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Bulletin of Pure and Applied Sciences- Math& Stat. (Started in 1982)
eISSN: 2320-3226
pISSN: 0970-6577
Impact Factor: 4.895 (2017)
DOI: 10.5958/2320-3226
Editor-in-Chief:  Prof. Dr. Lalit Mohan Upadhyaya
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Article Details

Bulletin of Pure and Applied Sciences- Math& Stat. (Started in 1982)
Year : 2020, Volume & Issue : BPAS-Maths & Stat 39E(1), JAN-JUN 2020
Page No. : 149-164, Article Type : Original Aticle
Article DOI : 10.5958/2320-3226.2020.00014.4 (Communicated, edited and typeset in Latex by Jyotindra C. Prajapati (Editor). Received March 21, 2019 / Revised February 26, 2020 / Accepted March 25, 2020. Online First Published on June 30, 2020)

On the integral representations of some of the Horns double and Srivastavas triple hypergeometric functions of matrix arguments

Lalit Mohan Upadhyaya1 , Ayman Shehata2 and A. Kamal3
Author’s Affiliation : 1. Department of Mathematics, Municipal Post Graduate College, Mussoorie, Dehradun, Uttarakhand-248179, India. 2. Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt. 2. Department of Mathematics, College of Science and Arts at Unaizah, Qassim University, Qassim, Kingdom of Saudi Arabia. 3. Department of Mathematics, College of Science and Arts at Muthnib, Qassim University, Qassim, Kingdom of Saudi Arabia. 3. Department of Mathematics, Faculty of Science, Port Said University, Port Said, Egypt. 1. E-mail: [email protected] , [email protected] 2. E-mail: [email protected] 3. Email: [email protected]

Corresponding Author : Lalit Mohan Upadhyaya,
E-Mail:-[email protected], [email protected]


Abstract

We propose to define the Horns double hypergeometric functions $H_3$  and $H_4$  of matrix arguments and deduce some integral representations for these two functions. Utilizing the first authors definitions (Upadhyaya, Lalit Mohan and Dhami, H.S.,  Matrix generalizations of multiple hypergeometric functions;  #1818, Nov. 2001, IMA Preprint Series, University of Minnesota, Minneapolis, U.S.A. (Retrieved from the University of Minnesota Digital Conservancy, http://hdl.handle.net/11299/3706 ); Upadhyaya, Lalit Mohan, Matrix Generalizations of Multiple Hypergeometric  Functions by Using Mathais Matrix Transform Techniques (Ph.D. Thesis, Kumaun  University,  Nainital, Uttarakhand, India),   #1943, Nov. 2003, IMA Preprint Series, University of  Minnesota, Minneapolis,   U.S.A.

(https://www.ima.umn.edu/sites/default/files/1943.pdf       

http://www.ima.umn.edu/preprints/abstracts/1943ab.pdf

http://www.ima.umn.edu/preprints/nov2003/1943.pdf  

http://hdl.handle.net/11299/3955

https://zbmath.org/?q=an:1254.33008

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.192.2172&rank=52).

(Retrieved from the University of Minnesota Digital Conservancy, http://hdl.handle.net/11299/3955) of the Srivastavas triple hypergeometric functions $H_A$  and $H_B$  of matrix arguments,  we further establish a number of integral representations for these two Srivastavas triple hypergeometric functions, which generalize some of the recent results of Choi, Hasanov and Turaev (Choi, Junesang, Hasanov, Anvar and Turaev, Mamasali,  Integral representations for Srivastavas hypergeometric function $H_B$,  J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math.,   Vol. 19, No. 2 (May 2012), (2012), 137-145:   http://dx.doi.org/10.7468/jksmeb.2012.19.2.137 ; Choi, Junesang, Hasanov, Anvar and Turaev, Mamasali,  Integral representations for Srivastavas hypergeometric function $H_A$,  Honam Mathematical J.,   Vol. 34, No. 1, (2012), 113-124:  http://dx.doi.org/10.5831/HMJ.2012.34.1.113 ; Choi, Junesang, Hasanov, Anvar and Turaev, Mamasali, Decomposition formulas and integral representations for some Exton hypergeometric functions, Journal of the Chungcheong Mathematical Society.,   Vol. 24, No. 4 (December 2011), (2011), 745-758) for these two of the Horns double and the  Srivastavas triple hypergeometric functions. For proving our results for these functions of matrix arguments we invoke the Mathais matrix transform technique for real symmetric positive definite matrices as arguments. We conclude by stating  the corresponding parallel results for these Horns double and the Srivastavas triple hypergeometric functions, when their argument matrices are complex Hermitian positive definite, with the remark that these parallel  results can be easily proved by following our given lines of proofs and by employing the corresponding known results available in the literature.

 

Keywords

Hypergeometric functions, Horns double hypergeometric functions, Srivastavas triple hypergeometric functions, Extons triple hypergeometric function, matrix argument, matrix transform, real positive definite, Hermitian positive definite. 2020 Mathematics Subject Classification: Primary: 33C05, 33C10, 33C15, 33C20, 33C99. Secondary: 60E, 62H, 44A05.
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