Applications of the formula of Faà di Bruno: combinatorial identities and monotonic functions
type: Item
instance of: scholarly article
title: Applications of the formula of Faà di Bruno: combinatorial identities and monotonic functions
zbMATH Open document ID: 1485.05003
author: Horst Alzer
author: Omran Kouba
published in: Results in Mathematics
publication date: 2021-08-10T00:00:00Z
review text: 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 \(\lambda \leq 4s\).
reviewed by: Rui Duarte
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 11B73
Mathematics Subject Classification ID: 26A48
Mathematics Subject Classification ID: 33B15
zbMATH DE Number: 7380201
zbMATH Keywords: formula of Faà di Bruno
zbMATH Keywords: combinatorial identities
zbMATH Keywords: Lah numbers
zbMATH Keywords: Stirling numbers
zbMATH Keywords: harmonic numbers
zbMATH Keywords: infinite series
zbMATH Keywords: absolutely monotonic functions
zbMATH Keywords: completely monotonic functions
MaRDI profile type: Publication
full work available at URL: s00025-021-01448-9
OpenAlex ID: W3185789754
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DOI: 10.1007/S00025-021-01448-9
describes a project that uses: Digital Library of Mathematical Functions
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description: scientific article; zbMATH DE number 7380201