zbMATH — the first resource for mathematics

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.

11B68 Bernoulli and Euler numbers and polynomials
11B37 Recurrences
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.