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In mathematics, the Bernoulli numbers
Bn were first discovered in connection with the closed forms of the sums
-
for various fixed values of n. The closed forms are always polynomials in m of degree n+1 and are called Bernoulli polynomials. The
coefficients of the Bernoulli polynomials are closely related to the
Bernoulli numbers, as follows:
-
For example, taking n to be 1, we have 0 + 1 + 2 + ... + (m−1) = 1/2 (B0
m2 + 2 B1 m1) = 1/2 (m2 − m).
The Bernoulli numbers were first studied by Jakob Bernoulli, after
whom they were named by Abraham de Moivre.
Bernoulli numbers may be calculated by using the following recursive
formula:
-
plus the initial condition that B0 = 1.
The Bernoulli numbers may also be defined using the technique of generating functions. Their exponential generating function is x/(ex − 1),
so that:
-
for all values of x of absolute value less than 2π
(the radius of convergence of this power series).
Sometimes the lower-case bn is used in order to distinguish these from the Bell numbers.
The first few Bernoulli numbers (sequences A027641 and
A027642 in OEIS) are listed below.
| n |
Bn |
| 0 |
1 |
| 1 |
−1/2 |
| 2 |
1/6 |
| 3 |
0 |
| 4 |
−1/30 |
| 5 |
0 |
| 6 |
1/42 |
| 7 |
0 |
| 8 |
−1/30 |
| 9 |
0 |
| 10 |
5/66 |
| 11 |
0 |
| 12 |
−691/2730 |
| 13 |
0 |
| 14 |
7/6 |
It can be shown that Bn = 0 for all odd n other than 1. The appearance of the peculiar
value B12 = −691/2730 appears to rule out the possibility of a simple closed form for Bernoulli numbers.
The Bernoulli numbers also appear in the Taylor series expansion of
the tangent and hyperbolic tangent functions, in the Euler-Maclaurin formula, and in expressions of certain values of the Riemann zeta function.
In note G of Ada Byron's notes on the analytical engine from 1842 an algorithm for computer generated Bernoulli numbers was
described for the first time.
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