没有公式的伯努利双纽线

Faulhaber and Bernoulli

In this note we will use Faulhaber's Formula to explain why the odd Bernoulli numbers are equal to zero.

简明的方法总结了无穷系列的公式微分研究

Methodus succincta summas serierum infinitarum per formulas differentiales investigandi

本文给出了关于Bernoulli数和黎曼ζ函数的更正式的级数。他给出了伯努利数的生成函数。这篇论文是从拉丁原文翻译成德语的。

费尔马公式、伯努利数、自然数幂和与互质数及函数方程 $f(x)+x^k=f(x+1)$

Faulhaber's formula, Bernoulli numbers, power sums of natural numbers and totatives and the functional equation $f(x)+x^k=f(x+1)$

本文研究了伯努利数与自然数幂和的关系，并通过函数方程 $f(x)+x^k=f(x+1)$ 探讨互质数幂和的表达式。

A Proof of Symmetry of the Power Sum Polynomials using a Novel Bernoulli Numbers Identity

The problem of finding formulas for sums of powers of natural numbers has been of interest to mathematicians for many centuries. Among these is Faulhaber's well-known formula expressing the power sums as polynomials whose coefficients involve Bernoulli numbers. In this paper we give an elementary proof that the sum of $p$-th powers of the first $n$ natural numbers can be expressed as a polynomial in $n$ of degree $p+1$. We also prove a novel identity involving Bernoulli numbers and use it to show symmetry of this polynomial.

Definitions, notations and proofs for Bernoulli numbers

This is a collection of definitions, notations and proofs for the Bernoulli numbers $B_n$ appearing in formulas for the sum of integer powers, some of which can be found scattered in the large related historical literature in French, English and German. We provide elementary proofs for the original convention with ${\mathcal B}_1=1/2$ and also for the current convention with $B_1=-1/2$, using only the binomial theorem and the concise Blissard symbolic (umbral) notation.