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Zbl 0887.11011
Kaneko, Masanobou
Poly-Bernoulli numbers.
(English)
[J] J. Théor. Nombres Bordx. 9, No.1, 221-228 (1997). ISSN 1246-7405

For every integer $k$, the poly-Bernoulli number $B_n^{(k)}$, $n=0,1,2$, is defined by $${1\over z} \text {Li}_k(z) \mid_{z=1 -e^{-x}} =\sum^\infty_{n=0} B_n^{(k)} {x^n\over n!},$$ where $\text {Li}_k(z)$ denotes the formal power series $\sum^\infty_{m=1} z^m/m^k$. When $k=1$, $B^{(1)}_n$ is the usual Bernoulli number. In the note under review the author gives an explicit formula for $B_n^{(k)}$ using the Stirling numbers of the second kind and shows the nice symmetric expression $$B_n^{(-k)} =B_k^{(-n)}.$$ As an application, he proves a von Staudt-type theorem in case of $k=2$ and a theorem of Vandiver on congruences for $B_n^{(1)}$.
[Helmut Müller (Hamburg)]
MSC 2000:
*11B68 Bernoulli numbers, etc.
11B73 Bell and Stirling numbers
11A07 Congruences, etc.

Keywords: poly-Bernoulli numbers; Stirling numbers of the second kind; von Staudt-type theorem; theorem of Vandiver; congruences

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