On Exton's Triple Hypergeometric Functions of Matrix Arguments-II

Abstract

In the present   paper we generalize a result of Mac Robert T.M. [12] for the case of matrix arguments. We also define   the   Exton's  triple hypergeometric function of matrix arguments and utilizing our earlier definition of the Exton's  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  and  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 it 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) 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 Refereed Research  Journals of Mathematics of the Country and they seem , to this author, to have managed to obtain their Ph.D. Degrees from some University (ies) of  the  Country at least to some extent on this basis.  When this author made written representations to the concerned University and the concerned Research Journal against this totally unacceptable and unethical academic malpractice about six years ago and requested the concerned University and Research Journal to order a very high level probe into this plagiarism scam and redress this author’s grievances - nothing so far has been received by him in writing in this connection from any quarters, therefore, this approach is followed here.