Bernoulli number

English

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Bernoulli number

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
  ${\frac {t}{e^{t}-1}}=\sum _{n=0}^{\infty }B_{n}{\frac {t^{n}}{n!}}$
- 2002, M. Aslam Chaudhry, Syed M. Zubair, On a Class of Incomplete Gamma Functions with Applications‎^[1], CRC Press (Chapman & Hall/CRC), page 287:
  In the first three sections, we present standard results about the Bernoulli numbers and polynomials and the Riemann zeta functions and its zeros.
- 2007, C. Edward Sandifer, How Euler Did It, Mathematical Association of America, page 175:
  This done, he extends occurrence of Bernoulli numbers in the expansion of $\textstyle {\frac {1}{2}}\cot({\frac {1}{2}}x)$ to the more general form $\textstyle {\frac {\pi }{n}}\cot({\frac {m\pi }{n}})$ and uses that to relate Bernoulli numbers to the values of $\zeta (2n)$ . 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 $n$ greater than 1, $B_{n}=0,$ and many formulae involve only "even-index" Bernoulli numbers. Consequently, some authors ignore these values and write $B_{n}$ to mean what, properly speaking, is $B_{2n}$ .
A sign convention affects the value assigned for $n=1$ $n=1$ . The modern (NIST) convention is that $\textstyle B_{1}=-{\frac {1}{2}}$ $\textstyle B_{1}=-{\frac {1}{2}}$ . An older convention, used by Leonhard Euler and some older textbooks, has that $\textstyle B_{1}^{*}=+{\frac {1}{2}}$ $\textstyle B_{1}^{*}=+{\frac {1}{2}}$ .
- The modified symbol $\textstyle B^{*}$ indicates the older convention is being used.
- Alternatively, the notations $B^{-}$ and $B^{+}$ can be used (where $\textstyle B_{1}^{-}=-{\frac {1}{2}}$ and $\textstyle B_{1}^{+}=+{\frac {1}{2}}$ ).
The Bernoulli numbers may be regarded as special values of the Bernoulli polynomials $B_{n}(x)$ , with $B_{n}=B_{n}(0)$ and $B_{n}^{*}=B_{n}^{*}(0)$ .
Note that the notations for Bernoulli numbers and Bernoulli polynomials are very similar.
Note as well that the letter $B$ 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 $\textstyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}$ . The partial sum $\textstyle S_{n}=\sum _{k=1}^{n}{\frac {1}{k^{2}}}$ differs from the limit value by $\textstyle E_{n}=\sum _{k=0}^{\infty }{\frac {B_{k}}{n^{k}}}$ .
- The Euler-Maclaurin formula.