# Certain Integral Representations for Hypergeometric Functions of Four Variables

### Abstract

In the present work, we first introduce five new quadruple hypergeometric series and then we give integral representations of Euler type and Laplace type for these new hypergeometric series, which we denote by

### References

Bin-Saad, Maged G., Younis, Jihad A. (2018). Some Integral Representations for Certain Quadruple

Hypergeometric Functions. MATLAB J., 1 , 61-68.

Bin-Saad, Maged G., Younis, Jihad A. and R. Aktaş (2018). Integral representations for certain quadruple

hypergeometric series. Far East J. Math. Sci., 103, 21-44.

Erdélyi , A., Magnus, W., Oberhettinger, F. and Tricomi, F.G. (1953). Higher Transcendental Functions.

Vol. I, McGraw-Hill Book Company, New York, Toronto and London.

Exton, H. (1976). Multiple hypergeometric functions and applications. Halsted Press, New York, London,

Sydney and Toronto.

Exton, H. (1982). Hypergeometric functions of three variables. J. Indian Acad. Math., 4, 113-119.

Frankl, F.I. (1973). Selected Works in Gas Dynamics. Nauka, Moscow, (in Russian).

Jihad A. Younis (2017). Operational representations for quadruple hypergeometric series and their

applications. M.Sc. Thesis, Dept. of Math., Aden University .

Lauricella, G. (1893). Sull funzioni ipergeometric a più variabili. Rend. Cric. Mat. Palermo 7, 111-158.

Lohöfer, G. (1989). Theory of an electromagnetically deviated metal sphere. I: Absorbed power. SIAM J.

Appl. Math., 49, 567–581.

Niukkanen, A.W. (1983). Generalised hypergeometric series arising( ,..., ) 1 N N F x xin physical and quantum chemical applications. J. Phys. A: Math. Gen., 16, 1813–1825.

Opps, S.B., Saad, N. and Srivastava, H.M. (2005). Some reduction and transformation formulas for the

Appell hypergeometric function 2 F . J. Math. Anal. Appl., 302, 180–195.

Padmanabham, P.A and Srivastava, H.M. (2000). Summation formulas associated with the Lauricella function (r ) A F . Appl. Math. Lett. 13 (1), 65–70.

Saran, S. (1954). Hypergeometric functions of three variables. Ganita 5, 2, 77-91.

Sharma, C. and Parihar, C.L. (1989). Hypergeometric functions of four variables . J. Indian Acad. Math., 11, 121-133.

Srivastava, H.M. and Choi, J. (2001). Series Associated with the Zeta and Related Functions. Kluwer

Academic Publishers, Dordrecht, Boston and London.

Srivastava, H.M. and Choi, J.(2012). Zeta and q-Zeta Functions and Associated Series and Integrals.

Elsevier Science Publishers, Amsterdam, London and New York.

Srivastava, H.M. (1985). A class of generalised multiple hypergeometric series arising in physical and

quantum chemical applications. J. Phys. A: Math. Gen., 18, L227–L234.

Srivastava, H.M. and Karlsson, P.W. (1985). Multiple Gaussian Hypergeometric Series. Halsted Press,

Bristone, London, New York and Toronto.

Srivastava, H.M. and Manocha, H.L. (1984). A treatise on generating functions. Ellis Horwood Lt1.,

Chichester.

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