# Bernoulli number

Let $B_{r}$ be the $r$th Bernoulli polynomial. Then the $r$th 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:

 $\sum_{r=0}^{\infty}B_{r}\frac{y^{r}}{r!}=\frac{y}{e^{y}-1}$

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:

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

Since $\operatorname{tanh}$ is an odd function, one can see that $B_{2r+1}=0$ for $r\geq 1$. Numerically, $B_{0}=1,B_{1}=-\frac{1}{2},B_{2}=\frac{1}{6},B_{4}=-\frac{1}{30},\cdots$

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