ON EXTON'S TRIPLE HYPERGEOMETRIC FUNCTIONS OF MATRIX ARGUMENTS-II

摘要

In the present paper we generalize a result of Mac Robert T.M.[12] for the case of matrix arguments We also define the Extori's X_4 triple hypergeometric function of matrix arguments and utilizing our earlier definition of the Exron's X_3 triple hypergeometric function of matrix arguments [10] and this generalized result of Mac Robert T.M. [12], we generalize the results of Mathur Radha [11] for the Exton's triple hypergeometric functions X_3 and X_4 of matrix arguments by using the Mathai's matrix transform technique. At the end of the paper we also state corresponding results when the argument matrices are Hermitian positive definite (i.e. the corresponding results for complex matrix arguments) for which the steps of the proofs are parallel to those given by us here. We explicitly mention here that if has become most urgent for us to state these parallel results in order to foil any attempts of plagiarism, because in the past many results of the present author which were given and proved by this author for real symmetric positive definite matrices in his Ph.D. Thesis [9, Matrix Generalizations of Multiple Hypergeometric Functions By Using Mathai's Matrix Transform Techniques (Ph.D. Thesis. Kumaun University. Nainital, Uttarakhand, India, (2004)) #1943. IMA Preprint Series, University of Minnesota, Minneapolis, U.S.A. (2003) (https://www.ima.umn.edu/sites/default/files/1943.pdf; http://hdl.handle.net/11299/3955 hnps://zbmath.org/?q=an:1254.33008. http://citescerx.ist.psu.edu/viewdoc/summary'? doi=10.1.1.192.2172&rank=52 and his other Preprints as mentioned in the references of this Ph.D. thesis of his) have been plagiarized by some authors who have published the corresponding parallel results for complex matrix arguments in their names and the name(s) of their Research Supervisor(s) simply by making the trivial modifications and following verbatim and copying blindly the steps of proofs of this author verbatim and getting them published in some Referced Research J