Bernoulli numberBernoulliNumber2001-10-15 20:44:232007-12-20 08:14:33Definitionnumber theory\usepackage{amssymb}
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%\usepackage{xypic}Let $B_r$ be the $r$th Bernoulli polynomial. Then the $r$th {\bf 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:
\[
\sum_{r=0}^{\infty} B_r \frac{y^r}{r!} = \frac{y}{e^y-1}
\]
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 \PMlinkname{sum of the $r$th powers of the first $n$ positive integers}{SumOfKthPowersOfTheFirstNPositiveIntegers}. They also occur in the Maclaurin expansion for the tangent function and in the Euler-Maclaurin summation formula.