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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

Differences of fractional order

Authors: J. B. Díaz and T. J. Osler
Journal: Math. Comp. 28 (1974), 185-202
MSC: Primary 39A05; Secondary 26A33
MathSciNet review: 0346352
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Abstract: Derivatives of fractional order, $ {D^\alpha }f$, have been considered extensively in the literature. However, little attention seems to have been given to finite differences of fractional order, $ {\Delta ^\alpha }f$. In this paper, a definition of differences of arbitrary order is presented, and $ {\Delta ^\alpha }f$ is computed for several specific functions f (Table 2.1). We find that the operator $ {\Delta ^\alpha }$ 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 generate series expansions involving the special functions.

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Additional Information

PII: S 0025-5718(1974)0346352-5
Keywords: Finite difference, fractional differences, fractional derivatives, Leibniz rule, special functions, Mellin-Barnes contour integrals
Article copyright: © Copyright 1974 American Mathematical Society