Bernoulli number
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 is an odd function, one can see that for . Numerically,
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 th powers of the first 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 |