Factorial functions and stirling numbers of fractional orders

Abstract

The aim of this paper is to study the binomial coefficients ( xn ), the factorial polynomials [x]n and [x]n, the Stirling numbers of first and second kind, namely s(n,k) and S(n,k), in the case that n ∈ ℕ is replaced by real α ∈ ℝ. In the course of the paper, the Vandermonde convolution formula is presented in an infinite series frame, the binomial coefficient function ( xa ), α ∈ ℝ, is sampled in terms of the binomial coefficients ( xk ) for k ∈ ℕo, Bell numbers of fractional orders are introduced. Emphasis is placed on the fractional order Stirling numbers s(α,k) and S(α,k), first studied here. Some applications of the S(α,k) are given.