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Identities for Bernoulli polynomials and Bernoulli numbers

Archiv der Mathematik, 2014
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Alzer, Horst, Kwong, Man Kam
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Bernoulli Numbers

The Fibonacci Quarterly, 1968
This paper is of an expository nature and is concerned mainly with the arithmetic properties of the Bernoulli numbers. Following an introductory section which reviews the basic formulas for the Bernoulli and Euler numbers and polynomials, the following topics are discussed: the Staudt-Clausen theorem, Kummer's congruences and some related properties ...
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Bernoulli Polynomials and Bernoulli Numbers

2002
In this chapter, we introduce a sequence of polynomials that is closely related to the h-antiderivative of polynomials and has many important applications.
Victor Kac, Pokman Cheung
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Generalized Bernoulli-Hurwitz numbers and the universal Bernoulli numbers

Russian Mathematical Surveys, 2011
The three fundamental properties of the Bernoulli numbers, namely, the von Staudt-Clausen theorem, von Staudt's second theorem, and Kummer's original congruence, are generalized to new numbers that we call generalized Bernoulli-Hurwitz numbers. These are coefficients in the power series expansion of a higher-genus algebraic function with respect to a ...
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The Integrality of the Values of Bernoulli Polynomials and of Generalised Bernoulli Numbers

Bulletin of the London Mathematical Society, 1997
\textit{G. Almkvist} and \textit{A. Meurman} [C. R. Math. Acad. Sci., Soc. R. Can. 13, 104-108 (1991; Zbl 0731.11014)] proved a result on the values of the Bernoulli polynomials at rational values of the argument. Subsequently \textit{B. Sury} [Bull. Lond. Math. Soc. 25, 327-329 (1993; Zbl 0807.11014)] and \textit{K. Bartz} and \textit{J. Rutkowski} [C.
Clarke, Francis, Slavutskii, I. Sh.
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Bernoulli Polynomials and Bernoulli Numbers

1973
The summing of the first n natural numbers, or Squares, or cubes, is a rather elementary problem in number theory and leads to the well known formulae $$\eqalign{ & \sum\limits_{n = 1}^N n \, = \,{{N(N + 1)} \over 2}, \cr & \sum\limits_{n = 1}^N {{n^2}} \, = \,{{N(N + 1)(2N + 1)} \over 6}, \cr & \sum\limits_{n = 1}^N {{n^3}} \, = \,{{{N^2}{{(N + 1)}
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Reciprocity Relations for Bernoulli Numbers

The American Mathematical Monthly, 2008
(2008). Reciprocity Relations for Bernoulli Numbers. The American Mathematical Monthly: Vol. 115, No. 3, pp. 237-244.
Takashi Agoh, Karl Dilcher
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Some Congruences for the Bernoulli Numbers

American Journal of Mathematics, 1953
Kongruenzrelationen Bernoullischer Zahlen sind seit Kummer mehrfach aufgestellt und untersucht worden. Nach Vandiver gilt z. B.: \[ B^{a(p -1)} (B^{p-1} - 1)^r\equiv 0(p^{r-1}),\quad (a>0,\;r>0,\;a+r r\) folgt \[ B^c (B^b-1)^r \equiv 0 \pmod {p^{r-1}, p^{r-3}, p^{r-h}}, \] je nachdem \(r < p -1\), \(r = p-1\), \(r\geq p\) ist, wobei \((r+1)/p\leq h ...
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q-Bernoulli and Eulerian Numbers

Transactions of the American Mathematical Society, 1954
In einer früheren Arbeit [Duke Math. J. 15, 987--1000 (1948; Zbl 0032.00304)] definierte der Verf. die rationalen Funktionen \(\eta_m\) der Unbestimmten \(q\) durch die symbolischen Formeln (in welchen nach der Entwicklung \(\eta^m\) durch \(\eta_m\) ersetzt wird) \((q\eta+1)^m=\eta^m\) \((m>1)\), \(\eta_0=1\), \(\eta_1=0\) und die polynome \(\eta_m(x)\
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Faster computation of Bernoulli numbers

Journal of Algorithms, 1992
The author presents an algorithm, based on the classical formula \[ B_{2k}=(-1)^{k+1}2(2k!)\zeta(2k)(2\pi)^{-2k}, \] to compute the \(2k\)th Bernoulli number \(B_{2k}\), defined by \(X/(e^ X- 1)=\sum_{n\geq 1}B_ n X^ n/n!\). The space requirement of this algorithm is \({\mathcal O}(n\log\log n)\) bits and it involves \({\mathcal O}(n^ 2\log^ 2n\log\log
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