Stirling’s series and Bernoulli numbers. (English) Zbl 0743.11012

For \(n=2,3,\ldots\) the following infinite system of recurrences for the Bernoulli numbers \(B_ m\) is shown: \[ B_ m={1\over n(1-n^ m)}\sum^{m-1}_{k=0}n^ k{m\choose k}B_ k\sum^{n- 1}_{j=1}j^{m-k}. \] The proof follows by direct computations from the usual generating function of the Bernoulli numbers.
Reviewer: R.F.Tichy (Graz)


11B68 Bernoulli and Euler numbers and polynomials
05A15 Exact enumeration problems, generating functions
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