Bernoulli number

English Alternative forms en 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 Takakazu 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 en 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 en 2002 M. Aslam Chaudhry; Syed M. Zubair On a Class of Incomplete Gamma Functions with Applications CRC Press (Chapman & Hall/CRC) https://books.google.com.au/books?id=xfTKBQAAQBAJ&pg=PA287&dq=%22Bernoulli+number%22%7C%22Bernoulli+numbers%22&hl=en&sa=X&ved=0ahUKEwiNspDu8MrcAhVIwrwKHYVoBOwQ6AEI6QEwJg#v=onepage&q=%22Bernoulli%20number%22%7C%22Bernoulli%20numbers%22&f=false 287 In the first three sections, we present standard results about the Bernoulli numbers and polynomials and the Riemann zeta functions and its zeros. en 2007 C. Edward Sandifer How Euler Did It w:Mathematical Association of America https://books.google.com.au/books?id=sohHs7ExOsYC&pg=PA175&dq=%22Bernoulli+number%22%7C%22Bernoulli+numbers%22&hl=en&sa=X&ved=0ahUKEwiNspDu8MrcAhVIwrwKHYVoBOwQ6AEIZDAN#v=onepage&q=%22Bernoulli%20number%22%7C%22Bernoulli%20numbers%22&f=false 175 This done, he extends occurrence of Bernoulli numbers in the expansion of to the more general form and uses that to relate Bernoulli numbers to the values of . To end the theoretical parts of his exposition, he gives some of the properties of the Bernoulli polynomials and notes that Bernoulli numbers grow faster than any geometric series. Usage notes For odd values of greater than 1, and many formulae involve only "even-index" Bernoulli numbers. Consequently, some authors ignore these values and write to mean what, properly speaking, is . A sign convention affects the value assigned for . The modern (NIST) convention is that . An older convention, used by Leonhard Euler and some older textbooks, has that . The modified symbol indicates the older convention is being used. Alternatively, the notations and can be used (where and ). The Bernoulli numbers may be regarded as special values of the Bernoulli polynomials , with and . Note that the notations for Bernoulli numbers and Bernoulli polynomials are very similar. Note as well that the letter 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 . The partial sum differs from the limit value by . The Euler-Maclaurin formula. Translations any number in a sequence that appears in numerous contexts See also en Bernoulli polynomial References 2013 Chris Budd How to add up quickly plus.maths.org http://plus.maths.org/content/how-add-quickly 2013-09-08 https://web.archive.org/web/20251115054959/https://plus.maths.org/content/how-add-quickly 15 November 2025 Further reading Bernoulli polynomial Faulhaber's formula Euler–Maclaurin formula Particular values of the Riemann zeta function Genocchi number Poly-Bernoulli number Hurwitz zeta function Bernoulli numbers on Encyclopedia of Mathematics Bernoulli number on NLab nLab Bernoulli Number on MathWorld Wolfram MathWorld en eponyms