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Poly-Bernoulli numbers. (English) Zbl 0887.11011
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 $B_n^{(1)}$.

MSC:
 11B68 Bernoulli and Euler numbers and polynomials 11B73 Bell and Stirling numbers 11A07 Congruences; primitive roots; residue systems
Full Text:
References:
 [1] Gould , H.W. : Explicit formulas for Bernoulli numbers , Amer. Math. Monthly 79 ( 1972 ), 44 - 51 . MR 306102 | Zbl 0227.10010 · Zbl 0227.10010 · doi:10.2307/2978125 [2] Ireland , K. and Rosen , M. : A Classical Introduction to Modern Number Theory , second edition. Springer GTM 84 ( 1990 ) MR 1070716 | Zbl 0712.11001 · Zbl 0712.11001 [3] Jordan , Charles: Calculus of Finite Differences , Chelsea Publ. Co. , New York , ( 1950 ) MR 183987 | Zbl 0041.05401 · Zbl 0041.05401 [4] Vandiver , H.S. : On developments in an arithmetic theory of the Bernoulli and allied numbers , Scripta Math. 25 ( 1961 ), 273 - 303 MR 142497 | Zbl 0100.26901 · Zbl 0100.26901