Congruences concerning Bernoulli numbers and Bernoulli polynomials
type: Item
instance of: scholarly article
title: Congruences concerning Bernoulli numbers and Bernoulli polynomials
zbMATH Open document ID: 0990.11008
DOI: 10.1016/S0166-218X(00)00184-0
published in: Discrete Applied Mathematics
publication date: 2002-08-17T00:00:00Z
review text: Let \(B_n(x)\), resp. \(B_n\), denote the classical Bernoulli polynomial, resp. number. In the paper under review the author proves some generalizations of Kummer's congruence by determining \[ \frac{B_{k(p-1)+b}(x)}{(k(p-1)+b)}\pmod{p^n} \] where \(p\) is an odd prime, \(x\) a \(p\)-integral rational number and \(p-1\nmid b\), while Kummer considered only the case \(n=1\). As an application he proves explicit formulae for \[ \sum_{x=1}^{p-1}\frac 1{x^k}\pmod{p^3}, \] and, for example, \[ (p-1)!\equiv \frac{pB_{2p-2}}{2p-2}-\frac{pB_{p-1}}{p-1}- \frac 12 \left(\frac{pB_{p-1}}{p-1}\right)^2\pmod{p^3},\;p>3. \] In a similar way he considers also generalized Bernoulli numbers \(B_{n,\chi}\), where \(\chi\) denotes a Dirichlet character.
reviewed by: Helmut Müller
Mathematics Subject Classification ID: 11B68
zbMATH DE Number: 1512456
zbMATH Keywords: Bernoulli numbers
zbMATH Keywords: Bernoulli polynomials
author: Zhihong Sun
MaRDI profile type: Publication
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full work available at URL: s0166-218x(00)00184-0
OpenAlex ID: W2166445918
description: scientific article; zbMATH DE number 1512456