Applications of the formula of Faà di Bruno: combinatorial identities and monotonic functions (Q2044611)

Lah numbers are defined for nonnegative integers \(n\) and \(k\) by \[ L(0,0)=1, \quad L(n,0)=L(0,k)=0, \quad L(n,k) = \frac{(n-1)!}{(k-1)!} \binom{n}{k}, \] for \(n, k \in \mathbb{N}\). These numbers satisfy the recurrence relation \[ L(n+1,k) = (n+k) L(n,k) + L(n,k-1). \] The authors present some identities involving Lah numbers and Stirling numbers of the first kind, Lah numbers and harmonic numbers, and harmonic numbers. In addition, the authors show that for all \(s>0\), \(t \in [-1,1]\), and \(\lambda \in \mathbb{R}\), the function \(F_{\lambda,s,t}\), defined by \[ F_{\lambda,s,t} (x) = \frac{x^\lambda}{(x-1)^{2s} (x^2-2tx+1)^s}, \] satisfies the condition \((-1)^n {F_{\lambda,s,t}}^{(n)} (x) \geq 0\), for all \(n \in \mathbb{N}_0\) and \(x \in (1,\infty)\), that is, \(F_{\lambda,s,t}\) is completely monotonic on \((1,\infty)\), if and only if \(\lambda \leq 4s\).