Shortened recurrence relations for Bernoulli numbers
type: Item
instance of: scholarly article
title: Shortened recurrence relations for Bernoulli numbers
zbMATH Open document ID: 1171.11010
author: Takashi Agoh
author: Karl Dilcher
published in: Discrete Mathematics
publication date: 2009-04-09T00:00:00Z
review text: The authors start with two little known results of {Saalschütz} giving recurrence relations for the Bernoulli numbers \(B_{2n}\). The first of these results actually is contained in the second one which reads as \[ \sum_{k=m}^n (-1)^{k+1} (2^{2k}-2) C(n-m,k-m) B_{2k}=\sum_{j=0}^m (-1)^j\frac{4^{n-j}(n-j)!}{2(n-j)+1} C^{(n)}(m,j) \] with certain numbers \(C(n,m)\) and \(C^{(n)}(m,j)\) which may be given explicitly and where \(0\leq j\leq m\leq n\). The case \(m=0\) gives the first identity. Then the authors express these numbers \(C(n,m)\), \(C^{(n)}(m,j)\) in terms of Stirling numbers of both kinds. The also discuss results by \textit{P. G. Todorov} [J. Math. Anal. Appl. 104, 309--350 (1984; Zbl 0552.10007)] of a similar taste involving Stirling numbers \(s(k,l)\) and \(S(m,j)\) of the first and second kind respectively. Finally, using generating functions for the numbers \(S(n,k)\) they get for \(1\leq m\leq n\) and \(k\geq 0\) the formula \[ \sum_{j=m}^n \binom{n+k}{j-m} S(n-j+k+m,k+m) B_j=\frac{(-1)^m}{k+m} \sum_{j=1}^{m+1} \frac{N(n,m,k,j)}{\binom{k+m-1}{j-1}}, \] where \(N(n,m,k,j)=(n+k) S(m+1,j) S(n+k-1,k+m-j)-m S(m,j)S(n+k,k+m-j)\). From this they also derive some other formulas.
reviewed by: Jens Schwaiger
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 11B73
Mathematics Subject Classification ID: 05A19
zbMATH DE Number: 5542383
zbMATH Keywords: Bernoulli numbers
zbMATH Keywords: Stirling numbers
zbMATH Keywords: recurrence relations
MaRDI profile type: Publication
full work available at URL: j.disc.2008.01.030
OpenAlex ID: W2149824583
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DOI: 10.1016/J.DISC.2008.01.030
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description: scientific article; zbMATH DE number 5542383