Differences of fractional order

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.