Editing Bernoulli number (section)

== Definitions ==
Many characterizations of the Bernoulli numbers have been found in the last 300 years, and each could be used to introduce these numbers.  Here only four of the most useful ones are mentioned:

* a recursive equation,
* an explicit formula,
* a generating function,
* an integral expression.

For the proof of the [[Logical equivalence|equivalence]] of the four approaches, see {{harvp|Ireland|Rosen|1990}} or {{harvp|Conway|Guy|1996}}.

=== Recursive definition ===
The Bernoulli numbers obey the sum formulas{{r|Weisstein2016}}
: <math> \begin{align} \sum_{k=0}^{m}\binom {m+1} k B^{-{}}_k &= \delta_{m, 0} \\ \sum_{k=0}^{m}\binom {m+1} k B^{+{}}_k &= m+1 \end{align}</math>
where <math>m=0,1,2...</math> and {{math|''δ''}} denotes the [[Kronecker delta]]. 

The first of these is sometimes written<ref>Jordan (1950) p 233</ref> as the formula (for m > 1)
<math display=block>(B+1)^m-B_m=0,</math>
where the power is expanded formally using the binomial theorem and <math>B^k</math> is replaced by <math>B_k</math>.

Solving for <math>B^{\mp{}}_m</math> gives the recursive formulas<ref>Ireland and Rosen (1990) p 229</ref>
: <math>\begin{align}
  B_m^{-{}} &= \delta_{m, 0} - \sum_{k=0}^{m-1} \binom{m}{k} \frac{B^{-{}}_k}{m - k + 1} \\
  B_m^+ &= 1 - \sum_{k=0}^{m-1} \binom{m}{k} \frac{B^+_k}{m - k + 1}.
\end{align}</math>

=== Explicit definition ===
In 1893 [[Louis Saalschütz]] listed  a total of 38 explicit formulas for the Bernoulli numbers,{{r|Saalschütz1893}} usually giving some reference in the older literature. One of them is (for <math>m\geq 1</math>):
:<math>\begin{align}
  B^-_m &= \sum_{k=0}^m \frac1{k+1} \sum_{j=0}^k \binom{k}{j} (-1)^j j^m \\
  B^+_m &= \sum_{k=0}^m \frac1{k+1} \sum_{j=0}^k \binom{k}{j} (-1)^j  (j + 1)^m.
\end{align}</math>

=== Generating function ===
The exponential [[generating function]]s are
:<math>\begin{alignat}{3}
  \frac{t}{e^t - 1}    &= \frac{t}{2} \left( \operatorname{coth} \frac{t}{2} -1 \right) &&= \sum_{m=0}^\infty \frac{B^{-{}}_m t^m}{m!}\\
  \frac{te^t}{e^t - 1} = \frac{t}{1 - e^{-t}} &= \frac{t}{2} \left( \operatorname{coth} \frac{t}{2} +1 \right) &&= \sum_{m=0}^\infty \frac{B^+_m t^m}{m!}.
\end{alignat}</math>
where the substitution is <math>t \to - t</math>. The two generating functions only differ by ''t''.
{{collapse top|title=Proof}}
If we let <math>F(t)=\sum_{i=1}^\infty f_it^i</math> and <math>G(t)=1/(1+F(t))=\sum_{i=0}^\infty g_it^i</math> then

:<math>G(t)=1-F(t)G(t).</math>

Then <math>g_0=1</math> and for <math>m>0</math> the m{{sup|th}} term in the series for <math>G(t)</math> is:

:<math>g_mt^i=-\sum_{j=0}^{m-1}f_{m-j}g_jt^m</math>

If

:<math>F(t)=\frac{e^t-1}t-1=\sum_{i=1}^\infty \frac{t^i}{(i+1)!}</math>

then we find that

:<math>G(t)=t/(e^t-1)</math>

:<math>\begin{align}
m!g_m&=-\sum_{j=0}^{m-1}\frac{m!}{j!}\frac{j!g_j}{(m-j+1)!}\\
&=-\frac 1{m+1}\sum_{j=0}^{m-1}\binom{m+1}jj!g_j\\
\end{align}</math>

showing that the values of <math>i!g_i</math> obey the recursive formula for the Bernoulli numbers <math>B^-_i</math>.
{{collapse bottom}}

The (ordinary) generating function
: <math> z^{-1} \psi_1(z^{-1}) = \sum_{m=0}^{\infty} B^+_m z^m</math>

is an [[asymptotic series]]. It contains the [[trigamma function]] {{math|''ψ''<sub>1</sub>}}.

=== Integral Expression ===
From the generating functions above, one can obtain the following integral formula for the even Bernoulli numbers: 

:<math>B_{2n} = 4n (-1)^{n+1} \int_0^{\infty} \frac{t^{2n-1}}{e^{2 \pi t} -1 } \mathrm{d} t </math>