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Zbl 0844.11019
Howard, F.T.
Applications of a recurrence for the Bernoulli numbers.
(English)
[J] J. Number Theory 52, No.1, 157-172 (1995). ISSN 0022-314X; ISSN 1096-1658/e

The author provides an easy proof of the recurrence $$B_m= {1\over {n(1- n^m)}} \sum^{m-1}_{k =0} n^k {m \choose k} B_k \sum^{n-1}_{j=1} j^{m-k},$$ where $\{B_m\}$ are the Bernoulli numbers. The author uses this formula to present proofs of theorems on Bernoulli numbers due to Staudt-Clausen, Carlitz, Frobenius and Ramanujan. An analogous recurrence for Genocchi numbers is given, which is used to present new proofs of theorems on Genocchi numbers of Lehmer, Ramanujan and Kummer. In some cases, earlier results are extended.
[M.Wyneken (Flint)]
MSC 2000:
*11B68 Bernoulli numbers, etc.

Keywords: recurrence; Bernoulli numbers; Genocchi numbers

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