A Study of the S-Generalized Gauss Hypergeometric Function and Its Associated Integral Transforms

H. M. Srivastava^{1, 2,} , Rashmi Jain³ and M. K. Bansal³

¹Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada

²China Medical University, Taichung 40402, Taiwan, Republic of China

³Department of Mathematics, Malaviya National Institute of Technology, Jaipur 302017, Rajasthan, India

Cite this paper:
H. M. Srivastava, Rashmi Jain and M. K. Bansal. A Study of the S-Generalized Gauss Hypergeometric Function and Its Associated Integral Transforms. Turkish Journal of Analysis and Number Theory. 2015; 3(5):116-119. doi: 10.12691/tjant-3-5-1

Abstract

The aim of the present paper is to further investigate the S-generalized Gauss hypergeometric function which was recently introduced by Srivastava et al. [8]. In the course of our study, we first present an integral representation, the Mellin transform and a complex integral representation of the S-generalized Gauss hypergeometric function. Next, we introduce a new integral transform whose kernel is the S-generalized Gauss hypergeometric function and point out its three special cases which are also believed to be new. We specify that the well-known Gauss hypergeometric function transform follows as a simple special case of our integral transforms. Finally, we establish an inversion formula for the integral transform which we have introduced in this investigation.

Keywords:
S-Generalized Gauss hypergeometric function Integral representation Complex integral representation Mellin transform Integral transform Inversion formula

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]	M. A. Chaudhry, A. Qadir, H. M. Srivastava and R. B. Paris, Extended hypergeometric and conuentm hypergeometric functions, Appl. Math. Comput. 159 (2004), 589-602.
[2]	L. Debnath and D. Bhatta, Integral Transforms and Their Applications, Third edition, Chapman and Hall (CRC Press), Taylor and Francis Group, London and New York, 2014.
[3]	S.-D. Lin, H. M. Srivastava and M.-M. Wong, Some applications of Srivastava's theorem involving a certain family of generalized and extended hypergeometric polynomials, Filomat 29 (2015), 1811-1819.
[4]	S.-D. Lin, H. M. Srivastava and J.-C. Yao, Some classes of generating relations associated with a family of the generalized Gauss type hypergeometric functions, Appl. Math. Inform. Sci. 9 (2015), 1731-1738.
[5]	E. Özergin, Some Properties of Hypergeometric Functions, Ph.D. Thesis, Eastern Mediterranean University, Gazimağusa, North Cyprus, Turkey, 2011.
[6]	E. Özergin, M. A. Özarslan and A. Altın, Extension of gamma, beta and hypergeometric functions, J. Comput. Appl. Math. 235 (2011), 4601-4610.
[7]	R. K. Parmar, A new generalization of Gamma, Beta, hypergeometric and conuent hypergeometric functions, Matematiche (Catania) 69 (2013), 33-52.
[8]	H. M. Srivastava, P. Agarwal and S. Jain, Generating functions for the generalized Gauss hypergeometric functions, Appl. Math. Comput. 247 (2014), 348-352.
[9]	H. M. Srivastava, M. A. Chaudhry and R. P. Agarwal, The incomplete Pochhammer symbols and their applications to hypergeometric and related functions, Integral Transforms Spec. Funct. 23 (2012), 659-683.
[10]	H. M. Srivastava and J. Choi, Seires Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrechet, Boston and London, 2001.
[11]	H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.
[12]	H. M. Srivastava, K. C. Gupta and S. P. Goyal, The H-Functions of One and Two Variables with Applications, South Asian Publishers, New Delhi and Madras, 1982.
[13]	H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1985.
[14]	H. M. Srivastava, M.-J. Luo, and R. K. Raina, A new integral transform and its applications, Acta Math. Sci. Ser. B Engl. Ed. 35 (2015), 1386-1400.
[15]	H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984.