Explicit expressions and integral representations for the Stirling numbers. A probabilistic approach

Abstract

We show that the Stirling numbers of the first and second kind can be represented in terms of moments of appropriate random variables. This allows us to obtain, in a unified way, a variety of old and new explicit expressions and integral representations of such numbers.

On the Uniqueness Conjecture for the Maximum Stirling Numbers of the Second Kind

“…To show Theorem 1, we use a probabilistic representation of S(n, k) by means of a multinomial law, which is close to the classical representation in terms of occupancy problems (see Pitman [9] for more details). Indeed, we give in Theorem 2 (the main result in this note) a closed form expression for the difference S(n, k) − S(n, k − 1) in terms of multinomial probabilites and the function g n (x) defined in (1). This allows us to characterize Wegner's conjecture on the one hand and to give a short proof of Theorem 1, on the other.…”

“…Unless otherwise stated, we assume from now on that n, k ∈ N. Let (U j ) j≥1 be a sequence of independent identically distributed random variables having the uniform distribution on [0, 1]. Sun [11] (see also [1]) gave the following probabilistic representation…”