Generalized Stirling Functions of Second Kind and Representations of Fractional Order Differences via Derivatives

Abstract

The paper is devoted to the study of generalized Stirling functions of second kind by using difference and differentiation operators of fractional order. The constructions under considerations give extensions of the classical Stirling numbers of second kind S(n, k) to functions S(n, β), whereby the second parameter k becomes any complex β, as well as to functions S(α, β), whereby also the first parameter n becomes any complex α. The main properties of these new functions, including recurrence relations are established. A chief application is a generalization of a basic formula of combinatorial and numerical analysis, namely expressing higher order differences in terms of derivatives, to the fractional instance: essentially a fractional order difference Δ β f(x) leads to S(n, β) in the above sum. Three concrete examples are presented.