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A 2-adic formula for Bernoulli numbers of the second kind and for the Nörlund numbers. (English) Zbl 1182.11014
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.

11B68Bernoulli and Euler numbers and polynomials
11B65Binomial coefficients, etc.
Full Text: DOI
[1] Adelberg, A.: 2-adic congruences of nörlund numbers and of Bernoulli numbers of the second kind, J. number theory 73, 47-58 (1998) · Zbl 0926.11010 · doi:10.1006/jnth.1998.2278
[2] Carlitz, L.: A note on Bernoulli and Euler polynomials of the second kind, Scripta math. 25, 323-330 (1961) · Zbl 0118.06501
[3] Comtet, L.: Advanced combinatorics, (1974) · Zbl 0283.05001
[4] Howard, F. T.: Nörlund’s number $Bn(n)$, , 355-366 (1993) · Zbl 0805.11023
[5] Jordan, C.: Calculus of finite differences, (1965) · Zbl 0154.33901
[6] Koblitz, N.: P-adic numbers, p-adic analysis, and zeta functions, (1977) · Zbl 0364.12015
[7] Merlini, D.; Sprugnoli, R.; Verri, M. C.: The Cauchy numbers, Discrete math. 306, 1906-1920 (2006) · Zbl 1098.05008 · doi:10.1016/j.disc.2006.03.065
[8] Nörlund, N. E.: Differenzenrechnung, (1924)