scholarly journals Fractional Calculus of the Extended Hypergeometric Function

2020 ◽  
Vol 5 (1) ◽  
pp. 369-384
Author(s):  
Recep Şahin ◽  
Oğuz Yağcı
Keyword(s):  
Kinetic Equation ◽  
Fractional Calculus ◽  
Fractional Derivative ◽  
Hypergeometric Function ◽  
Generating Functions ◽  
Integral Transforms ◽  
Gauss Hypergeometric Function ◽  
Fractional Kinetic Equation ◽  
Fractional Derivative Operators

AbstractHere, our aim is to demonstrate some formulae of generalization of the extended hypergeometric function by applying fractional derivative operators. Furthermore, by applying certain integral transforms on the resulting formulas and develop a new futher generalized form of the fractional kinetic equation involving the generalized Gauss hypergeometric function. Also, we obtain generating functions for generalization of extended hypergeometric function..

scholarly journals Fractional Calculus involving (p, q)-Mathieu Type Series

2020 ◽  
Vol 5 (2) ◽  
pp. 15-34 ◽  
Author(s):  
Daljeet Kaur ◽  
Praveen Agarwal ◽  
Madhuchanda Rakshit ◽  
Mehar Chand
Keyword(s):  
Kinetic Equation ◽  
Fractional Calculus ◽  
Fractional Integral ◽  
Kinetic Equations ◽  
Integral Transforms ◽  
Type Series ◽  
Integral Formulas ◽  
Fractional Kinetic Equation ◽  
Fractional Calculus Operators ◽  
Fractional Kinetic Equations

AbstractAim of the present paper is to establish fractional integral formulas by using fractional calculus operators involving the generalized (p, q)-Mathieu type series. Then, their composition formulas by using the integral transforms are introduced. Further, a new generalized form of the fractional kinetic equation involving the series is also developed. The solutions of fractional kinetic equations are presented in terms of the Mittag-Leffler function. The results established here are quite general in nature and capable of yielding both known and new results.

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.

INTEGRAL TRANSFORM OF K4-FUNCTION

2021 ◽  
Vol 21 (2) ◽  
pp. 429-436
Author(s):  
SEEMA KABRA ◽  
HARISH NAGAR
Keyword(s):  
Laplace Transform ◽  
Hypergeometric Function ◽  
Integral Transform ◽  
Integral Transforms ◽  
Gauss Hypergeometric Function ◽  
Wright Function ◽  
Generalized Wright Function ◽  
Special Cases ◽  
Euler Transform

In this present work we derived integral transforms such as Euler transform, Laplace transform, and Whittaker transform of K4-function. The results are given in generalized Wright function. Some special cases of the main result are also presented here with new and interesting results. We further extended integral transforms derived here in terms of Gauss Hypergeometric function.

scholarly journals A New Extension of the τ-Gauss Hypergeometric Function and Its Associated Properties

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 996 ◽  
Author(s):  
Hari Mohan Srivastava ◽  
Asifa Tassaddiq ◽  
Gauhar Rahman ◽  
Kottakkaran Sooppy Nisar ◽  
Ilyas Khan
Keyword(s):  
Fractional Calculus ◽  
Hypergeometric Function ◽  
Mellin Transform ◽  
Gauss Hypergeometric Function ◽  
Extended Version ◽  
Pochhammer Symbol ◽  
Basic Properties ◽  
Derivative Formulas

In this article, we define an extended version of the Pochhammer symbol and then introduce the corresponding extension of the τ-Gauss hypergeometric function. The basic properties of the extended τ-Gauss hypergeometric function, including integral and derivative formulas involving the Mellin transform and the operators of fractional calculus, are derived. We also consider some new and known results as consequences of our proposed extension of the τ-Gauss hypergeometric function.

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 A remark on local fractional calculus and ordinary derivatives

2016 ◽  
Vol 14 (1) ◽  
pp. 1122-1124 ◽  
Author(s):  
Ricardo Almeida ◽  
Małgorzata Guzowska ◽  
Tatiana Odzijewicz
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Short Note ◽  
General Definition ◽  
Local Fractional Derivative ◽  
Fundamental Properties ◽  
Local Fractional Calculus ◽  
Definition Of

AbstractIn this short note we present a new general definition of local fractional derivative, that depends on an unknown kernel. For some appropriate choices of the kernel we obtain some known cases. We establish a relation between this new concept and ordinary differentiation. Using such formula, most of the fundamental properties of the fractional derivative can be derived directly.

Mkhitar Djrbashian and his contribution to Fractional Calculus

2020 ◽  
Vol 23 (6) ◽  
pp. 1797-1809
Author(s):  
Sergei Rogosin ◽  
Maryna Dubatovskaya
Keyword(s):  
Cauchy Problem ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Approximation Theory ◽  
Fractional Derivatives ◽  
Complex Analysis ◽  
Integral Transforms ◽  
Modern Theory ◽  
Survey Paper ◽  
The Cauchy Problem

Abstract This survey paper is devoted to the description of the results by M.M. Djrbashian related to the modern theory of Fractional Calculus. M.M. Djrbashian (1918-1994) is a well-known expert in complex analysis, harmonic analysis and approximation theory. Anyway, his contributions to fractional calculus, to boundary value problems for fractional order operators, to the investigation of properties of the Queen function of Fractional Calculus (the Mittag-Leffler function), to integral transforms’ theory has to be understood on a better level. Unfortunately, most of his works are not enough popular as in that time were published in Russian. The aim of this survey is to fill in the gap in the clear recognition of M.M. Djrbashian’s results in these areas. For same purpose, we decided also to translate in English one of his basic papers [21] of 1968 (joint with A.B. Nersesian, “Fractional derivatives and the Cauchy problem for differential equations of fractional order”), and were invited by the “FCAA” editors to publish its re-edited version in this same issue of the journal.

Recursion formulas for multivariable hypergeometric functions

2015 ◽  
Vol 08 (04) ◽  
pp. 1550082 ◽  
Author(s):  
Vivek Sahai ◽  
Ashish Verma
Keyword(s):  
Hypergeometric Function ◽  
Radiation Field ◽  
Hypergeometric Functions ◽  
Integral Transforms ◽  
Recursion Formulas ◽  
Appl Math ◽  
Appell Functions

Recently, Opps, Saad and Srivastava [Recursion formulas for Appell’s hypergeometric function [Formula: see text] with some applications to radiation field problems, Appl. Math. Comput. 207 (2009) 545–558] presented the recursion formulas for Appell’s function [Formula: see text] and also gave its applications to radiation field problems. Then Wang [Recursion formulas for Appell functions, Integral Transforms Spec. Funct. 23(6) (2012) 421–433] obtained the recursion formulas for Appell functions [Formula: see text] and [Formula: see text]. In our investigation here, we derive the recursion formulas for 14 three-variable Lauricella functions, three Srivastava’s triple hypergeometric functions and four [Formula: see text]-variable Lauricella functions.

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