Editing Bernoulli number (section)

== Applications of the Bernoulli numbers ==

=== Asymptotic analysis ===
Arguably the most important application of the Bernoulli numbers in mathematics is their use in the [[Euler–Maclaurin formula]]. Assuming that {{mvar|f}} is a sufficiently often differentiable function the Euler–Maclaurin formula can be written as{{sfnp|Graham|Knuth|Patashnik|1989|loc=9.67}}

: <math>\sum_{k=a}^{b-1} f(k) = \int_a^b f(x)\,dx + \sum_{k=1}^m \frac{B^-_k}{k!} (f^{(k-1)}(b)-f^{(k-1)}(a))+R_-(f,m).</math>

This formulation assumes the convention {{math|1=''B''{{su|p=−|b=1}} = −{{sfrac|1|2}}}}. Using the convention {{math|1=''B''{{su|p=+|b=1}} = +{{sfrac|1|2}}}} the formula becomes

: <math>\sum_{k=a+1}^{b} f(k) = \int_a^b f(x)\,dx + \sum_{k=1}^m \frac{B^+_k}{k!} (f^{(k-1)}(b)-f^{(k-1)}(a))+R_+(f,m).</math>

Here <math>f^{(0)}=f</math> (i.e. the zeroth-order derivative of <math>f</math> is just <math>f</math>). Moreover, let <math>f^{(-1)}</math> denote an [[antiderivative]] of <math>f</math>. By the [[fundamental theorem of calculus]],

: <math>\int_a^b f(x)\,dx = f^{(-1)}(b) - f^{(-1)}(a).</math>

Thus the last formula can be further simplified to the following succinct form of the Euler–Maclaurin formula

: <math> \sum_{k=a+1}^{b} f(k)= \sum_{k=0}^m \frac{B_k}{k!} (f^{(k-1)}(b)-f^{(k-1)}(a))+R(f,m). </math>

This form is for example the source for the important Euler–Maclaurin expansion of the zeta function

: <math> \begin{align}
  \zeta(s) & =\sum_{k=0}^m \frac{B^+_k}{k!} s^{\overline{k-1}} + R(s,m) \\
           & = \frac{B_0}{0!}s^{\overline{-1}} + \frac{B^+_1}{1!} s^{\overline{0}} + \frac{B_2}{2!} s^{\overline{1}} +\cdots+R(s,m) \\
           & = \frac{1}{s-1} + \frac{1}{2} + \frac{1}{12}s + \cdots + R(s,m).
\end{align} </math>

Here {{math|''s''<sup>{{overline|''k''}}</sup>}} denotes the [[Pochhammer symbol|rising factorial power]].{{sfnp|Graham|Knuth|Patashnik|1989|loc=2.44, 2.52}}

Bernoulli numbers are also frequently used in other kinds of [[asymptotic expansion]]s. The following example is the classical Poincaré-type asymptotic expansion of the [[digamma function]] {{math|''ψ''}}.

:<math>\psi(z) \sim \ln z - \sum_{k=1}^\infty \frac{B^+_k}{k z^k} </math>

=== Sum of powers ===
{{main|Faulhaber's formula}}
Bernoulli numbers feature prominently in the [[Closed-form expression|closed form]] expression of the sum of the {{math|''m''}}th powers of the first {{math|''n''}} positive integers. For {{math|''m'', ''n'' ≥ 0}} define

:<math>S_m(n) = \sum_{k=1}^n k^m = 1^m + 2^m + \cdots + n^m. </math>

This expression can always be rewritten as a [[polynomial]] in {{math|''n''}} of degree {{math|''m'' + 1}}. The [[coefficient]]s of these polynomials are related to the Bernoulli numbers by '''Bernoulli's formula''':
: <math>S_m(n) = \frac{1}{m + 1} \sum_{k=0}^m \binom{m + 1}{k} B^+_k n^{m + 1 - k} = m! \sum_{k=0}^m \frac{B^+_k n^{m + 1 - k}}{k! (m+1-k)!} ,</math>
where {{math|<big><big>(</big></big>{{su|p=''m'' + 1|b=''k''|a=c}}<big><big>)</big></big>}} denotes the [[binomial coefficient]].

For example, taking {{math|''m''}} to be 1 gives the [[triangular number]]s {{math|0, 1, 3, 6, ...}} {{OEIS2C|id=A000217}}.

:<math> 1 + 2 + \cdots + n = \frac{1}{2} (B_0 n^2 + 2 B^+_1 n^1) = \tfrac12 (n^2 + n).</math>

Taking {{math|''m''}} to be 2 gives the [[square pyramidal number]]s {{math|0, 1, 5, 14, ...}} {{OEIS2C|id=A000330}}.

: <math>1^2 + 2^2 + \cdots + n^2 = \frac{1}{3} (B_0 n^3 + 3 B^+_1 n^2 + 3 B_2 n^1) = \tfrac13 \left(n^3 + \tfrac32 n^2 + \tfrac12 n\right).</math>

Some authors use the alternate convention for Bernoulli numbers and state Bernoulli's formula in this way:
: <math>S_m(n) = \frac{1}{m + 1} \sum_{k=0}^m (-1)^k \binom{m + 1}{k} B^{-{}}_k n^{m + 1 - k}.</math>

Bernoulli's formula is sometimes called [[Faulhaber's formula]] after [[Johann Faulhaber]] who also found remarkable ways to calculate [[sums of powers]].

Faulhaber's formula was generalized by V. Guo and J. Zeng to a [[q-analog|{{mvar|q}}-analog]].{{r|GuoZeng2005}}

===Taylor series===
The Bernoulli numbers appear in the [[Taylor series]] expansion of many [[trigonometric functions]] and [[hyperbolic function]]s.

<math display="block">\begin{align}
\tan x &= \hphantom{{1\over x}} \sum_{n=1}^\infty \frac{(-1)^{n-1} 2^{2n} (2^{2n}-1) B_{2n} }{(2n)!}\; x^{2n-1}, && \left|x \right| < \frac \pi 2. \\
\cot x &= {1\over x} \sum_{n=0}^\infty \frac{(-1)^n B_{2n} (2x)^{2n}}{(2n)!}, & 0 < & |x| < \pi. \\
\tanh x &= \hphantom{{1\over x}} \sum_{n=1}^\infty \frac{2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}\;x^{2n-1}, && |x| < \frac \pi 2. \\
\coth x &= {1\over x} \sum_{n=0}^\infty \frac{B_{2n} (2x)^{2n}}{(2n)!}, & 0 < & |x| < \pi.
\end{align}</math>

===Laurent series===
The Bernoulli numbers appear in the following [[Laurent series]]:{{sfnp|Arfken|1970|p=463}}

[[Digamma function]]: <math> \psi(z)= \ln z- \sum_{k=1}^\infty \frac {B_k^{+{}}} {k z^k} </math>

=== Use in topology ===
The [[Kervaire–Milnor formula]] for the order of the cyclic group of diffeomorphism classes of [[exotic sphere|exotic {{math|(4''n'' − 1)}}-spheres]] which bound [[parallelizable manifold]]s involves Bernoulli numbers. Let {{math|''ES''<sub>''n''</sub>}} be the number of such exotic spheres for {{Math|''n'' ≥ 2}}, then

:<math>\textit{ES}_n = (2^{2n-2}-2^{4n-3}) \operatorname{Numerator}\left(\frac{B_{4n}}{4n} \right) .</math>

The [[Hirzebruch signature theorem#L genus and the Hirzebruch signature theorem|Hirzebruch signature theorem]] for the [[Hirzebruch signature theorem#L genus and the Hirzebruch signature theorem|{{mvar|L}} genus]] of a [[Smooth manifold|smooth]] [[Orientability|oriented]] [[closed manifold]] of [[dimension]] 4''n'' also involves Bernoulli numbers.