Congruences of \(p\)-adic integer order Bernoulli numbers
type: Item
instance of: scholarly article
title: Congruences of \(p\)-adic integer order Bernoulli numbers
zbMATH Open document ID: 0866.11013
author: Arnold Adelberg
published in: Journal of Number Theory
publication date: 1997-07-20T00:00:00Z
review text: 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. 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. In the Erratum a correct statement of Theorem 1 (iii) is given as well as two minor misprint corrections.
reviewed by: József Sándor
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 11S80
zbMATH DE Number: 935011
zbMATH Keywords: \(p\)-adic analysis
zbMATH Keywords: Bernoulli polynomials of higher order
zbMATH Keywords: irreducibility theorems
zbMATH Keywords: \(p\)-adic integer order Bernoulli numbers
zbMATH Keywords: congruences
zbMATH Keywords: Stirling numbers
MaRDI profile type: Publication
full work available at URL: jnth.1996.0103
OpenAlex ID: W2088486782
DOI: 10.1006/JNTH.1996.0103
description: scientific article; zbMATH DE number 935011