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Bernoulli number∗
alozano†
2013-03-21 12:25:34
Let Br be the rth Bernoulli polynomial. Then the rth Bernoulli number
is
Br := Br (0).
This means, in particular, that the Bernoulli numbers are given by an exponential generating function in the following way:
∞
X
r=0
Br
yr
y
= y
r!
e −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:
y
y ey + 1 y
=
− = (y/2)(tanh(y/2) − 1).
ey − 1
2 ey − 1 2
Since tanh is an odd function, one can see that B2r+1 = 0 for r ≥ 1. Numerically,
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B0 = 1, B1 = − 21 , B2 = 16 , B4 = − 30
,···
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.
∗ hBernoulliNumberi
created: h2013-03-21i by: halozanoi version: h30219i Privacy setting: h1i hDefinitioni h11B68i h49J24i h49J22i h49J20i h49J15i
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