On the Exton’s triple hypergeometric function X2 of matrix arguments ∗

Lalit Mohan  Upadhyaya; Ayman Shehata; A  Kamal

doi:10.48165/

Authors

Lalit Mohan Upadhyaya Department of Mathematics, Municipal Post Graduate College, Mussoorie, Dehradun, Uttarakhand-248179, India.
Ayman Shehata Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt.
A Kamal Department of Mathematics, Faculty of Science, Port Said University, Port Said, Egypt.

DOI:

https://doi.org/10.48165/

Keywords:

Hypergeometric functions, Exton’s triple hypergeometric function, matrix argument, matrix transform, real positive definiE, Hermitian positive definite

Abstract

We define the Exton’s triple hypergeometric function X2 of matrix argu ments and establish some integral representations for this function which generalize the corresponding results of Choi, Hasanov and Turaev (Choi, Junesang, Hasanov, Anvar and Turaev, Mamasali, Certain integral representations of Euler type for the Exton function X2, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math., 17(4), (2010), 347–354) for the matrix arguments case.

References

Upadhyaya, Lalit Mohan and Dhami, H.S. (Nov.2001). Matrix generalizations of multiple hyperge ometric functions; #1818, IMA Preprint Series, University of Minnesota, Minneapolis, U.S.A. (Re trieved from the University of Minnesota Digital Conservancy, http://hdl.handle.net/11299/ 3706).

Upadhyaya, Lalit Mohan (Nov. 2003): Matrix Generalizations of Multiple Hypergeometric Functions by Using Mathai’s Matrix Transform Techniques (Ph.D. Thesis, Kumaun University, Nainital, Uttarakhand, India) #1943, 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.pdfhttp://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). [3] Mathai, A.M. (1992). Jacobians of Matrix Transformations- I; Centre for Mathematical Sciences, Trivandrum, India.

Mathai, A.M. (1993). Hypergeometric Functions of Several Matrix Arguments; Centre for Math ematical Sciences, Trivandrum, India.

Slater, L.J. (1966). Generalized Hypergeometric Functions, Cambridge University Press, Cam bridge.

Exton, H. (1982). Hypergeometric functions of three variables, J. Indian Acad. Math., 4, 113–119. [7] Mathai, A.M. (1997). Jacobians of Matrix Transformations and Functions of Matrix Argument, World Scientific Publishing Co. Pte. Ltd., Singapore.

Mathai A.M. (1993). Lauricella functions of real symmetric positive definite matrices, Indian J. Pure Appl. Math., 24(9), 513–531.

Mathai A.M. (1993). Appell’s and Humbert’s functions of matrix arguments, Linear Algebra Appl., 183, 201–221.

Mathai, A.M. and Provost, Serge B. (2005). Some complex matrix-variate statistical distributions on rectangular matrices, Linear Algebra and its Applications, 410, 198-ff216.

Upadhyaya, Lalit Mohan (2017). On Exton’s triple hypergeometric functions of matrix arguments ffI, Bulletin of Pure and Applied Sciences, Section-E, Mathematics & Statistics, Vol. 36(E), No.1, 31–36. (Article DOI : 10.5958/2320-3226.2017.00003.0) https://www.bpasjournals.com

Upadhyaya, Lalit Mohan (2017). On Exton’s triple hypergeometric functions of ma trix arguments ffII, Bulletin of Pure and Applied Sciences, Section-E, Mathematics & Statistics, Vol. 36(E), No.2, 207-217. (Article DOI : 10.5958/2320-3226.2017.00023.6) https://web.a.ebscohost.com/abstract?direct=true&profile=ehost&scope=site&authtype= crawler&jrnl=09706577&AN=128769173&h=VKjrBgoGK8zPkmFuWAH0mPQ9

http://www.indianjournals.com/ijor.aspx?target=ijor:bpasms&volume=36e&issue=2& article=013

http://www.ijour.net/ijor.aspx?target=ijor:bpasms&volume=36e&issue=2&article=013 https://www.bpasjournals.com

Exton, H. (1976). Multiple Hypergeometric Functions and Applications, Ellis Horwood Limited, Publishers, Chichester , Sussex, England. Halsted Press: A Divison of John Wiley and Sons, Chichester, New York, Brisbane.

Choi, Junesang, Hasanov, Anvar and Turaev, Mamasali (2010). Certain integral representations of Euler type for the Exton function X2 , J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math., Vol. 17, No. 4 (November 2010), 347–354.

Mathai, A.M. (1978). Some results on functions of matrix arguments, Mathematische Nachrichten, 84, 171–177.

Srivastava, H.M. and Karlsson, P.W. (1985). Multiple Gaussian Hypergeometric Series, Ellis Hor wood Limited, Publishers, Chichester, Sussex, England. Halsted Press: A Divison of John Wiley and Sons, Chichester, New York, Brisbane.

Upadhyaya, Lalit Mohan (2019). Introducing the Upadhyaya integral transform, Bulletin of Pure and Applied Sciences, Section-E, Mathematics and Statistics, Vol. 38(E), No. 1, 471–510. doi 10.5958/2320-3226.2019.00051.1 https://www.bpasjournals.com/