Bernoulli number

Bernoulli numbersgeneralized Bernoulli numberAn identityBernoulliBernoulli's numbersConnection with Stirling numbers of the first kindgeneralized Bernoulli numbersgenerating function for the Bernoulli numbersSeidel triangle
[[File:Bernoulli_numbers_graphs.svg|thumb|Graphs of modern Bernoulli numbers,wikipedia
223 Related Articles

Faulhaber's formula

Bernoulli's formulaBernoulli's functionclosed form
The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of powers of the first positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function.
because the Bernoulli number

Riemann zeta function

zeta functionRiemann zeta-functionRiemann's zeta function
The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of powers of the first positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function. By means of the zeta functional equation and the gamma reflection formula the following relation can be obtained :
th Bernoulli number.

Ars Conjectandi

1713
Seki's discovery was posthumously published in 1712 in his work Katsuyo Sampo; Bernoulli's, also posthumously, in his Ars Conjectandi of 1713.
He incorporated fundamental combinatorial topics such as his theory of permutations and combinations (the aforementioned problems from the twelvefold way) as well as those more distantly connected to the burgeoning subject: the derivation and properties of the eponymous Bernoulli numbers, for instance.

Euler–Maclaurin formula

Euler–Maclaurin summation formulaEuler–Maclaurin summationEuler's summation formula
The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of powers of the first positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function.
where B_k is the kth Bernoulli number (with B_1=1/2) and R_p is an error term which depends on n, m, p, and f and is usually small for suitable values of p.

Computer program

programprogramscomputer programs
As a result, the Bernoulli numbers have the distinction of being the subject of the first published complex computer program.
The translation contained Note G which completely detailed a method for calculating Bernoulli numbers using the Analytical Engine.

Analytical Engine

Analytic EngineBabbage engineBabbage machine
Ada Lovelace's note G on the Analytical Engine from 1842 describes an algorithm for generating Bernoulli numbers with Babbage's machine.
These programs treat polynomials, iterative formulas, Gaussian elimination, and Bernoulli numbers.

Seki Takakazu

Seki KōwaKowa SekiTakakazu Seki
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 Kōwa.
For example, he is credited with the discovery of Bernoulli numbers.

Ada Lovelace

Ada ByronAda Lovelace DayAugusta Ada King (née Byron), Countess of Lovelace
Ada Lovelace's note G on the Analytical Engine from 1842 describes an algorithm for generating Bernoulli numbers with Babbage's machine.
The notes are around three times longer than the article itself and include (in Note G), in complete detail, a method for calculating a sequence of Bernoulli numbers using the Analytical Engine, which might have run correctly had it ever been built (only Babbage's Difference Engine has been built, completed in London in 2002).

Taylor series

Taylor expansionMaclaurin seriesTaylor polynomial
The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of powers of the first positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function.
are the Bernoulli numbers.

Jacob Bernoulli

Jakob BernoulliBernoulliJames Bernoulli
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 Kōwa.
The Bernoulli numbers appear in the book in a discussion of the exponential series.

Gamma function

ΓgammaEuler gamma function
By means of the zeta functional equation and the gamma reflection formula the following relation can be obtained :
are the Bernoulli numbers.

Kummer–Vandiver conjecture

Vandiver's conjectureVandiver conjecture
modulo p, where p is a prime; for example to test whether Vandiver's conjecture holds for p, or even just to determine whether p is an irregular prime.
The first factor h 1 is well understood and can be computed easily in terms of Bernoulli numbers, and is usually rather large.

Charles Babbage

BabbageBabbage, CharlesBabbage engines
Ada Lovelace's note G on the Analytical Engine from 1842 describes an algorithm for generating Bernoulli numbers with Babbage's machine.
Ada Lovelace, who corresponded with Babbage during his development of the Analytical Engine, is credited with developing an algorithm that would enable the Engine to calculate a sequence of Bernoulli numbers.

Summation

sumsumssigma notation
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
: where B_k denotes a Bernoulli number (that is Faulhaber's formula).

Trigamma function

trigamma
It contains the trigamma function
, i.e. the Bernoulli numbers of the second kind.

Asymptotic expansion

asymptotic seriesasymptotic expansionsasymptotic
is an asymptotic series.

Trigonometric functions

cosinetrigonometric functiontangent
The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of powers of the first positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function.

Bernoulli polynomials

Bernoulli polynomialEuler polynomialsEuler polynomial
The Bernoulli numbers are special values of the Bernoulli polynomials B_n(x), with and.
for n ≥ 0, where B k are the Bernoulli numbers, and E k are the Euler numbers.

Von Staudt–Clausen theorem

von Staudt-Clausen theoremVon-Staudt Clausen theorem
) which is an explicit formula for Bernoulli numbers and can be used to prove Von-Staudt Clausen theorem.
In number theory, the von Staudt–Clausen theorem is a result determining the fractional part of Bernoulli numbers, found independently by and.

Agoh–Giuga conjecture

Giuga's conjecturehis conjecture
For example, the Agoh–Giuga conjecture postulates that p is a prime number if and only if
In number theory the Agoh–Giuga conjecture on the Bernoulli numbers B k postulates that p is a prime number if and only if

Generating function

exponential generating functionordinary generating functiongenerating functions
The exponential generating functions are

Digamma function

Gauss's digamma theoremdigammadigamma-function
The following example is the classical Poincaré-type asymptotic expansion of the digamma function
:where B k is the kth Bernoulli number and ζ is the Riemann zeta function.

Regular prime

irregular primegave a criterionirregular
modulo p, where p is a prime; for example to test whether Vandiver's conjecture holds for p, or even just to determine whether p is an irregular prime.
Regular primes may be defined via the divisibility of either class numbers or of Bernoulli numbers.

Herbrand–Ribet theorem

Herbrand-Ribet theorem
is congruent to −1 modulo p. Divisibility properties of the Bernoulli numbers are related to the ideal class groups of cyclotomic fields by a theorem of Kummer and its strengthening in the Herbrand-Ribet theorem, and to class numbers of real quadratic fields by Ankeny–Artin–Chowla.
It is a strengthening of Ernst Kummer's theorem to the effect that the prime p divides the class number of the cyclotomic field of p-th roots of unity if and only if p divides the numerator of the n-th Bernoulli number B n

Hyperbolic function

hyperbolic tangenthyperbolichyperbolic cosine
The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of powers of the first positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function.