scholarly journals A New Extension of Extended Caputo Fractional Derivative Operator

2018 ◽  
Author(s):  
Gauhar Rahman ◽  
Kottakkaran Sooppy Nisar ◽  
Shahid Mubeen
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Hypergeometric Functions ◽  
Special Functions ◽  
Beta Function ◽  
Elementary Functions ◽  
Research Papers ◽  
Derivative Operator ◽  
Mellin Transforms ◽  
Generating Relations

Recently, different extensions of the fractional derivative operator are found in many research papers. The main aim of this paper is to establish an extension of the extended Caputo fractional derivative operator. The extension of an extended fractional derivative of some elementary functions derives by considering an extension of beta function which includes the Mittag-Leffler function in the kernel. Further, an extended fractional derivative of some familiar special functions, the Mellin transforms of newly defined Caputo fractional derivative operator and the generating relations for extension of extended hypergeometric functions also presented in this study.

scholarly journals On Extended Caputo Fractional Derivative Operator

2017 ◽  
Author(s):  
Gauhar Rahman ◽  
Kottakkaran Sooppy Nisar ◽  
Muhammad Arshad
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Hypergeometric Functions ◽  
Special Functions ◽  
Elementary Functions ◽  
Derivative Operator ◽  
Mellin Transforms ◽  
Generating Relations

The main objective of this present paper is to introduce further extension of extended Caputo fractional derivative operator and establish the extension of an extended fractional derivative of some known elementary functions. Also, we investigate the extended fractional derivative of some familiar special functions, the Mellin transforms of newly defined Caputo fractional derivative operator and generating relations for extension of extended hypergeometric functions.

scholarly journals A new extension and applications of Caputo fractional derivative operator

Analysis ◽  
2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Gauhar Rahman ◽  
Kottakkaran Sooppy Nisar ◽  
Muhammad Arshad
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Hypergeometric Functions ◽  
Mellin Transform ◽  
Special Functions ◽  
Elementary Functions ◽  
Derivative Operator ◽  
Generating Relations

AbstractThe main objective of this paper is to introduce a further extension of the extended Caputo fractional derivative operator and establish the extension of an extended fractional derivative of some known elementary functions. Additionally, we investigate the extended fractional derivative of some familiar special functions, the Mellin transform of the newly defined Caputo fractional derivative operator and generating relations for the extensions of extended hypergeometric functions.

scholarly journals An Extension of Caputo Fractional Derivative Operator by Use of Wiman’s Function

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2238
Author(s):  
Rahul Goyal ◽  
Praveen Agarwal ◽  
Alexandra Parmentier ◽  
Clemente Cesarano
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Hypergeometric Functions ◽  
Special Functions ◽  
Derivative Operator ◽  
The Family ◽  
Generating Relations ◽  
Fractional Derivative Operators ◽  
Extended Operator ◽  
Two Parameter

The main aim of this work is to study an extension of the Caputo fractional derivative operator by use of the two-parameter Mittag–Leffler function given by Wiman. We have studied some generating relations, Mellin transforms and other relationships with extended hypergeometric functions in order to derive this extended operator. Due to symmetry in the family of special functions, it is easy to study their various properties with the extended fractional derivative operators.

scholarly journals Extended Riemann-Liouville type fractional derivative operator with applications

2017 ◽  
Vol 15 (1) ◽  
pp. 1667-1681 ◽  
Author(s):  
P. Agarwal ◽  
Juan J. Nieto ◽  
M.-J. Luo
Keyword(s):  
Fractional Derivative ◽  
Hypergeometric Functions ◽  
Beta Function ◽  
Gauss Hypergeometric Function ◽  
Modified Bessel Function ◽  
Elementary Functions ◽  
Liouville Type ◽  
Derivative Operator ◽  
Generating Relations ◽  
Liouville Fractional Derivative

Abstract The main purpose of this paper is to introduce a class of new extended forms of the beta function, Gauss hypergeometric function and Appell-Lauricella hypergeometric functions by means of the modified Bessel function of the third kind. Some typical generating relations for these extended hypergeometric functions are obtained by defining the extension of the Riemann-Liouville fractional derivative operator. Their connections with elementary functions and Fox’s H-function are also presented.

scholarly journals On Generalized k-Fractional Derivative Operator

2017 ◽  
Author(s):  
Gauhar Rahman ◽  
Kottakkaran Sooppy Nisar ◽  
Shahid Mubeen
Keyword(s):  
Fractional Derivative ◽  
Hypergeometric Functions ◽  
Mellin Transform ◽  
Beta Function ◽  
Fractional Operator ◽  
Derivative Operator ◽  
Definition Of

The main objective of this paper is to introduce k-fractional derivative operator by using the definition of k-beta function. We establish some results related to the newly defined fractional operator such as Mellin transform and relations to k-hypergeometric and k-Appell's functions. Also, we investigate the k-fractional derivative of k-Mittag-Leffler and Wright hypergeometric functions.

scholarly journals Some Incomplete Hypergeometric Functions and Incomplete Riemann-Liouville Fractional Integral Operators

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 483 ◽  
Author(s):  
Mehmet Ali Özarslan ◽  
Ceren Ustaoğlu
Keyword(s):  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Recurrence Relations ◽  
Beta Function ◽  
Integral Operators ◽  
Integral Representations ◽  
Fractional Integral Operators ◽  
Derivative Formulas ◽  
Generating Relations ◽  
Transformation Formulas

Very recently, the incomplete Pochhammer ratios were defined in terms of the incomplete beta function B y ( x , z ) . With the help of these incomplete Pochhammer ratios, we introduce new incomplete Gauss, confluent hypergeometric, and Appell’s functions and investigate several properties of them such as integral representations, derivative formulas, transformation formulas, and recurrence relations. Furthermore, incomplete Riemann-Liouville fractional integral operators are introduced. This definition helps us to obtain linear and bilinear generating relations for the new incomplete Gauss hypergeometric functions.

scholarly journals On Some Formulas for the k-Analogue of Appell Functions and Generating Relations via k-Fractional Derivative

2020 ◽  
Vol 4 (4) ◽  
pp. 48 ◽  
Author(s):  
Övgü Gürel Yılmaz ◽  
Rabia Aktaş ◽  
Fatma Taşdelen
Keyword(s):  
Fractional Derivative ◽  
Hypergeometric Functions ◽  
Close Relation ◽  
Generating Relations ◽  
Transformation Formulas ◽  
Appell Functions

Our present investigation is mainly based on the k-hypergeometric functions which are constructed by making use of the Pochhammer k-symbol in Diaz et al. 2007, which are one of the vital generalizations of hypergeometric functions. In this study, we focus on the k-analogues of F1Appell function introduced by Mubeen et al. 2015 and the k-generalizations of F2 and F3 Appell functions indicated in Kıymaz et al. 2017. we present some important transformation formulas and some reduction formulas which show close relation not only with k-Appell functions but also with k-hypergeometric functions. Employing the theory of Riemann–Liouville k-fractional derivative from Rahman et al. 2020, and using the relations which we consider in this paper, we acquire linear and bilinear generating relations for k-analogue of hypergeometric functions and Appell functions.

scholarly journals Extended k-Gamma and k-Beta Functions of Matrix Arguments

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1715
Author(s):  
Ghazi S. Khammash ◽  
Praveen Agarwal ◽  
Junesang Choi
Keyword(s):  
Gamma Function ◽  
Hypergeometric Functions ◽  
Special Functions ◽  
Integral Formula ◽  
Beta Function ◽  
Integral Representations ◽  
Matrix Argument ◽  
Functional Relations

Various k-special functions such as k-gamma function, k-beta function and k-hypergeometric functions have been introduced and investigated. Recently, the k-gamma function of a matrix argument and k-beta function of matrix arguments have been presented and studied. In this paper, we aim to introduce an extended k-gamma function of a matrix argument and an extended k-beta function of matrix arguments and investigate some of their properties such as functional relations, inequality, integral formula, and integral representations. Also an application of the extended k-beta function of matrix arguments to statistics is considered.

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