Turkish Journal of Analysis and Number Theory
ISSN (Print): 2333-1100 ISSN (Online): 2333-1232 Website: http://www.sciepub.com/journal/tjant
Open Access
Journal Browser
Go
Turkish Journal of Analysis and Number Theory. 2015, 3(5), 116-119
DOI: 10.12691/tjant-3-5-1
Open AccessArticle

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

H. M. Srivastava1, 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

Pub. Date: December 03, 2015

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

Creative CommonsThis 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.