# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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.

##### MSC:
 11B68 Bernoulli and Euler numbers and polynomials 11B65 Binomial coefficients, etc.
Full Text:
##### References:
 [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)