Leibniz Rule for Fractional Derivatives Generalized and an Application to Infinite Series
01 May 1970-Siam Journal on Applied Mathematics (Society for Industrial and Applied Mathematics)-Vol. 18, Iss: 3, pp 658-674
About: This article is published in Siam Journal on Applied Mathematics.The article was published on 1970-05-01. It has received 211 citations till now. The article focuses on the topics: General Leibniz rule & Product rule.
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TL;DR: The authors presents a review of definitions of fractional order derivatives and integrals that appear in mathematics, physics, and engineering, and presents a discussion of the relationship between fractional derivatives and integral derivatives.
Abstract: This paper presents a review of definitions of fractional order derivatives and integrals that appear in mathematics, physics, and engineering.
387 citations
TL;DR: A new formula for the fractional derivative with Mittag-Leffler kernel is established, in the form of a series of Riemann–Liouville fractional integrals, which brings out more clearly the non-locality of fractional derivatives and is easier to handle for certain computational purposes.
246 citations
TL;DR: A systematic account of the investigations carried out by various authors in the field of fractional calculus and its applications and several interesting results are considered.
231 citations
TL;DR: The fractional derivative operator as mentioned in this paper is an extension of the familiar derivative operator $D^n $ to arbitrary (integer, rational, irrational, or complex) values of n. The most important representation
Abstract: The fractional derivative operator is an extension of the familiar derivative operator $D^n $ to arbitrary (integer, rational, irrational, or complex) values of n. The most important representation...
178 citations
TL;DR: In this paper, a Leibniz rule for the fractional difference of the product of two functions is discovered and used to gen- erate series expansions involving the special functions.
Abstract: Derivatives of fractional order, D af, have been considered extensively in the literature. However, little attention seems to have been given to finite differences of frac- tional order, A af. In this paper, a definition of differences of arbitrary order is presented, and A af is computed for several specific functions f (Table 2.1). We find that the operator A a is closely related to the contour integral which defines Meijer's G-function. A Leibniz rule for the fractional difference of the product of two functions is discovered and used to gen- erate series expansions involving the special functions.
162 citations
Cites background from "Leibniz Rule for Fractional Derivat..."
...70, (6)], unless a is a positive integer or zero....
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780 citations
TL;DR: Rodrigues's formula can be applied also to (1.1) and 1.3) as discussed by the authors, but here the situation is slightly more involved in that the integrals with respect to σ^2 are of fractional order and their inversion requires the knowledge of differentiation and integration.
Abstract: Rodrigues’s formula can be applied also to (1.1) and (1.3) but here the situation is slightly more involved in that the integrals with respect to σ^2 are of fractional order and their inversion requires the knowledge of differentiation and integration of fractional order. In spite of this complication the method has its merits and seems more direct than that employed in [1] and [3]. Moreover, once differentiation and integration of fractional order are used, it seems appropriate to allow a derivative of fractional order with respect to σ^-1 to appear so that the ultraspherical polynomial in (1.3) may be replaced by an (associated) Legendre function. This will be done in the present paper.
83 citations
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TL;DR: In this paper, the conditions générales d'utilisation (http://www.numdam.org/conditions) of the agreement with the Scuola Normale Superiore di Pisa are described.
Abstract: © Scuola Normale Superiore, Pisa, 1961, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
13 citations
"Leibniz Rule for Fractional Derivat..." refers background in this paper
...The usual starting point for a definition of fractional derivative taken in recent papers [1], [4], [5], [8], is the Riemann-Liouville fractional integral...
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...Bassam [1] presented another derivation of (1....
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