Differences of fractional order

Derivatives of fractional order, <!-- MATH ${D^\alpha }f$ --> , have been considered extensively in the literature. However, little attention seems to have been given to finite differences of fractional order, <!-- MATH ${\Delta ^\alpha }f$ --> . In this paper, a definition of differences of arbitrary order is presented, and <!-- MATH ${\Delta ^\alpha }f$ --> is computed for several specific functions f (Table 2.1). We find that the operator <!-- MATH ${\Delta ^\alpha }$ --> 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.