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On Stirling numbers and Euler sums. (English) Zbl 0877.39001
In this paper, rich in content, the author first offers a relation between the Stirling numbers of the first kind, defined by \(S(0,0)=1, S(n,0)=0, (n>0)\), \[ S(n,k)=(n-1)S(n-1,k)+S(n-1,k-1), (n>1) \] and the “\(r\)-th order harmonic numbers” \(\sum_{k=1}^{n}k^{-r}\). The proof consists of verification for \(k=1,2,3,4\), followed by “it follows then that the general formula ... is ...”.
Further offerings are connections to zeta and polygamma functions, to Euler sums, and to values at 1 of hypergeometric functions with all but at most one of the coefficient pairs equal.
[Close acquaintance with several special functions and their properties is presupposed with or without special mention].

11B73 Bell and Stirling numbers
11M41 Other Dirichlet series and zeta functions
33B15 Gamma, beta and polygamma functions
Full Text: DOI
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