scholarly journals A Study of Extensions of Classical Summation Theorems for the Series $$_{3}F_{2}$$ and $$_{4}F_{3}$$ with Applications

2021 ◽  
Vol 76 (2) ◽  
Author(s):  
Mohamed M. Awad ◽  
Wolfram Koepf ◽  
Asmaa O. Mohammed ◽  
Medhat A. Rakha ◽  
Arjun K. Rathie
Keyword(s):  
Hypergeometric Functions ◽  
Generalized Hypergeometric Functions ◽  
Special Cases ◽  
Laplace Type

AbstractVery 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.

scholarly journals On a New Class of Laplace-Type Integrals Involving Generalized Hypergeometric Functions

Axioms ◽  
2019 ◽  
Vol 8 (3) ◽  
pp. 87 ◽  
Author(s):  
Wolfram Koepf ◽  
Insuk Kim ◽  
Arjun K. Rathie
Keyword(s):  
Hypergeometric Functions ◽  
Generalized Hypergeometric Functions ◽  
New Class ◽  
Special Cases ◽  
Laplace Type

In the theory of generalized hypergeometric functions, classical summation theorems for the series 2 F 1 , 3 F 2 , 4 F 3 , 5 F 4 and 7 F 6 play a key role. Very recently, Masjed-Jamei and Koepf established generalizations of the above-mentioned summation theorems. Inspired by their work, the main objective of the paper is to provide a new class of Laplace-type integrals involving generalized hypergeometric functions p F p for p = 2 , 3 , 4 , 5 and 7 in the most general forms. Several new and known cases have also been obtained as special cases of our main findings.

scholarly journals Certain Unified Integrals Associated with Product of M-Series and Incomplete H-functions

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1191 ◽  
Author(s):  
Manish Kumar Bansal ◽  
Devendra Kumar ◽  
Ilyas Khan ◽  
Jagdev Singh ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Hypergeometric Functions ◽  
Generalized Hypergeometric Functions ◽  
Special Cases ◽  
Fox’S H Function

In this paper, we established some interesting integrals associated with the product of M-series and incomplete H-functions, which are expressed in terms of incomplete H-functions. Next, we give some special cases by specializing the parameters of M-series and incomplete H-functions (for example, Fox’s H-Function, Incomplete Fox Wright functions, Fox Wright functions and Incomplete generalized hypergeometric functions) and also listed few known results. The results obtained in this work are general in nature and very useful in science, engineering and finance.

Hypergeometric Mellin transforms

1955 ◽  
Vol 51 (4) ◽  
pp. 577-589 ◽  
Author(s):  
L. J. Slater
Keyword(s):  
Hypergeometric Functions ◽  
Generalized Hypergeometric Functions ◽  
Mellin Transforms ◽  
Special Cases

The Mellin transforms of generalized hypergeometric functions are discussed in this paper, and it is shown how some of the most general integrals of the Mellin type can be deduced from them. Four general theorems are considered and a number of special cases are given in detail.

scholarly journals EVALUATION OF A NEW CLASS OF EULERIAN'S TYPE INTEGRALS INVOLVING GENERALIZED HYPERGEOMETRIC FUNCTIONS

2019 ◽  
pp. 855
Author(s):  
Insuk Kim ◽  
Gradimir V. Milovanović ◽  
Arjun Rathie
Keyword(s):  
Hypergeometric Functions ◽  
Generalized Hypergeometric Functions ◽  
New Class ◽  
Special Cases

Very recently Masjed-Jamei and Koepf [Axioms 2018, 7 (2), 38] established interesting and useful generalizations of various classical summation theorems for the${}_{2}F_{1}$, ${}_{3}F_{2}$, ${}_{4}F_{3}$, ${}_{5}F_{4}$ and ${}_{6}F_{5}$ generalized hypergeometric series.The main aim of this paper is to establish eleven Eulerian's type integrals involving generalized hypergeometric functions by employing these theorems. Several special cases (known and unknown) have also been given.

scholarly journals Some applications of Srivastava’s theorem involving a certain family of generalized and extended hypergeometric polynomials

Filomat ◽  
2015 ◽  
Vol 29 (8) ◽  
pp. 1811-1819 ◽  
Author(s):  
Shy-Der Lin ◽  
H.M. Srivastava ◽  
Mu-Ming Wong
Keyword(s):  
Generating Functions ◽  
Hypergeometric Functions ◽  
Integral Transforms ◽  
Pochhammer Symbol ◽  
Generalized Hypergeometric Functions ◽  
Fundamental Properties ◽  
Hypergeometric Polynomials ◽  
Special Cases

Recently, Srivastava et al. [H. M. Srivastava, M. A. Chaudhry and R. P. Agarwal, The incomplete Pochhammer symbols and their applications to hypergeometric and related functions, Integral Transforms Spec. Funct. 23 (2012), 659-683] introduced and initiated the study of many interesting fundamental properties and characteristics of a certain pair of potentially useful families of the so-called generalized incomplete hypergeometric functions. Ever since then there have appeared many closely-related works dealing essentially with notable developments involving various classes of generalized hypergeometric functions and generalized hypergeometric polynomials, which are defined by means of the corresponding incomplete and other novel extensions of the familiar Pochhammer symbol. Here, in this sequel to some of these earlier works, we derive several general families of hypergeometric generating functions by applying Srivastava?s Theorem. We also indicate various (known or new) special cases and consequences of the results presented in this paper.

scholarly journals Slip Effects on Fractional Viscoelastic Fluids

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
Muhammad Jamil ◽  
Najeeb Alam Khan
Keyword(s):  
Hypergeometric Functions ◽  
Fractional Derivatives ◽  
Fluid Motion ◽  
Maxwell Fluid ◽  
Newtonian Fluids ◽  
Generalized Hypergeometric Functions ◽  
General Solutions ◽  
Slip Effects ◽  
Special Cases ◽  
Wright Generalized Hypergeometric Functions

Unsteady flow of an incompressible Maxwell fluid with fractional derivative induced by a sudden moved plate has been studied, where the no-slip assumption between the wall and the fluid is no longer valid. The solutions obtained for the velocity field and shear stress, written in terms of Wright generalized hypergeometric functions , by using discrete Laplace transform of the sequential fractional derivatives, satisfy all imposed initial and boundary conditions. The no-slip contributions, that appeared in the general solutions, as expected, tend to zero when slip parameter is . Furthermore, the solutions for ordinary Maxwell and Newtonian fluids, performing the same motion, are obtained as special cases of general solutions. The solutions for fractional and ordinary Maxwell fluid for no-slip condition also obtained as limiting cases, and they are equivalent to the previously known results. Finally, the influence of the material, slip, and the fractional parameters on the fluid motion as well as a comparison among fractional Maxwell, ordinary Maxwell, and Newtonian fluids is also discussed by graphical illustrations.

Fractional Integration of Generalized Bessel Function of the First Kind

2011 ◽  
Author(s):  
Pradeep Malik ◽  
Saiful R. Mondal ◽  
A. Swaminathan
Keyword(s):  
Bessel Function ◽  
Hypergeometric Functions ◽  
Fractional Integration ◽  
Integral Transforms ◽  
Generalized Hypergeometric Functions ◽  
Fractional Integral Operators ◽  
Generalized Bessel Function ◽  
Hyperbolic Sine ◽  
Hyperbolic Cosine ◽  
Special Cases

Generalizing the classical Riemann-Liouville and Erde´yi-Kober fractional integral operators two integral transforms involving Gaussian hypergeometric functions in the kernel are considered. Formulas for composition of such integrals with generalized Bessel function of the first kind are obtained. Special cases involving trigonometric functions such as sine, cosine, hyperbolic sine and hyperbolic cosine are deduced. These results are established in terms of generalized Wright function and generalized hypergeometric functions.

scholarly journals Summation Formulas Obtained by Means of the Generalized Chain Rule for Fractional Derivatives

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
S. Gaboury ◽  
R. Tremblay
Keyword(s):  
General Formula ◽  
Hypergeometric Functions ◽  
Fractional Derivatives ◽  
Special Functions ◽  
Mathematical Physics ◽  
Chain Rule ◽  
Generalized Hypergeometric Functions ◽  
Summation Formulas ◽  
Several Variables ◽  
Special Cases

In 1970, several interesting new summation formulas were obtained by using a generalized chain rule for fractional derivatives. The main object of this paper is to obtain a presumably new general formula. Many special cases involving special functions of mathematical physics such as the generalized hypergeometric functions, the Appell F1 function, and the Lauricella functions of several variables FD(n) are given.

scholarly journals On several new Laplace transforms of generalized hypergeometric functions 2F2(x)

2021 ◽  
Vol 39 (4) ◽  
pp. 97-109
Author(s):  
Asmaa Orabi Mohammed ◽  
Medhat A. Rakha ◽  
Mohammed M. Awad ◽  
Arjun K. Rathie
Keyword(s):  
Laplace Transform ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Laplace Transforms ◽  
Theoretical Physics ◽  
Generalized Hypergeometric Function ◽  
Generalized Hypergeometric Functions ◽  
Special Cases ◽  
And Mathematics

By employing generalizations of Gauss's second, Bailey's and Kummer's summation theorems obtained earlier by Rakha and Rathie, we aim to establish unknown Laplace transform of six rather general formulas of generalized hypergeometric function 2F2[a,b;c,d;x]. The results obtained in this paper are simple, interesting, easily established and may be useful in theoretical physics, engineering and mathematics. Results obtained earlier by Kim et al. and Choi and Rathie follow special cases of our main findings.

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