scholarly journals The Marichev-Saigo-Maeda Fractional Calculus Operators Pertaining to the V -Function

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
S. Chandak ◽  
Biniyam Shimelis ◽  
Nigussie Abeye ◽  
A. Padma
Keyword(s):  
Fractional Calculus ◽  
Bessel Function ◽  
Hypergeometric Function ◽  
Exponential Function ◽  
Special Functions ◽  
Generalized Hypergeometric Function ◽  
Generalized Bessel Function ◽  
Fractional Calculus Operators

In the present paper, we establish some composition formulas for Marichev-Saigo-Maeda (MSM) fractional calculus operators with V -function as the kernel. In addition, on account of V -function, a variety of known results associated with special functions such as the Mittag-Leffler function, exponential function, Struve’s function, Lommel’s function, the Bessel function, Wright’s generalized Bessel function, and the generalized hypergeometric function have been discovered by defining suitable values for the parameters.

scholarly journals Unified integral associated with the generalized V-function

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
S. Chandak ◽  
S. K. Q. Al-Omari ◽  
D. L. Suthar
Keyword(s):  
Bessel Function ◽  
Hypergeometric Function ◽  
Exponential Function ◽  
Special Functions ◽  
General Nature ◽  
Generalized Hypergeometric Function ◽  
Generalized Bessel Function ◽  
Integral Formulas ◽  
Special Cases ◽  
Lommel Function

Abstract In this paper, we present two new unified integral formulas involving a generalized V-function. Some interesting special cases of the main results are also considered in the form of corollaries. Due to the general nature of the V-function, several results involving different special functions such as the exponential function, the Mittag-Leffler function, the Lommel function, the Struve function, the Wright generalized Bessel function, the Bessel function and the generalized hypergeometric function are obtained by specializing the parameters in the presented formulas. More results are also discussed in detail.

Extended Mittag-Leffler function and associated fractional calculus operators

2020 ◽  
Vol 27 (2) ◽  
pp. 199-209 ◽  
Author(s):  
Junesang Choi ◽  
Rakesh K. Parmar ◽  
Purnima Chopra
Keyword(s):  
Fractional Calculus ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Fractional Integrals ◽  
Generalized Hypergeometric Function ◽  
Pochhammer Symbol ◽  
Special Cases ◽  
Fractional Calculus Operators ◽  
Fractional Integrals And Derivatives ◽  
Appl Math

AbstractMotivated mainly by certain interesting recent extensions of the generalized hypergeometric function [H. M. Srivastava, A. Çetinkaya and I. Onur Kıymaz, A certain generalized Pochhammer symbol and its applications to hypergeometric functions, Appl. Math. Comput. 226 2014, 484–491] by means of the generalized Pochhammer symbol, we introduce here a new extension of the generalized Mittag-Leffler function. We then systematically investigate several properties of the extended Mittag-Leffler function including some basic properties, Mellin, Euler-Beta, Laplace and Whittaker transforms. Furthermore, certain properties of the Riemann–Liouville fractional integrals and derivatives associated with the extended Mittag-Leffler function are also investigated. Some interesting special cases of our main results are pointed out.

scholarly journals Unified Approach to Fractional Calculus Images of Special Functions—A Survey

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2260 ◽  
Author(s):  
Virginia Kiryakova
Keyword(s):  
Fractional Calculus ◽  
Hypergeometric Function ◽  
Special Functions ◽  
Great Part ◽  
General Setting ◽  
Generalized Hypergeometric Function ◽  
Unified Approach ◽  
Generalized Fractional Calculus ◽  
Special Cases ◽  
Google Search

Evaluation of images of special functions under operators of fractional calculus has become a hot topic with hundreds of recently published papers. These are growing daily and we are able to comment here only on a few of them, including also some of the latest of 2019–2020, just for the purpose of illustrating our unified approach. Many authors are producing a flood of results for various operators of fractional order integration and differentiation and their generalizations of different special (and elementary) functions. This effect is natural because there are great varieties of special functions, respectively, of operators of (classical and generalized) fractional calculus, and thus, their combinations amount to a large number. As examples, we mentioned only two such operators from thousands of results found by a Google search. Most of the mentioned works use the same formal and standard procedures. Furthermore, in such results, often the originals and the images are special functions of different kinds, or the images are not recognized as known special functions, and thus are not easy to use. In this survey we present a unified approach to fulfill the mentioned task at once in a general setting and in a well visible form: for the operators of generalized fractional calculus (including also the classical operators of fractional calculus); and for all generalized hypergeometric functions such as pΨq and pFq, Fox H- and Meijer G-functions, thus incorporating wide classes of special functions. In this way, a great part of the results in the mentioned publications are well predicted and appear as very special cases of ours. The proposed general scheme is based on a few basic classical results (from the Bateman Project and works by Askey, Lavoie–Osler–Tremblay, etc.) combined with ideas and developments from more than 30 years of author’s research, and reflected in the cited recent works. The main idea is as follows: From one side, the operators considered by other authors are cases of generalized fractional calculus and so, are shown to be (m-times) compositions of weighted Riemann–Lioville, i.e., Erdélyi–Kober operators. On the other side, from each generalized hypergeometric function pΨq or pFq (p≤q or p=q+1) we can reach, from the final number of applications of such operators, one of the simplest cases where the classical results are known, for example: to 0Fq−p (hyper-Bessel functions, in particular trigonometric functions of order (q−p)), 0F0 (exponential function), or 1F0 (beta-distribution of form (1−z)αzβ). The final result, written explicitly, is that any GFC operator (of multiplicity m≥1) transforms a generalized hypergeometric function into the same kind of special function with indices p and q increased by m.

scholarly journals Solvability of some Abel-type integral equations involving the Gauss hypergeometric function as kernels in the spaces of summable functions

2001 ◽  
Vol 43 (2) ◽  
pp. 291-320 ◽  
Author(s):  
R. K. Raina ◽  
H. M. Srivastava ◽  
A. A. Kilbas ◽  
M. Saigo
Keyword(s):  
Fractional Calculus ◽  
Hypergeometric Function ◽  
Integral Equations ◽  
Gauss Hypergeometric Function ◽  
Generalized Fractional Calculus ◽  
Type Integral ◽  
Fractional Calculus Operators ◽  
Generalized Fractional Calculus Operators

AbstractThis paper is devoted to the study of the solvability of certain one-and multidimensional Abel-type integral equations involving the Gauss hypergeometric function as their kernels in the space of summable functions. The multidimensional equations are considered over certain pyramidal domains and the results obtained are used to present the multidimensional pyramidal analogues of generalized fractional calculus operators and their properties.

scholarly journals Convergence and Extended Linear Generating Function for Generalized Hypergeometric Function

2019 ◽  
Vol 8 (2S11) ◽  
pp. 3577-3579
Keyword(s):  
Hypergeometric Function ◽  
Generating Function ◽  
Special Functions ◽  
Generalized Hypergeometric Function ◽  
Classical Polynomials ◽  
The Subject

The subject of Special functions has a lot importance during the last few decades. The intend of this work is to test the convergence and to introduce the extended linear generating relation for the generalized hypergeometric function. The result is followed by its applications to the classical polynomials.

scholarly journals The Marichev-Saigo-Maeda Fractional-Calculus Operators Involving the (p,q)-Extended Bessel and Bessel-Wright Functions

2021 ◽  
Vol 5 (4) ◽  
pp. 210
Author(s):  
Hari M. Srivastava ◽  
Eman S. A. AbuJarad ◽  
Fahd Jarad ◽  
Gautam Srivastava ◽  
Mohammed H. A. AbuJarad
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Bessel Function ◽  
Fractional Integral ◽  
Hadamard Product ◽  
Gauss Hypergeometric Function ◽  
Wright Function ◽  
Special Cases ◽  
Fractional Calculus Operators ◽  
Fractional Derivative Operators

The goal of this article is to establish several new formulas and new results related to the Marichev-Saigo-Maeda fractional integral and fractional derivative operators which are applied on the (p,q)-extended Bessel function. The results are expressed as the Hadamard product of the (p,q)-extended Gauss hypergeometric function Fp,q and the Fox-Wright function rΨs(z). Some special cases of our main results are considered. Furthermore, the (p,q)-extended Bessel-Wright function is introduced. Finally, a variety of formulas for the Marichev-Saigo-Maeda fractional integral and derivative operators involving the (p,q)-extended Bessel-Wright function is established.

scholarly journals Transformation formulas of incomplete hypergeometric functions via fractional calculus operators

2021 ◽  
Vol 13(62) (2) ◽  
pp. 571-580
Author(s):  
Kamlesh Jangid ◽  
Sunil Dutt Purohit ◽  
Daya Lal Suthar
Keyword(s):  
Fractional Calculus ◽  
Hypergeometric Function ◽  
Present Article ◽  
Hypergeometric Functions ◽  
Fractional Calculus Operators ◽  
Transformation Formulas

The desire for present article is to derive from the application of fractional calculus operators a transformation that expresses a potentially useful incomplete hypergeometric function in various forms of a countable sum of lesser-order functions. Often listed are numerous (known or new) specific cases and implications of the findings described herein

scholarly journals Some properties of Wright-type generalized hypergeometric function via fractional calculus

2014 ◽  
Vol 2014 (1) ◽  
pp. 119 ◽  
Author(s):  
Snehal B Rao ◽  
Jyotindra C Prajapati ◽  
Amitkumar D Patel ◽  
Ajay K Shukla
Keyword(s):  
Fractional Calculus ◽  
Hypergeometric Function ◽  
Generalized Hypergeometric Function

scholarly journals Composition Formulae for the k-Fractional Calculus Operators Associated with k-Wright Function

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
D. L. Suthar
Keyword(s):  
Fractional Calculus ◽  
Hypergeometric Function ◽  
Fractional Order ◽  
Wright Function ◽  
Special Cases ◽  
Fractional Calculus Operators ◽  
Fractional Order Integral

In this article, the k-fractional-order integral and derivative operators including the k-hypergeometric function in the kernel are used for the k-Wright function; the results are presented for the k-Wright function. Also, some of special cases related to fractional calculus operators and k-Wright function are considered.

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