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Zbl 1182.11014
Young, Paul Thomas
A 2-adic formula for Bernoulli numbers of the second kind and for the Nörlund numbers.
(English)
[J] J. Number Theory 128, No. 11, 2951-2962 (2008). ISSN 0022-314X; ISSN 1096-1658/e

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.
[Yilmaz Simsek (Antalya)]
MSC 2000:
*11B68 Bernoulli numbers, etc.
11B65 Binomial coefficients, etc.

Keywords: Bernoulli numbers of the second kind; Cauchy numbers of the first kind; Nörlund numbers

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