| Version current |
Version 8 |
| Let $B_r$ be the $r$th Bernoulli polynomial. Then the $r$th {\bf Bernoulli number} is |
Let $B_r$ be the $r$th Bernoulli polynomial. Then the $r$th {\bf Bernoulli number} is |
| \[ |
\[ |
| B_r := B_r(0). |
B_r := B_r(0). |
| \] |
\] |
|
|
| This means, in particular, that the Bernoulli numbers are given by an exponential generating function in the following way: |
This means, in particular, that the Bernoulli numbers are given by an exponential generating function in the following way: |
| \[ |
\[ |
|
\sum_{r=0}^{\infty} B_r \frac{y^r}{r!} = \frac{y}{e^y-1}
|
\sum_{i=0}^{\infty} B_r \frac{y^r}{r!} = \frac{y}{e^y-1}
|
| \] |
\] |
| and, in fact, the Bernoulli numbers are usually defined as the coefficients that appear in such expansion. |
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: |
Observe that this generating function can be rewritten: |
| \[ |
\[ |
| \frac{y}{e^y-1} = \frac{y}{2}\frac{e^y+1}{e^y-1} - \frac{y}{2} = (y/2)(\operatorname{tanh}(y/2) -1). |
\frac{y}{e^y-1} = \frac{y}{2}\frac{e^y+1}{e^y-1} - \frac{y}{2} = (y/2)(\operatorname{tanh}(y/2) -1). |
| \] |
\] |
| Since $\operatorname{tanh}$ is an odd function, one can see that $B_{2r+1}=0$ for $r \geq 1$. Numerically, $B_0 = 1, B_1 = -\frac{1}{2}, B_2 = \frac{1}{6}, B_4 = -\frac{1}{30}, \cdots$ |
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. |
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. |
