Results 231 to 240 of about 10,813 (266)

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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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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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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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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Formulas for Bernoulli Numbers and Polynomials

Results in Mathematics
Special polynomials and numbers possess much importance in multifarious areas of sciences such as physics, mathematics, applied sciences, engineering, and other related research fields covering differential equations, number theory, functional analysis, quantum mechanics, mathematical analysis, mathematical physics, and so on.
Ulrich Abel, Horst Alzer
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On the numbers of ward and bernoulli

Proceedings of the Indian Academy of Sciences - Section A, 1936
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On Miki's identity for Bernoulli numbers

Journal of Number Theory, 2005
Ira M Gessel
exaly  

An identity of symmetry for the Bernoulli polynomials

Discrete Mathematics, 2008
Sheng-Liang Yang
exaly