Bernoulli number

English

Alternative forms

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Etymology

Named after Swiss mathematician Template:W (1654–1705), who discovered the numbers independently of and at about the same time as Japanese mathematician Template:W.

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

1. Template:Lb 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, Template:W, 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

     $\frac{t}{e^t-1}=\sum _{n=0}^{\mathrm{\infty }}{B}_n\frac{t^n}{n!}$

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   • Template:Quote-book

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 $n=1$. The modern (Template:W) convention is that $B_1=-\frac{1}{2}$. An older convention, used by Template:W and some older textbooks, has that $B_1^{*}=+\frac{1}{2}$.
  • The modified symbol $B^{*}$ indicates the older convention is being used.
  • Alternatively, the notations $B^{-}$ and $B^{+}$ can be used (where $B_1^{-}=-\frac{1}{2}$ and $B_1^{+}=+\frac{1}{2}$).
• 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 Template:Ws and Template:W.
• 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 ${\sum }_{n=1}^{\mathrm{\infty }}\frac{1}{n^2}=\frac{{\pi }^2}{6}$. The partial sum $S_n={\sum }_{k=1}^n\frac{1}{k^2}$ differs from the limit value by $E_n={\sum }_{k=0}^{\mathrm{\infty }}\frac{B_k}{n^k}$.
  • The Template:W.

See also

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References

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Further reading

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• Bernoulli numbers on Template:W
• Bernoulli number on Template:W
• Bernoulli Number on Template:W