Kaneko, Masanobou Poly-Bernoulli numbers. (English) Zbl 0887.11011 J. Théor. Nombres Bordx. 9, No. 1, 221-228 (1997). 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 \(B_n^{(1)}\). Reviewer: Helmut Müller (Hamburg) Cited in 9 ReviewsCited in 107 Documents MSC: 11B68 Bernoulli and Euler numbers and polynomials 11B73 Bell and Stirling numbers 11A07 Congruences; primitive roots; residue systems Keywords:poly-Bernoulli numbers; Stirling numbers of the second kind; von Staudt-type theorem; theorem of Vandiver; congruences PDFBibTeX XMLCite \textit{M. Kaneko}, J. Théor. Nombres Bordx. 9, No. 1, 221--228 (1997; Zbl 0887.11011) Full Text: DOI Numdam EuDML EMIS Digital Library of Mathematical Functions: §24.16(iii) Other Generalizations ‣ §24.16 Generalizations ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.