# 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:

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

Since $\mathrm{tanh}$ 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 |