A 2-adic formula for Bernoulli numbers of the second kind and for the Nörlund numbers
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.
scientific article; zbMATH DE number 5541701