Bernoulli number

	fraction	decimal
0	1	+1.000000000
1	±	±0.500000000
2		+0.166666666
3	0	+0.000000000
4	−	−0.033333333
5	0	+0.000000000
6		+0.023809523
7	0	+0.000000000
8	−	−0.033333333
9	0	+0.000000000
10		+0.075757575
11	0	+0.000000000
12	−	−0.253113553
13	0	+0.000000000
14		+1.166666666
15	0	+0.000000000
16	−	−7.092156862
17	0	+0.000000000
18		+54.97117794
19	0	+0.000000000
20	−	−529.1242424

In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function.
The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by and ; they differ only for, where and. For every odd,. For every even, is negative if is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoulli polynomials, with and.
The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and independently by Japanese mathematician Seki Takakazu. Seki's discovery was posthumously published in 1712 in his work Katsuyō Sanpō; Bernoulli's, also posthumously, in his Ars Conjectandi of 1713. Ada Lovelace's note G on the Analytical Engine from 1842 describes an algorithm for generating Bernoulli numbers with Babbage's machine; it is disputed whether Lovelace or Babbage developed the algorithm. As a result, the Bernoulli numbers have the distinction of being the subject of the first published complex computer program.

Notation

The superscript used in this article distinguishes the two sign conventions for Bernoulli numbers. Only the term is affected:

with is the sign convention prescribed by NIST and many modern textbooks.
with was used in the older literature, and by Donald Knuth following Peter Luschny's "Bernoulli Manifesto".

In the formulas below, one can switch from one sign convention to the other with the relation, or for integer = 2 or greater, simply ignore it.
Since for all odd, and many formulas only involve even-index Bernoulli numbers, a few authors write "" instead of. This article does not follow that notation.

History

Early history

The Bernoulli numbers are rooted in the early history of the computation of sums of integer powers, which have been of interest to mathematicians since antiquity.
Methods to calculate the sum of the first positive integers, the sum of the squares and of the cubes of the first positive integers were known, but there were no real 'formulas', only descriptions given entirely in words. Among the great mathematicians of antiquity to consider this problem were Pythagoras, Archimedes, Aryabhata, Al-Karaji and Ibn al-Haytham.
During the late sixteenth and early seventeenth centuries mathematicians made significant progress. In the West Thomas Harriot of England, Johann Faulhaber of Germany, Pierre de Fermat and fellow French mathematician Blaise Pascal all played important roles.
Thomas Harriot seems to have been the first to derive and write formulas for sums of powers using symbolic notation, but even he calculated only up to the sum of the fourth powers. Johann Faulhaber gave formulas for sums of powers up to the 17th power in his 1631 Academia Algebrae, far higher than anyone before him, but he did not give a general formula.
Blaise Pascal in 1654 proved Pascal's identity relating to the sums of the th powers of the first positive integers for.
The Swiss mathematician Jacob Bernoulli was the first to realize the existence of a single sequence of constants which provide a uniform formula for all sums of powers.
The joy Bernoulli experienced when he hit upon the pattern needed to compute quickly and easily the coefficients of his formula for the sum of the th powers for any positive integer can be seen from his comment. He wrote:
Bernoulli's result was published posthumously in Ars Conjectandi in 1713. Seki Takakazu independently discovered the Bernoulli numbers and his result was published a year earlier, also posthumously, in 1712. However, Seki did not present his method as a formula based on a sequence of constants.
Bernoulli's formula for sums of powers is the most useful and generalizable formulation to date. The coefficients in Bernoulli's formula are now called Bernoulli numbers, following a suggestion of Abraham de Moivre.
Bernoulli's formula is sometimes called Faulhaber's formula after Johann Faulhaber who found remarkable ways to calculate sum of powers but never stated Bernoulli's formula. According to Knuth a rigorous proof of Faulhaber's formula was first published by Carl Jacobi in 1834. Knuth's in-depth study of Faulhaber's formula concludes:
In the above Knuth meant ; instead using the formula avoids subtraction:

Reconstruction of "Summæ Potestatum"

The Bernoulli numbers / were introduced by Jacob Bernoulli in the book Ars Conjectandi published posthumously in 1713. The main formula can be seen in the second half of the corresponding . The constant coefficients denoted,, and by Bernoulli are mapped to the notation which is now prevalent as,,,. The expression means – the small dots are used as grouping symbols. Using today's terminology these expressions are falling factorial powers. The factorial notation as a shortcut for was introduced much later in 1808 by Christian Kramp. The integral symbol ∫ on the left hand side goes back to Gottfried Wilhelm Leibniz in 1675 who used it as a long letter for "summa". The letter on the left hand side is not an index of summation but gives the upper limit of the range of summation which is to be understood as. Putting things together, for positive, today a mathematician is likely to write Bernoulli's formula as:
This formula suggests setting when switching from the so-called 'archaic' enumeration which uses only the even indices 2, 4, 6... to the modern form. Most striking in this context is the fact that the falling factorial has for the value. Thus Bernoulli's formula can be written
if, recapturing the value Bernoulli gave to the coefficient at that position.
The formula for on page 97 of Bernoulli's Ars Conjectandi contains an error at the last term; it should be instead of.

Definitions

Many characterizations of the Bernoulli numbers have been found in the last 300 years, and each could be used to introduce these numbers. Here only four of the most useful ones are mentioned:

a recursive equation,
an explicit formula,
a generating function,
an integral expression.

For the proof of the equivalence of the four approaches, see or.

Recursive definition

The Bernoulli numbers obey the sum formulas
where and denotes the Kronecker delta.
The first of these is sometimes written as the formula
where the power is expanded formally using the binomial theorem and is replaced by.
Solving for gives the recursive formulas

Explicit definition

In 1893 Louis Saalschütz listed a total of 38 explicit formulas for the Bernoulli numbers, usually giving some reference in the older literature. One of them is :

Generating function

The exponential generating functions are
where the substitution is. The arithmetic difference between the generating functions for and is t.
If we let and then
Then and for the m term in the series for is:
If
then we find that
showing that the values of obey the recursive formula for the Bernoulli numbers.
The generating function
is an asymptotic series. It contains the trigamma function.

Integral Expression

From the generating functions above, one can obtain the following integral formula for the even Bernoulli numbers:

Bernoulli numbers and the Riemann zeta function

The Bernoulli numbers can be expressed in terms of the Riemann zeta function:
Here the argument of the zeta function is 0 or negative. As is zero for negative even integers, if n>1 is odd, is zero.
By means of the zeta functional equation and the gamma reflection formula the following relation can be obtained:
Now the argument of the zeta function is positive.
It then follows from and Stirling's formula that

Efficient computation of Bernoulli numbers

In some applications it is useful to be able to compute the Bernoulli numbers through modulo, where is a prime; for example to test whether Vandiver's conjecture holds for, or even just to determine whether is an irregular prime. It is not feasible to carry out such a computation using the above recursive formulae, since at least arithmetic operations would be required. Fortunately, faster methods have been developed which require only operations.
David Harvey describes an algorithm for computing Bernoulli numbers by computing modulo for many small primes, and then reconstructing via the Chinese remainder theorem. Harvey writes that the asymptotic time complexity of this algorithm is and claims that this implementation is significantly faster than implementations based on other methods. Using this implementation Harvey computed for. Harvey's implementation has been included in SageMath since version 3.1. Prior to that, Bernd Kellner computed to full precision for in December 2002 and Oleksandr Pavlyk for with Mathematica in April 2008.