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Bernoulli Polynomials and Bernoulli Numbers
2002In 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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Commentarii mathematici Universitatis Sancti Pauli = Rikkyo Daigaku sugaku zasshi, 1999
The authors continue the investigations of the second author [J. Théor. Nombres Bordx. 9, 221-228 (1997; Zbl 0887.11011)]. For each integer \(k\) the poly-Bernoulli numbers \(B_n^{(k)}\), \(n=0,1,2\ldots\) are defined by the generating series \[ \frac{\text{ Li}_k(1-e^{-x})}{1-e^{-x}}=\sum_{n=0}^{\infty}B_n^{(k)} \frac{x^n}{n!}, \] where \(\text{ Li}_k(
Tsuneo, Arakawa, Masanobu, Kaneko
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The authors continue the investigations of the second author [J. Théor. Nombres Bordx. 9, 221-228 (1997; Zbl 0887.11011)]. For each integer \(k\) the poly-Bernoulli numbers \(B_n^{(k)}\), \(n=0,1,2\ldots\) are defined by the generating series \[ \frac{\text{ Li}_k(1-e^{-x})}{1-e^{-x}}=\sum_{n=0}^{\infty}B_n^{(k)} \frac{x^n}{n!}, \] where \(\text{ Li}_k(
Tsuneo, Arakawa, Masanobu, Kaneko
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Bernoulli Polynomials and Bernoulli Numbers
1973The 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, 2011The 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, 1954In 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, 1992The 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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Some Congruences for the Bernoulli Numbers
American Journal of Mathematics, 1953Kongruenzrelationen 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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Formulas for Bernoulli Numbers and Polynomials
Results in MathematicsSpecial 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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