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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
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References:

[1] Gould, H.W.: Explicit formulas for Bernoulli numbers, Amer. Math. Monthly79 (1972), 44-51. · Zbl 0227.10010
[2] Ireland, K. and Rosen, M.: A Classical Introduction to Modern Number Theory, second edition. Springer GTM84 (1990) · Zbl 0712.11001
[3] Jordan, Charles:Calculus of Finite Differences, Chelsea Publ. Co., New York, (1950) · 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 · Zbl 0100.26901
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