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In this paper, we introduce the k-Horn's hypergeometric function H9,k and we investigate its limit formulas, integral representations, differentiation formulas, infinite sums, recursion formulas, the Laplace, Mellin, fractional Fourier, double Laplace and double Mellin transforms for the k-Horn's hypergeometric function H9,k. Finally, we discuss the fractional integration and the k-fractional differentiation. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the k-Horn's hypergeometric function H9,k and we investigate its limit formulas, integral representations, differentiation formulas, infinite sums, recursion formulas, the Laplace, Mellin, fractional Fourier, double Laplace and double Mellin transforms for the k-Horn's hypergeometric function H9,k. Finally, we discuss the fractional integration and the k-fractional differentiation. [ABSTRACT FROM AUTHOR]

In this paper we introduce an extension of the Horn's hypergeometric function H11. Furthermore, we investigate the limit formulas, integral representations, differentiation formulas, infinite sums, recursion formulas, Laplace, Mellin and fractional Fourier transforms for the extended Horn's hypergeometric function H11. Finally, we discuss double Laplace and double Mellin transforms of this function. [ABSTRACT FROM AUTHOR]

In this paper we introduce an extension of the Horn's hypergeometric function H11. Furthermore, we investigate the limit formulas, integral representations, differentiation formulas, infinite sums, recursion formulas, Laplace, Mellin and fractional Fourier transforms for the extended Horn's hypergeometric function H11. Finally, we discuss double Laplace and double Mellin transforms of this function. [ABSTRACT FROM AUTHOR]

It is almost two years now when the Upadhyaya transform was introduced by the first author (Upadhyaya, Lalit Mohan, Introducing the Upadhyaya integral transform, Bull. Pure Appl. Sci. Sect. E Math. Stat., 38(E)(1), 471-510, doi 10.5958/2320-3226.2019.00051.1 https://www.researchgate.net/publication/334033797)) as the most powerful, versatile and robust generalization and unification of a number of variants of the classical Laplace transform which have appeared in the mathematics research literature during the years 1993 to 2019. In this paper we provide an update on the Upadhyaya transform, where we explain the definition the one-dimensional Upadhyaya transform and its n-dimensional generalization in more detail and we show that how many other various variants of the classical Laplace transform, that have come to our notice since then and most of which are introduced into the mathematics research literature during the past two years by a number of authors after the advent of the Upadhyaya transform, follow as a special case of the Upadhyaya transform. We find the Upadhyaya transform of some trigonometric and hyperbolic functions, the sine integral, the generalized hypergeometric function and the Bessel function of the first kind in order to exemplify the vast power of the Upadhyaya transform and we also correct a minor typo in the aforementioned paper of the first author. [ABSTRACT FROM AUTHOR]

We define the Exton's triple hypergeometric function X2 of matrix arguments 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. [ABSTRACT FROM AUTHOR]

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.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)) of the Srivastava's triple hypergeometric functions HA and HB 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 HB, 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 hypergeometric function HA, 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 Srivastava'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. [ABSTRACT FROM AUTHOR]

We establish some integral representations for the Srivastava's triple hypergeometric function HB of matrix arguments which generalize some of the recent results of Choi, Hasanov and Turaev [12] for this function. We prove our results by employing the Mathai's matrix transform technique for real symmetric positive definite matrices as arguments. Towards the end of the paper we also give the corresponding results when the argument matrices are complex Hermitian positive definite. [ABSTRACT FROM AUTHOR]

We prove some Eulerian integrals involving the hypergeometric functions of single matrix argument. The integrals studied here involve the 0F1, 1F2, 2 F3, 1F1, 3F3, 2F1 and the 3 2 F functions of matrix argument. We apply the Mathai's matrix transform technique to prove our results which provide a generalization of the corresponding results available in the literature for these functions for the case of scalar variables. At the end we also give the corresponding results when the argument matrices are complex Hermitian positive definite. [ABSTRACT FROM AUTHOR]

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 X4 triple hypergeometric function of matrix arguments and utilizing our earlier definition of the Exton's X3 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 X3 and X4 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) (https://www.ima.umn.edu/sites/default/files/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 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. [ABSTRACT FROM AUTHOR]

The present paper carries ahead my previous study [10] of a basic analogue of a Kampé de Fériet function in two variables which follows as a particular case of a generalized Kampé de Fériet function appearing in eq.(282,283), p. 349 of Srivastava and Karlsson [7]. A summation formula and some transformations of this formula have been deduced here for the function under study. [ABSTRACT FROM AUTHOR]

In the present paper I have defined the Horn's G1 and H2 double hypergeometric functions of matrix arguments with real symmetric strictly negative definite and real symmetric strictly positive definite matrices as the arguments and have established integral representations for them using the Mathai's matrix transform technique. [ABSTRACT FROM AUTHOR]

In the present paper I have deduced a summation formula for the Horn's double hypergeometric function H7 for special values of the arguments along with its different possible forms. [ABSTRACT FROM AUTHOR]

In the present paper I have discussed a basic analogue of the Lauricella-Saran triple hypergeometric function FG and have deduced some summation formulas for it. [ABSTRACT FROM AUTHOR]