A simple explicit formula for the generalized Bernoulli numbers. (Une formule simple explicite des nombres de Bernoulli généralisés.)(French)Zbl 0606.10008

Generalizing a standard result for ordinary Bernoulli numbers, the author shows that the generalized Bernoulli numbers $$B_n^{(r)}$$ (whose exponential generating function is $$t^r/(e^t-1)^r)$$ are given by $B_n^{(r)}=\sum^n_{\nu =0}(-1)^{\nu} \left( \begin{matrix} r+n\\ n-\nu \end{matrix} \right) \left( \begin{matrix} r+\nu -1\\ \nu \end{matrix} \right) \left( \begin{matrix} n+\nu \\ \nu \end{matrix} \right)^{-1} S(n+\nu,\nu),$ where the $$S(n,k)$$ are Stirling numbers of the second kind.
Reviewer: P. A. B. Pleasants

MSC:

 11B68 Bernoulli and Euler numbers and polynomials 11B73 Bell and Stirling numbers 05A10 Factorials, binomial coefficients, combinatorial functions 05A19 Combinatorial identities, bijective combinatorics