The Bernoulli numbers are a sequence of signed rational numbers that can be defined by the identity

These numbers arise in the series expansions of trigonometric functions, and are extremely important in number theory and analysis.

There are actually two definitions for the Bernoulli numbers. To distinguish them, the Bernoulli numbers as defined in modern usage (National Bureau of Standards convention) are written , while the Bernoulli numbers encountered in older literature are written (Gradshteyn and Ryzhik 2000). In each case, the Bernoulli numbers are a special case of the Bernoulli polynomials or with and .

Bernoulli numbers defined by the modern definition are denoted and sometimes called "even-index" Bernoulli numbers. These are the Bernoulli numbers returned, by example, by the Mathematica function BernoulliB[n].

The Bernoulli number can be defined by the contour integral

where the contour encloses the origin, has radius less than (to avoid the poles at ), and is traversed in a counterclockwise direction (Arfken 1985, p. 413).

The first few Bernoulli numbers are

(Sloane's A000367 and A002445), with

for n = 1, 2, ....

The numbers of digits in the numerator of for the n = 2, 4, ... are 1, 1, 1, 1, 1, 3, 1, 4, 5, 6, 6, 9, 7, 11, ... (Sloane's A068399), while the numbers of digits in the corresponding denominators are 1, 2, 2, 2, 2, 4, 1, 3, 3, 3, 3, 4, 1, 3, 5, 3, ... (Sloane's A092904). Both are plotted above.

The running maxima of denominators are 1, 6, 30, 42, 66, 2730, 14322, 1919190, ... (Sloane's A100194), which occur for n = 2, 4, 6, 8, 12, 14, 32, 38, ... (Sloane's A100195).

The number of digits in the numerator to for n = 0, 1, ... are 1, 1, 83, 1779, 27691, ... (Sloane's A103233).

for 1806, but for no other n (Kellner 2005).

The only known Bernoulli numbers having prime numerators occur for n = 10, 12, 14, 16, 18, 36, and 42 (Sloane's A092132), corresponding to 5, -691, 7, -3617, 43867, -26315271553053477373, and 1520097643918070802691 (Sloane's A092133), with no other primes for (E. W. Weisstein, Apr. 17, 2005).

The following table summarizes record computations of the nth Bernoulli number . The results were checked using Kummer congruences with different prime moduli and (except for ) are downloadable from http://www.bernoulli.org/. The program Calcbn for Windows and Linux can be used to calculate Bernoulli numbers up to index extremely quickly (Kellner).

The denominator of (mod 1) is given by the von Staudt-Clausen theorem, which also implies that the denominator of is squarefree (Hardy and Wright 1979). Another curious property is that the fraction part of in decimal has a decimal period which divides n, and there is a single digit before that period (Conway 1996).

Consider the generating function

which converges uniformly for and all x (Castellanos 1988). Taking the partial derivative gives

The solution to this differential equation can be found using separation of variables as

so integrating gives

But integrating (20) explicitly gives

so

Solving for T(t) and plugging back into (19) then gives

(Castellanos 1988). Setting x = 0 and adding to both sides then gives

Letting then gives

for .

The Bernoulli numbers may also be calculated from

The Bernoulli numbers are given by the double sum

where is a binomial coefficient. They also satisfy the sum

as well as the interesting sums

(Lehmer 1935, Carlitz 1968), as well as the nice sum identity

(Gosper).

An asymptotic expansion for the even Bernoulli numbers is

Bernoulli numbers appear in expressions of the form , where p = 1, 2, .... Bernoulli numbers also appear in the series expansions of functions involving , , , , , , , , and .

An analytic solution exists for even orders,

for n = 1, 2, ..., where is the Riemann zeta function. Another intimate connection with the Riemann zeta function is provided by the identity

An integral in terms of the Euler polynomial is given by

where is an Euler polynomial (J. Crepps, pers. comm., Apr. 2002).

Bernoulli first used the Bernoulli numbers while computing . He used the property of the figurate number triangle that

along with a form for which he derived inductively to compute the sums up to n = 10 (Boyer 1968, p. 85). For , the sum is given by

where the notation means the quantity in question is raised to the appropriate power k, and all terms of the form are replaced with the corresponding Bernoulli numbers . Note that it is common (e.g., Carlitz 1965) to simply write with the understanding that after expansion, is replaced by .

Written explicitly in terms of a sum of powers,

where

It is also true that the coefficients of the terms sum to 1,

which Bernoulli stated without proof.

Ramanujan gave a number of curious infinite sum identities involving Bernoulli numbers (Berndt 1994).

The older definition of the Bernoulli numbers, no longer in widespread use, defines using the equations

for , or
for (Whittaker and Watson 1990, p. 125). The Bernoulli numbers may be calculated from the integral

and analytically from

for n = 1, 2, ..., where is the Riemann zeta function.

The Bernoulli numbers are a superset of the archaic ones since

The first few Bernoulli numbers are

 

Agoh's Conjecture, Bernoulli Function, Bernoulli Number of the Second Kind, Bernoulli Polynomial, Debye Functions, Euler-Maclaurin Integration Formulas, Euler Number, Figurate Number Triangle, Genocchi Number, Integer Sequence Primes, Irregular Prime, Modified Bernoulli Number, Pascal's Triangle, Riemann Zeta Function, von Staudt-Clausen Theorem