Bernoulli number

Let Br be the rth Bernoulli polynomialDlmfDlmfPlanetmathPlanetmath. Then the rth Bernoulli numberMathworldPlanetmathPlanetmath is


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


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:


Since tanh is an odd functionMathworldPlanetmath, one can see that B2r+1=0 for r1. Numerically, B0=1,B1=-12,B2=16,B4=-130,

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 rth powers of the first n positive integers ( 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