On the integral representations of some of the Horn's double and Srivastava's triple hypergeometric functions of matrix arguments^*

We propose to define the Horn's double hypergeometric functions H₃ and H₄ of matrix arguments and deduce some integral representations for these two functions. Utilizing the first author's 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 Mathai's 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.pdfhttp://www.ima.umn.edu/preprints/abstracts/1943ab.pdfhttp://www.ima.umn.edu/preprints/nov2003/1943.pdf http://hdl.handle.net/11299/3955https://zbmath.org/?q=an:1254.33008http://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 Srivastava's triple hypergeometric functions H_A and H_B of matrix arguments, we further establish a number of integral representations for these two Srivastava's 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 Srivastava's 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 Srivastava's hyper-geometric 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 Horn's double and the Sri-vastava's triple hypergeometric functions. For proving our results for these functions of matrix arguments we invoke the Mathai's matrix transform technique for real symmetric positive definite matrices as arguments. We conclude by stating the corresponding parallel results for these Horn's double and the Srivastava's 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.