Some identities and an explicit formula for Bernoulli and Stirling numbers. (English) Zbl 1291.11051

Summary: The authors establish eight identities which reveal that the functions \(\frac{1}{(1 - e^{{\pm}t})^k}\) and the derivatives \(\left(\frac{1}{e^{{\pm}t} - 1}\right)^{(i)}\) can be expressed by each other by linear combinations with coefficients involving the combinatorial numbers and Stirling numbers of the second kind, find an explicit formula for Bernoulli numbers in terms of Stirling numbers of the second kind, and present two identities for Stirling numbers of the second kind.


11B68 Bernoulli and Euler numbers and polynomials
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
33B10 Exponential and trigonometric functions
39B22 Functional equations for real functions
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