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Convolution identities for Stirling numbers of the first kind. (English) Zbl 1219.11035

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.

MSC:

11B73 Bell and Stirling numbers
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