Bernoulli number

== English == === Alternative forms === Bernoulli's number (dated, chiefly in plural) === Etymology === Named after Swiss mathematician Jacob Bernoulli (1654–1705), who discovered the numbers independently of and at about the same time as Japanese mathematician Seki Kōwa. The numbers first appeared as coefficients in Bernoulli's formulae for the sum of the first n positive integers, each raised to a given power. The sequence was subsequently found in numerous other contexts. === Noun === Bernoulli number (plural Bernoulli numbers) (mathematical analysis, number theory) Any one of the rational numbers in a sequence of such that appears in numerous contexts, including formulae for sums of integer powers and certain power series expansions. 1993, Serge Lang, Complex Analysis, Springer, 3rd Edition, page 418, The assertion about the value of the zeta function at negative integers then comes immediately from the definition of the Bernoulli numbers in terms of the coefficients of a power series, namely t e t − 1 = ∑ n = 0 ∞ B n t n n ! {\displaystyle {\frac {t}{e^{t}-1}}=\sum _{n=0}^{\infty }B_{n}{\frac {t^{n}}{n!}}} ==== Usage notes ==== For odd values of n {\displaystyle n} greater than 1, B n = 0 , {\displaystyle B_{n}=0,} and many formulae involve only "even-index" Bernoulli numbers. Consequently, some authors ignore these values and write B n {\displaystyle B_{n}} to mean what, properly speaking, is B 2 n {\displaystyle B_{2n}} . A sign convention affects the value assigned for n = 1 {\displaystyle n=1} . The modern (NIST) convention is that B 1 = − 1 2 {\displaystyle \textstyle B_{1}=-{\frac {1}{2}}} . An older convention, used by Leonhard Euler and some older textbooks, has that B 1 ∗ = + 1 2 {\displaystyle \textstyle B_{1}^{*}=+{\frac {1}{2}}} . The modified symbol B ∗ {\displaystyle \textstyle B^{*}} indicates the older convention is being used. Alternatively, the notations B − {\displaystyle B^{-}} and B + {\displaystyle B^{+}} can be used (where B 1 − = − 1 2 {\displaystyle \textstyle B_{1}^{-}=-{\frac {1}{2}}} and B 1 + = + 1 2 {\displaystyle \textstyle B_{1}^{+}=+{\frac {1}{2}}} ). The Bernoulli numbers may be regarded as special values of the Bernoulli polynomials B n ( x ) {\displaystyle B_{n}(x)} , with B n = B n ( 0 ) {\displaystyle B_{n}=B_{n}(0)} and B n ∗ = B n ∗ ( 0 ) {\displaystyle B_{n}^{*}=B_{n}^{*}(0)} . Note that the notations for Bernoulli numbers and Bernoulli polynomials are very similar. Note as well that the letter B {\displaystyle B} is used also for Bell numbers and Bell polynomials. Places where Bernoulli numbers appear include: Bernoulli's formula for the sum of the mth powers of the first n positive integers (also called Faulhaber's formula, although Faulhaber did not explore the properties of the coefficients). Taylor series expansions of the tangent and hyperbolic tangent functions. Formulae for particular values of the Riemann zeta function. The residual error of partial sums of certain power series: In particular, consider the series ∑ n = 1 ∞ 1 n 2 = π 2 6 {\displaystyle \textstyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}} . The partial sum S n = ∑ k = 1 n 1 k 2 {\displaystyle \textstyle S_{n}=\sum _{k=1}^{n}{\frac {1}{k^{2}}}} differs from the limit value by E n = ∑ k = 0 ∞ B k n k {\displaystyle \textstyle E_{n}=\sum _{k=0}^{\infty }{\frac {B_{k}}{n^{k}}}} . The Euler-Maclaurin formula. ==== Translations ==== === See also === Bernoulli polynomial === References === Chris Budd (2013), “How to add up quickly”, in plus.maths.org‎[2], retrieved 8 September 2013 === Further reading === Bernoulli polynomial on Wikipedia.Wikipedia Faulhaber's formula on Wikipedia.Wikipedia Euler–Maclaurin formula on Wikipedia.Wikipedia Particular values of the Riemann zeta function on Wikipedia.Wikipedia Genocchi number on Wikipedia.Wikipedia Poly-Bernoulli number on Wikipedia.Wikipedia Hurwitz zeta function on Wikipedia.Wikipedia Bernoulli numbers on Encyclopedia of Mathematics Bernoulli number on nLab Bernoulli Number on Wolfram MathWorld