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# Bernoulli number

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 (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.

nBn
01
1-1/2
21/6
30
4-1/30
50
61/42
70
8-1/30
90
105/66
110
12-691/2730
130
147/6

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.