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Zbl 0866.11013
Adelberg, Arnold
Congruences of $p$-adic integer order Bernoulli numbers. (English)
[J] J. Number Theory 59, No.2, 374-388 (1996); erratum 65, No.1, 179 (1997). ISSN 0022-314X; ISSN 1096-1658/e

The $p$-adic integer order Bernoulli numbers $B_n^{(p)}$ of the title are defined by $$\biggl({t\over{e^t-1}}\biggr)^p\cdot e^{xt}=\sum^\infty_{n=0} B_n^{(p)}(x)\cdot {t^n\over n!},$$ with $B_n^{(p)}= B_n^{(p)}(0)$, where $p$ is an arbitrary $p$-adic integer. In this very well-written paper the author proves some new congruences for $B_n^{(p)}$. These generalize e.g. the classical Kummer congruences for ordinary Bernoulli numbers. Several results by F. T. Howard or L. Carlitz are also extended. \par The congruences (which are too complicated to be stated here) have ramifications for the Stirling numbers of the first and second kind. These are consequences of irreducibility theorems on certain Bernoulli polynomials of order divisible by $p$, as an application of the very general congruence properties obtained by the author for $B_n^{(p)}$ and related numbers.\par In the Erratum a correct statement of Theorem 1 (iii) is given as well as two minor misprint corrections.
[József Sándor (Jud.Harghita)]

MSC 2000:: *11B68 Bernoulli numbers, etc.
11S80 Other analytic theory of local fields

Keywords: $p$-adic analysis; Bernoulli polynomials of higher order; irreducibility theorems; $p$-adic integer order Bernoulli numbers; congruences; Stirling numbers

Cited in: Zbl 1015.11006

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