Bernoulli numbers

In mathematics, the Bernoulli numbers B_n were first discovered in connection with the closed forms of the sums

<math>\sum_{k=0}^{m-1} k^n = 0^n + 1^n + 2^n + \cdots + {(m-1)}^n <math>

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:

<math>\sum_{k=0}^{m-1} k^n = {1\over{n+1}}\sum_{k=0}^n{n+1\choose{k}} B_k m^{n+1-k}<math>

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:

<math>\sum_{j=0}^m{m+1\choose{j}}B_j = 0<math>

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/(e^x − 1), so that:

<math>

\frac{x}{e^x-1} = \sum_{n=0}^{\infin} B_n \frac{x^n}{n!} <math> for all values of x of absolute value less than 2π (the radius of convergence of this power series).

Sometimes the lower-case b_n is used in order to distinguish these from the Bell numbers.

The first few Bernoulli numbers (sequences and 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 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.