New properties of \(r\)-Stirling series
type: Item
instance of: scholarly article
title: New properties of \(r\)-Stirling series
zbMATH Open document ID: 1174.11026
published in: Acta Mathematica Hungarica
publication date: 2008-10-21T00:00:00Z
review text: The Stirling numbers of the first kind denoted \([{{n}\over {m}}]\) count the number of permutations of \(n\) elements with \(m\) cycles, while the Stirling numbers of the second kind \(\{{{n}\over{m}}\}\) count the number of partitions of a set with \(n\) elements into \(m\) nonempty disjoint subsets. For a positive integer \(r\), let \([{{n}\over {m}}]_r\) be the \(r\)-Stirling numbers of the first kind, which count the number of permutations of the set with \(n\) elements in \(m\) disjoint cycles such that the first \(r\) elements are in disjoint cycles, and similarly \(\{{{n}\over {m}}\}_r\) be the \(r\)-Stirling numbers of the second kind which count the number of partitions of a set with \(n\) elements into \(m\) disjoint subsets such that the first \(r\) elements are in disjoint distinct subsets. Putting \[ G_{k,m,q}^{1,r}(z)=\sum_{n=1}^{\infty}\left[{{n+k+r}\over {m+r}}\right]_r{{z^n}\over {n^q n!}} \] and \(G_{k,m,q}^{2,r}(z)\) for the analogous sum where the \(r\)-Stirling number of the first kind are replaced by the \(r\)-Stirling numbers of the second kind, in this paper, the author determines recurrences for the above numbers. For example, \[ G_{0,m,q}^{1,r}(z)={{1}\over {r-1}}\left(G_{0,m,q-1}^{1,r-1}+(r-1)G_{0,m,q}^{1,r-1}-G_{0,m-1,q}^{1,r}\right)(z) \] and for \(k\geq 1\), \[ G_{k,m,q}^{1,r}(z)=G_{k-1,m,q-1}^{1,r}(z)+(k+r-1)G_{k-1,m,q}^{1,r}(z)+G_{k-1,m-1,q}^{1,r}(z) \] (Theorem 1). Similar recurrences are given for \(G_{k,m,q}^{2,r}\) (Theorems 3 and 4). The last section of the paper deals with the asymptotic behaviors of the \(r\)-Stirling numbers of the first and second kind.
reviewed by: Florian Luca
Mathematics Subject Classification ID: 11B73
Mathematics Subject Classification ID: 33B30
zbMATH DE Number: 5355137
zbMATH Keywords: Stirling numbers
zbMATH Keywords: \(r\)-Stirling numbers
zbMATH Keywords: hyperharmonic numbers
zbMATH Keywords: Nielsen's generalized polylogarithms
zbMATH Keywords: polylogarithms
zbMATH Keywords: hypergeometric series
zbMATH Keywords: Lerch transcendent
author: István Mező
MaRDI profile type: Publication
full work available at URL: s10474-007-7047-9
OpenAlex ID: W2139048024
cites work: On Stirling numbers and Euler sums
cites work: Q4831463
cites work: On an Intriguing Integral and Some Series Related to ζ(4)
cites work: The \(r\)-Stirling numbers
cites work: Q4719273
cites work: On asymptotics, Stirling numbers, gamma function and polylogs
cites work: Q3682546
cites work: Automorphism-invariant representations of real reductive groups
cites work: Asymptotic Development of the Stirling Numbers of the First Kind
cites work: Asymptotic Estimates of Stirling Numbers
cites work: The asymptotic behavior of the Stirling numbers of the first kind
DOI: 10.1007/S10474-007-7047-9
Recommended article: Q5207927
Recommended article: A new Stirling series as continued fraction
Recommended article: Q3397168
Recommended article: The \(r\)-Stirling numbers
Recommended article: New series with Cauchy and Stirling numbers, Part 2
Recommended article: Q3390102
Recommended article: Stirling's Series Revisited
Recommended article: Q4954987
Recommended article: New series identities with Cauchy, Stirling, and harmonic numbers, and Laguerre polynomials
Recommended article: Dirichlet series and series with Stirling numbers
description: scientific article; zbMATH DE number 5355137