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

Todorov, Pavel G.

C. R. Acad. Sci., Paris, Sér. I 301, 665-666 (1985).

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

Keywords:

generalized Bernoulli numbers; Stirling numbers