A recurrence formula for the Bernoulli numbers
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.
convergents of continued fraction expansions
scientific article; zbMATH DE number 897687