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.
(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.