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

Deeba, Elias Y.; Rodriguez, Dennis M.

Am. Math. Mon. 98, No. 5, 423-426 (1991).

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)

MSC:

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

Keywords:

infinite system of recurrences; Bernoulli numbers; generating function