A 2-adic formula for Bernoulli numbers of the second kind and for the Nörlund numbers
type: Item
instance of: scholarly article
title: A 2-adic formula for Bernoulli numbers of the second kind and for the Nörlund numbers
zbMATH Open document ID: 1182.11014
author: Paul Thomas Young
published in: Journal of Number Theory
publication date: 2009-04-08T00:00:00Z
review text: The Bernoulli numbers of the second kind are defined by means of the following generating function \[ \frac{t}{\log(1+t)}=\sum_{n=0}^{\infty}b_{n}t^n. \] The numbers \(n!b_{n}\) have been called the Cauchy numbers of the first kind. These numbers satisfy the following relation \[ n!b_{n}=\int_0^1 x(x-1)(x-2)\dots (x-n+1)\,dx. \] The first few of these numbers are given by \(b_0=1, b_1=\frac{1}{2}, b_2=-\frac{1}{12}, b_3=\frac{1}{24}, b_4=-\frac{19}{720}, b_5=\frac{3}{160}\). These numbers are related to Euler's constant, \(\gamma\) and \(n\)th harmonic numbers, \(H_{n}\), that is \[ \sum_{n=1}^{\infty}(-1)^{n+1}\frac{b_n}{n}=\gamma \] and \[ 1+\sum_{n=1}^{\infty}(-1)^{n+1}\frac{H_n}{n}=\frac{\pi^2}{6}, \] where \[ H_{n}=\sum_{k=1}^n\frac{1}{k}. \] The Bernoulli numbers of higher-order are defined by means of the following generating function \[ (\frac{t}{e^t-1})^{s}=\sum_{n=0}^{\infty}B_{n}^{(s)}\frac{t^n}{n!}. \] For \(n=s\) the numbers \(B_{n}^{(n)}\) are called the Nörlund numbers or the Cauchy numbers of the second type, may be determined by the generating function \[ \frac{t}{(1+t)\log(1+t)}=\sum_{n=0}^{\infty}B_{n}^{(n)}\frac{t^n}{n!}. \] Relations between the numbers \(b_{n}\) and \(B_{n}^{(n)}\) are given by \[ B_{n}^{(n)}=n!\sum_{k=1}^n(-1)^{n-k}b_{k}, \] and \[ b_{n}=\frac{B_{n}^{(n)}}{n!}+\frac{B_{n-1}^{(n-1)}}{(n-1)!}. \] The author gives a formula expressing the Bernoulli numbers of the second kind as 2-adically convergent sums of traces of algebraic integers. By using this formula, the author proves the formulae and conjectures of Adelberg concerning the initial 2-adic digits of these numbers. He also gives many relations on these numbers and Nörlund numbers or the Cauchy numbers of the second type.
reviewed by: Yilmaz Simsek
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 11B65
zbMATH DE Number: 5541701
zbMATH Keywords: Bernoulli numbers of the second kind
zbMATH Keywords: Cauchy numbers of the first kind
zbMATH Keywords: Nörlund numbers
MaRDI profile type: Publication
full work available at URL: j.jnt.2008.03.002
OpenAlex ID: W2031848354
cites work: 2-adic congruences of Nörlund numbers and of Bernoulli numbers of the second kind
cites work: Q5728600
cites work: Q4769056
cites work: Q4281098
cites work: Q5535148
cites work: Q4138813
cites work: The Cauchy numbers
DOI: 10.1016/J.JNT.2008.03.002
description: scientific article; zbMATH DE number 5541701