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Your search keyword '"RAKHA, MEDHAT A."' showing total 95 results
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95 results on '"RAKHA, MEDHAT A."'

1. Generalized Extension of Watson's theorem for the series $_{3}F_{2}(1)$

Author

Rakha, Medhat A., Awad, Mohammed M., and Mohammed, Asmaa O.

Subjects
Mathematics - Classical Analysis and ODEs, 33C05, 33C20, 33C70
Abstract

The $_{3}F_{2}$ hypergeometric function plays a very significant role in the theory of hypergeometric and generalized hypergeometric series. Despite that $_{3}F_{2}$ hypergeometric function has several applications in mathematics, also it has a lot of applications in physics and statistics. The fundamental purpose of this research paper is to find out the explicit expression of the $_{3}F_{2}$ Watson's classical summation theorem of the form: \[ _{3}F_{2}\left[ \begin{array} [c]{ccccc}% a, & b, & c & & \\ & & & ; & 1\\ \frac{1}{2}(a+b+i+1), & 2c+j & & & \end{array} \right] \] with arbitrary $i$ and $j$, where for $i=j=0$, we get the well known Watson's theorem for the series $_{3}F_{2}(1)$., Comment: 13 pages

Published
2017

2. On certain hypergeometric identities deducible by using beta integral method

Author

Ibrahim, Adel K., Rakha, Medhat A., and Rathie, Arjun K.

Subjects
Mathematics - Complex Variables, Mathematics - Classical Analysis and ODEs, 33C05, 33C20, 33C70
Abstract

The aim of this research paper is to demonstrate how one can obtain eleven new and interesting hypergeometric identities (in the form of a single result) from the old ones by mainly applying the well known beta integral method which was used successfully and systematically by Krattenthaler and Rao in their well known, very interesting research papers. The results are derived with the help of generalization of a quadratic transformation formula due to Kummer very recently obtained by Kim, et al. . Several identities including one obtained earlier by Krattenthaler and Rao follow special cases of our main findings. The results established in this paper are simple, interesting, easily established and may be potentially useful., Comment: 8 pages

Published
2013

3. Remark On Two Results due to Ramanujan on Hypergeometric Series

Author

Rakha, Medhat A., Ibrahim, Adel K., and Rathie, Arjun K.

Subjects
Mathematics - Complex Variables
Abstract

During the course of verifying the results of Ramanujan on hypergeometric series, Berndt in his notebooks, Part II mentioned corrected forms of two of the Ramanujan's results. The aim of this short research note is to point out that one of the results obtained by Ramanujan is correct (and not of Berndt's result) and the second result (which is slightly differ from Ramanujan's result and Berndt's result) is given here in corrected form.

Published
2013

11. Another Method for Proving Certain Reduction Formulas for the Humbert Function ψ 2 Due to Brychkov et al. with an Application.

Author

Mohammed, Asmaa O., Kilicman, Adem, Awad, Mohamed M., Rathie, Arjun K., and Rakha, Medhat A.

Subjects
APPLIED mathematics, ENGINEERING mathematics, OPERATIONS research, HYPERGEOMETRIC functions, HYPERGEOMETRIC series, INTEGRAL representations
Abstract

Recently, Brychkov et al. established several new and interesting reduction formulas for the Humbert functions (the confluent hypergeometric functions of two variables). The primary objective of this study was to provide an alternative and simple approach for proving four reduction formulas for the Humbert function ψ 2 . We construct intriguing series comprising the product of two confluent hypergeometric functions as an application. Numerous intriguing new and previously known outcomes are also achieved as specific instances of our primary discoveries. It is well-known that the hypergeometric functions in one and two variables and their confluent forms occur naturally in a wide variety of problems in applied mathematics, statistics, operations research, physics (theoretical and mathematical) and engineering mathematics, so the results established in this paper may be potentially useful in the above fields. Symmetry arises spontaneously in the abovementioned functions. [ABSTRACT FROM AUTHOR]

Published
2022
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22. A Study of Extensions of Classical Summation Theorems for the Series 3F2 and 4F3 with Applications.

Author

Awad, Mohamed M., Koepf, Wolfram, Mohammed, Asmaa O., Rakha, Medhat A., and Rathie, Arjun K.

Abstract

Very recently, Masjed-Jamei & Koepf [Some summation theorems for generalized hypergeometric functions, Axioms, 2018, 7, 38, 10.3390/axioms 7020038] established some summation theorems for the generalized hypergeometric functions. The aim of this paper is to establish extensions of some of their summation theorems in the most general form. As an application, several Eulerian-type and Laplace-type integrals have also been given. Results earlier obtained by Jun et al. and Koepf et al. follow special cases of our main findings. [ABSTRACT FROM AUTHOR]

Published
2021
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25. On an extension of kummer-type II transformation

Author

Rakha, Medhat Awadalla and Rathie, Arjun Kumar

Subjects
Generalized hypergeometric series, Kummer’s type I and II transformations
Abstract

In the theory of hypergeometric and generalized hypergeometric series, Kummer’s type I and II transformations play an important role. In this short research paper, we aim to establish the explicit expression of e⁻ x/2 2F2 [a, d + n; x 2a + n, d;] for n = 3. For n = 0, we have the well known Kummer’s second transformation. For n = 1, the result was established by Rathie and Pogany [12] and later on by Choi and Rathie[2]. For n = 2, the result was recently established by Rakha, et al. [10]. The result is derived with the help of Kummer’s second transformation and its contiguous results recently obtained by Kim, et. al.[4]. The result established in this short research paper is simple, interesting, easily established and may be potentially useful. Publisher's Version

Published
2014

31. Extensions of Certain Classical Summation Theorems for the Series F, F, and F with Applications in Ramanujan's Summations

Author

Kim, Yong Sup, Rakha, Medhat A., and Rathie, Arjun K.

Subjects
Article Subject, lcsh:Mathematics, lcsh:QA1-939
Abstract

Motivated by the extension of classical Gauss's summation theorem for the series . As applications, certain very interesting summations due to Ramanujan have been generalized. The results derived in this paper are simple, interesting, easily established, and may be useful.F

Published
2010

49. On a reducibility of the Kampé de Fériet function.

Author

Rakha, Medhat A., Awad, Mohammed M., and Rathie, Arjun K.

Subjects
*HYPERGEOMETRIC functions, *POCHHAMMER-Chree frequency equations, *GAMMA functions, *DIFFERENTIAL equations, *TRANSCENDENTAL functions
Abstract

The main objective of this research note is to provide an interesting result for the reducibility of the Kampé de Fériet function. The result is derived with the help of two results for the terminating 3 F2 series very recently obtained by Rakha et al. A few interesting special cases have also been given. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Published
2015
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50. ON LAPLACE TRANSFORMS OF SOME SPECIAL 1F1(α;b;x).

Author

SHABNA, P. A., RAKHA, MEDHAT A., and AWAD, MOHAMED M.

Subjects
*LAPLACE transformation, *HYPERGEOMETRIC functions, *HYPERGEOMETRIC series, *INTEGERS, *GAMMA functions
Abstract

The aim of this research paper is to demonstrate how one can obtain a large number of transforms of some special confluent hypergeometric function 1F1(α,;b;x) by employing generalizations of Kummer's summation theorem, Gauss's second summation theorem and Bailey's summation theorem for the series 2F1 in the theory of hypergeometric series, recently added in the literature. The results in this paper may be useful in physics, engineering and mathematics. [ABSTRACT FROM AUTHOR]

Published
2013

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