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Properties and applications of the reciprocal logarithm numbers. (English) Zbl 1208.11032
The author investigates the sequence of rational numbers \((A_k)_{k\geq 0}\), which he calls the “reciprocal logarithm numbers” defined by the series expansion \(1/\ln(1+z) = \sum_{k=0}^\infty A_k z^{k-1}\). He derives several properties of these numbers such as recursions, integral representations and identities. More specifically, he points out the relationship of the \(A_k\)’s to Stirling numbers of the first kind and to Euler’s constant. A similar approach is then presented for the series expansion of \(1/\arctan x\).

MSC:
11B73 Bell and Stirling numbers
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
05A17 Combinatorial aspects of partitions of integers
05C90 Applications of graph theory
Software:
Mathematica; OEIS
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References:
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