Differences of fractional order

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