Leibniz Rule for Fractional Derivatives Generalized and an Application to Infinite Series

1970 ◽  
Vol 18 (3) ◽  
pp. 658-674 ◽  
Author(s):  
Thomas J. Osler
Keyword(s):  
Infinite Series ◽  
Fractional Derivatives ◽  
Leibniz Rule

scholarly journals An Unified Study of Some Multiple Integrals

2018 ◽  
pp. 37-53
Author(s):  
FY. AY. Ant
Keyword(s):  
Infinite Series ◽  
General Class ◽  
Fractional Derivatives ◽  
Fractional Integration ◽  
Compact Form ◽  
Leibniz Rule ◽  
General Nature ◽  
General Character ◽  
Unified Study ◽  
Multiple Integrals

The object of this paper is to derive three unified fractional derivatives formulae for the Saigo-Maeda operators of fractional integration. The first formula deals with the product of a general class of multivariable polynomials and the multivariable Aleph- function. The second concerns the multivariable polynomials and two multivariable Aleph-functions with the help of the Leibniz rule for fractional derivatives. The last relation also implies the product of a class of multivariable polynomials and the multivariable Aleph-function but it is obtained by the application of the first formula twice and it implicates two independents variables instead of one. The polynomials and the functions have their arguments of the type are quite general nature. These formulae, besides being on very general character have been put in a compact form avoiding the occurrence of infinite series and thus making them put in applications. Our findings provide unifications and extensions of some (known and new) results. We shall give several corollaries and particular cases.

Geometric interpretation of fractional-order derivative

2016 ◽  
Vol 19 (5) ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Differential Geometry ◽  
Fractional Order ◽  
Infinite Series ◽  
Fractional Derivatives ◽  
Geometric Interpretation ◽  
Order Derivative ◽  
Jet Bundles ◽  
Fractional Order Derivative ◽  
Integer Order ◽  
Derivatives Of

AbstractA new geometric interpretation of the Riemann-Liouville and Caputo derivatives of non-integer orders is proposed. The suggested geometric interpretation of the fractional derivatives is based on modern differential geometry and the geometry of jet bundles. We formulate a geometric interpretation of the fractional-order derivatives by using the concept of the infinite jets of functions. For this interpretation, we use a representation of the fractional-order derivatives by infinite series with integer-order derivatives. We demonstrate that the derivatives of non-integer orders connected with infinite jets of special type. The suggested infinite jets are considered as a reconstruction from standard jets with respect to order.

A new glance on the Leibniz rule for fractional derivatives

2018 ◽  
Vol 62 ◽  
pp. 244-249 ◽  
Author(s):  
K. Sayevand ◽  
J. Tenreiro Machado ◽  
D. Baleanu
Keyword(s):  
Fractional Derivatives ◽  
Leibniz Rule

scholarly journals Fractional derivatives of certain special functions

2001 ◽  
Vol 32 (2) ◽  
pp. 103-109
Author(s):  
V. B. L. Chaurasia ◽  
Anju Godika
Keyword(s):  
General Class ◽  
Fractional Derivatives ◽  
Special Functions ◽  
Leibniz Rule ◽  
Special Cases ◽  
Derivatives Of

The theorems relating to the fractional derivatives for the multivariable H-function [1,15,17] and a general class of multivariable polynomials [13] have been established in this paper. Use of well known generalized Leibniz rule has been made to derive some theorems. Certain special cases of the main theorems have also been discussed.

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