Bernoulli number

Let $B_r$ be the $r$th Bernoulli polynomial. Then the $r$th Bernoulli number is

$$B_r:=B_r(0).$$

This means, in particular, that the Bernoulli numbers are given by an exponential generating function in the following way:

$$\sum _{r=0}^{\mathrm{\infty }}{B}_r\frac{y^r}{r!}=\frac{y}{e^y-1}$$

and, in fact, the Bernoulli numbers are usually defined as the coefficients that appear in such expansion.

Observe that this generating function can be rewritten:

$$\frac{y}{e^y-1}=\frac{y}{2}\frac{e^y+1}{e^y-1}-\frac{y}{2}=(y/2)(\tanh (y/2)-1).$$

Since $\tanh$ is an odd function, one can see that $B_{2r+1}=0$ for $r\ge 1$. Numerically, $B_0=1,B_1=-\frac{1}{2},B_2=\frac{1}{6},B_4=-\frac{1}{30},\mathrm{\cdots }$

These combinatorial numbers occur in a number of contexts; the most elementary is perhaps that they occur in the formulas for the sum of the $r$th powers of the first $n$ positive integers (http://planetmath.org/SumOfKthPowersOfTheFirstNPositiveIntegers). They also occur in the Maclaurin expansion for the tangent function and in the Euler-Maclaurin summation formula.

Title	Bernoulli number
Canonical name	BernoulliNumber
Date of creation	2013-03-22 11:45:58
Last modified on	2013-03-22 11:45:58
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	14
Author	alozano (2414)
Entry type	Definition
Classification	msc 11B68
Classification	msc 49J24
Classification	msc 49J22
Classification	msc 49J20
Classification	msc 49J15
Related topic	GeneralizedBernoulliNumber
Related topic	BernoulliPolynomials
Related topic	SumOfKthPowersOfTheFirstNPositiveIntegers
Related topic	EulerMaclaurinSummationFormula
Related topic	ValuesOfTheRiemannZetaFunctionInTermsOfBernoulliNumbers
Related topic	TaylorSeriesViaDivision
Related topic	BernoulliPolynomialsAndNumbers
Related topic	EulerNumbers2