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 Certain Fractional Integral Formulas Involving the Product of Generalized Bessel Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
D. Baleanu ◽  
P. Agarwal ◽  
S. D. Purohit
Keyword(s):  
Bessel Function ◽  
Fractional Integral ◽  
Bessel Functions ◽  
Fractional Integration ◽  
Fractional Integrals ◽  
General Character ◽  
Generalized Bessel Function ◽  
Integral Formulas ◽  
Generalized Bessel Functions ◽  
Special Cases

We apply generalized operators of fractional integration involving Appell’s functionF3(·)due to Marichev-Saigo-Maeda, to the product of the generalized Bessel function of the first kind due to Baricz. The results are expressed in terms of the multivariable generalized Lauricella functions. Corresponding assertions in terms of Saigo, Erdélyi-Kober, Riemann-Liouville, and Weyl type of fractional integrals are also presented. Some interesting special cases of our two main results are presented. We also point out that the results presented here, being of general character, are easily reducible to yield many diverse new and known integral formulas involving simpler functions.

scholarly journals On Generalized Fractional Integral Operators and the Generalized Gauss Hypergeometric Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Dumitru Baleanu ◽  
Praveen Agarwal
Keyword(s):  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Integral Transform ◽  
Special Functions ◽  
Fractional Integration ◽  
Fractional Integral Operators ◽  
Integral Formulas ◽  
Special Cases ◽  
Generalized Fractional Integral ◽  
Generalized Fractional Integral Operators

A remarkably large number of fractional integral formulas involving the number of special functions, have been investigated by many authors. Very recently, Agarwal (National Academy Science Letters) gave some integral transform and fractional integral formulas involving theFpα,β·. In this sequel, here, we aim to establish some image formulas by applying generalized operators of the fractional integration involving Appell’s functionF3(·)due to Marichev-Saigo-Maeda. Some interesting special cases of our main results are also considered.

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 Pathway fractional integral operators of generalized k-wright function and k4-function

2017 ◽  
Vol 35 (2) ◽  
pp. 235 ◽  
Author(s):  
Dinesh Kumar ◽  
Ram Kishore Saxena ◽  
Jitendra Daiya
Keyword(s):  
Fractional Integral ◽  
Fractional Integration ◽  
Integral Operators ◽  
Wright Function ◽  
Finite Product ◽  
Fractional Integral Operators ◽  
Integral Formulas ◽  
Fractional Integration Operator ◽  
Composition Formula ◽  
Special Cases

In the present work we introduce a composition formula of the pathway fractional integration operator with finite product of generalized K-Wright function and K4-function. The obtained results are in terms of generalized Wright function.Certain special cases of the main results given here are also considered to correspond with some known and new (presumably) pathway fractional integral formulas.

scholarly journals Certain Integral Transform and Fractional Integral Formulas for the Generalized Gauss Hypergeometric Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Junesang Choi ◽  
Praveen Agarwal
Keyword(s):  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Integral Transform ◽  
Special Functions ◽  
Integral Transforms ◽  
Integral Formulas ◽  
Special Cases

A remarkably large number of integral transforms and fractional integral formulas involving various special functions have been investigated by many authors. Very recently, Agarwal gave some integral transforms and fractional integral formulas involving theFp(α,β)(·). In this sequel, using the same technique, we establish certain integral transforms and fractional integral formulas for the generalized Gauss hypergeometric functionsFp(α,β,m)(·). Some interesting special cases of our main results are also considered.

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.

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.

Fractional integral operators characterized by some new hypergeometric summation formulas

2017 ◽  
Vol 20 (2) ◽  
Author(s):  
Min-Jie Luo ◽  
Ravinder Krishna Raina
Keyword(s):  
Special Class ◽  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Integral Operators ◽  
Generalized Hypergeometric Functions ◽  
Fractional Integral Operators ◽  
Summation Formulas ◽  
Generalized Fractional Integral ◽  
Generalized Fractional Integral Operators

AbstractThe purpose of this paper is to study generalized fractional integral operators whose kernels involve a very special class of generalized hypergeometric functions

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 On some Multidimensional Fractional Integral Operators Involving Multivariable I-Function

2015 ◽  
Vol 11 ◽  
pp. 12-20
Author(s):  
Yashwant Singh ◽  
Nanda Kulkarni
Keyword(s):  
Fractional Integral ◽  
Integral Operators ◽  
Integral Transforms ◽  
Two Dimensional ◽  
Fractional Integral Operators ◽  
Basic Properties ◽  
Structural Relationships ◽  
Special Cases ◽  
Multidimensional Integral ◽  
The One

In the present paper, we study certain multidimensional fractional integral operators involving a general I-function in their kernel. We give five basic properties of these operators, and then establish two theorems and two corollaries, which are believed to be new. These basic theorems exhibit structural relationships between the multidimensional integral transforms. The one- and two-dimensional analogues of these results, which are new and of interest in themselves, can easily be deduced. Special cases of these latter theorems will give rise to certain known results obtained from time to time by several earlier authors.

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