Bernoulli number (original) (raw)

In mathematics, the Bernoulli numbers B n 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 (_B_0 _m_2 + 2 _B_1 _m_1) = 1/2 (_m_2 - 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 _B_0 = 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 (2π is 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 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 B n = 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.