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A recurrence formula for the Bernoulli numbers. (English) Zbl 0854.11012
Let \(B_n\) be the \(n\)-th Bernoulli number defined by the formal power series \[ {x\over {e^x -1}}= \sum^\infty_{n=0} B_n {x^n \over n!} \] and put \(\widetilde {B}_n:= (n+ 1)B_n\). In this note under review the author gives two proofs of the formula \[ \widetilde {B}_{2n}= -{1\over {n+1}} \sum^{n-1}_{i=0} {{n+1} \choose i} \widetilde {B}_{n+i}, \] one which uses convergents of continued fraction expansions of \(f(x)= (\sqrt {x}/ 2) \coth (\sqrt {x}/ 2)\) with a lot of recurrence relations, and a very short and elegant one (which is due to D. Zagier) defining a suitable convolution on the set of sequences.

MSC:
11B68 Bernoulli and Euler numbers and polynomials
11B37 Recurrences
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[1] Heine, E.: Ueber die Zaehler und Nenner der Naeherungswerthe von Kettenbruechen. Jour, fuer die reine und angew. Math., 57 231-247 (1860). · ERAM 057.1517cj
[2] Ireland, K. and Rosen, M.: A Classical Introduction to Modern Number Theory. 2nd ed., Springer, GTM 84 (1990). · Zbl 0712.11001
[3] Perron, O.: Die Lehre von den Kettenbruechen. Teubner (1929). B* (n > 1). · JFM 55.0262.09
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