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Taylor’s Series Generalized for Fractional Derivatives and Applications
The familiar Taylor’s series expansion of the function , $f(z)$ has for its general term $D^n f(z_0 ){{(z - z_0 )^n } / {n!}}$. A new generalization of Taylor’s series in which the general term isExpand
Differences of fractional order
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. InExpand
Fundamental properties of fractional derivatives via pochhammer integrals
In this paper, various representations of fractional differentiation are explored, and a definition using Pochhammer contour integrals emerges as deserving special emphasis. The analyticity ofExpand
Fractional Derivatives and Special Functions
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...
The Integral Analog of the Leibniz Rule
This paper demonstrates that the classical Leibniz rule for the derivative of the product of two functions
A Child's Garden of Fractional Derivatives
Tom Osler (osler@rowan.edu) is a professor of mathematics at Rowan University. He received his Ph.D. from the Courant Institute at New York University in 1970 and is the author of twenty-threeExpand
The Union of Vieta's and Wallis's Products for Pi
While (1) and (2) seem unrelated, they are both special cases of a more general double product (3). The first product in (3) consists of the first p factors of Vieta's original infinite product (1).Expand
A Further Extension of the Leibniz Rule to Fractional Derivatives and Its Relation to Parseval’s Formula
The familiar Leibniz rule for the Nth derivative of the product of two functions is $D^N uv = \sum {\left( {\begin{array}{*{20}c} N \\ n \\ \end{array} } \right)D^{N - n} uD^n v}$. A generalizationExpand