Probabilistic proofs of relations with Stirling numbers of the first kind by Dirichlet process (Q922581)

Let s(n,k) be the Stirling numbers of the first kind. The author gives a proof, based on properties of Dirichlet processes, for the identities \(\sum^{n}_{k=1}| s(n,k)| x^ k=x(x+1)...(x+n-1)\) and \(\sum^{n}_{k=1}s(n,k)x^ k=x(x-1)...(x-n+1).\) While it is customary to use the latter equation as the definition of s(n,k), the author defines s(n,k) by its explicit formula, i.e., as \((-1)^{n+k}\) times the sum of \(n!/k_ 1!k_ 2!... k_ n! 2^{k_ 2}... n^{k_ n}\) over all integers \(k_ j\geq 0\) with \(k_ 1+2k_ 2+...+nk_ n=n\) and \(k_ 1+k_ 2+...+k_ n=k\), and then proves the mentioned identities, emphasizing the probabilistic nature of the proof.