Agoh, Takashi; Dilcher, Karl Convolution identities for Stirling numbers of the first kind. (English) Zbl 1219.11035 Integers 10, No. 1, 101-109, A9 (2010). The paper derives new convolution identities for the Stirling numbers of the first kind, e.g. for nonnegative integers \(n,k\), and \(m\) \[ \sum_{r=0}^m {m\choose r} {s(n-r,k+m) \over (n-r)!}={1\over n!} \sum_{j=0}^m (-1)^j s(n-m,k+j)s(m+1,m+1-j) \] holds. As an interesting application, the paper obtains a linear recurrence for the Stirling numbers of the first kind: for integers \(n\geq 1\) and \(0\leq m\leq n\), \(0\leq k \leq n-m\) \[ \sum_{j=k+m}^n {j\choose k+m} m^{j-k-m}s(n,j)=n!\sum_{r=0}^m {m\choose r} {s(n-r,k+m) \over (n-r)!} \] holds. Reviewer: László A. Székely (Columbia) Cited in 3 Documents MSC: 11B73 Bell and Stirling numbers Keywords:Stirling numbers; convolution; recurrence relation PDFBibTeX XMLCite \textit{T. Agoh} and \textit{K. Dilcher}, Integers 10, No. 1, 101--109, A9 (2010; Zbl 1219.11035) Full Text: DOI EuDML