Applications of the Formula of Faà di Bruno: Combinatorial Identities and Monotonic Functions.

We show that the formula of Faà di Bruno for the nth derivative of a composite function can be applied to deduce various combinatorial identities involving Lah numbers, Stirling numbers and harmonic numbers. One of our results states that for all integers m, n with m ≥ n ≥ 1 , we have 2 n m n (2 H 2 m - n - H m - n) = ∑ k = 0 n / 2 (- 1) k m k 2 m - 2 k n - 2 k (2 H 2 (m - k) - H m - k) , <graphic href="25_2021_1448_Article_Equ43.gif"></graphic> where H k = 1 + 1 / 2 + ⋯ + 1 / k denotes the kth harmonic number. Moreover, we present connections between absolutely and completely monotonic functions and we provide a three-parameter class of completely monotonic functions. [ABSTRACT FROM AUTHOR]