W., Some generating relations for extended Appell’s and Lauricella’s hypergeometric functions. Rocky Mountain J. Math., 44, 1987-2007, 2014.
[6]  Singh, M., Pundhir, S and Singh, M. P., Generating function of certain hypergeometric functions by means of fractional calculus. International J. Comput. Eng. Res. (IJCER), 11, 40-47, 2017.
[7]  Bin-Saad, M. G and Younis, J. A., On connections between certain class of new quadruple and known triple hypergeometric series. Tamap Journal of Mathematics and Statistics, Volume 2019.
[8]  Bin-Saad, M. G and Younis, J. A., Certain quadruple hypergeometric series and their integral representations. Appl. Appl. Math., 14, 1085-1098, 2019.
[9]  Bin-Saad, M. G and Younis, J. A., Certain integrals associated with hypergeometric functions of four variables. Earthline J. Math. Sci., 2, 325-341, 2019.
[10]  Bin-Saad, M. G and Younis, J. A., Certain integral representations of some quadruple hypergeometric series. Palestine J. Math., 9, 132-141, 2020.
[11]  Bin-Saad, M. G and Younis, J. A., Some integrals connected with a new quadruple hypergeometric series, Universal J. Math. Appl., 3, 19-27, 2020.
[12]  Srivastava, M. H and Karlsson, P. W., Multiple Gaussian Hypergeometric Series. Ellis Horwood Lt1., Chichester, 1984.
[13]  Lauricella, G., Sull funzioni ipergeometric a pi variabili. Rend. Cric. Mat. Palermo, 7, 111-158, 1893.
[14]  Exton, H., Hypergeometric functions of three variables. J. Indian Acad. Math., 4, 113-119, 1982.
[15]  Sharma, C and Parihar, C. L., Hypergeometric functions of four variables (I). J. Indian Acad. Math., 11, 121-133, 1989.
[16]  Erdélyi, A., Magnus, W., Oberhettinger, F and Tricomi, F. G., Higher Transcendental Functions. Vol. I, McGraw-Hill Book Company, New York, Toronto and London, 1953.
[17]  Rainville, E. D., Special Functions. The Macmillan Company: New York, NY, USA, 1960; Reprinted by Chelsea Publishing Company, Bronx, NY, USA, 1971.
[18]  Miller, K. S and Ross, B., An Introduction to Fractional Calculus and Fractional Differential Equations. Wiley, New York, 1993.
[19]  Bin-Saad, M. G and Hussein, M. A., Operational images and relations of two and three variable hypergeometric series. J. Progr. Res. Math., 2, 39-46, 2015.