Results 161 to 170 of about 146,038 (187)

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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Convolutions of Bernoulli and Euler Polynomials

Sarajevo Journal of Mathematics
By means of the generating function technique, several convolution identities are derived for the polynomials of Bernoulli and Euler.   2000 Mathematics Subject Classification.
CHU, Wenchang, ZHOU R. R.
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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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Convolution and Reciprocity Formulas for Bernoulli Polynomials

Integers, 2011
AbstractWe prove a new convolution identity for sums of products of two Bernoulli polynomials. This can be rewritten to obtain a reciprocity relation for a related sum. The proof uses some results on Stirling numbers of both kinds which are of independent interest. In particular, a class of polynomials related to the Stirling numbers of the second kind
Takashi Agoh, Karl Dilcher
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A New Formula for the Bernoulli Polynomials

Results in Mathematics, 2010
The author defines \(r\)-Whitney numbers \(w_{m,r}(n,k)\) and \(W_{m,r}(n,k)\) of the first and second kind by the equalities \[ m^nx^{\underline{n}}=\sum_{k=0}^nw_{m,r}(n,k)(mx+r)^k \] and \[ (mx+r)^n=\sum_{k=0}^nm^kW_{m,r}(n,k)x^{\underline{k}} \] with \(x^{\underline{n}}=x(x-1)\cdots(x-n+1)\) denoting falling factorials. These numbers are also given
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Probabilistic Bernoulli and Euler Polynomials

Russian Journal of Mathematical Physics
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Kim, T., Kim, D. S.
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Generalization of Bernoulli polynomials

International Journal of Mathematical Education in Science and Technology, 2002
The Bernoulli polynomials are generalized and some properties of the resulting generalizations are presented.
Qi, Feng, Guo, Bai-Ni
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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 Generalized Bivariate (p,q)-Bernoulli–Fibonacci Polynomials and Generalized Bivariate (p,q)-Bernoulli–Lucas Polynomials

Symmetry, 2023
Waseem Ahmad Khan, Can Kizilateş, Hao Guan   +2 more
exaly  

Remarks on some relationships between the Bernoulli and Euler polynomials

Applied Mathematics Letters, 2004
Àkos Pintér
exaly