Editing Bernoulli number (section)

==The relation to the Euler numbers and {{pi}}==

The [[Euler number]]s are a sequence of integers intimately connected with the Bernoulli numbers. Comparing the
asymptotic expansions of the Bernoulli and the Euler numbers shows that the Euler numbers {{math|''E''<sub>2''n''</sub>}} are in magnitude approximately {{math|{{sfrac|2|π}}(4<sup>2''n''</sup> − 2<sup>2''n''</sup>)}} times larger than the Bernoulli numbers {{math|''B''<sub>2''n''</sub>}}. In consequence:

: <math> \pi \sim 2 (2^{2n} - 4^{2n}) \frac{B_{2n}}{E_{2n}}. </math>

This asymptotic equation reveals that {{pi}} lies in the common root of both the Bernoulli and the Euler numbers. In fact {{pi}} could be computed from these rational approximations.

Bernoulli numbers can be expressed through the Euler numbers and vice versa. Since, for odd {{mvar|n}}, {{math|''B''<sub>''n''</sub> {{=}} ''E''<sub>''n''</sub> {{=}} 0}} (with the exception {{math|''B''<sub>1</sub>}}), it suffices to consider the case when {{mvar|n}} is even.

:<math>\begin{align}
 B_n &= \sum_{k=0}^{n-1}\binom{n-1}{k} \frac{n}{4^n-2^n}E_k & n&=2, 4, 6, \ldots \\[6pt]
 E_n &= \sum_{k=1}^n \binom{n}{k-1} \frac{2^k-4^k}{k} B_k & n&=2,4,6,\ldots
\end{align}</math>

These conversion formulas express a connection between the Bernoulli and the Euler numbers. But more important, there is a deep arithmetic root common to both kinds of numbers, which can be expressed through a more fundamental sequence of numbers, also closely tied to {{pi}}. These numbers are defined for {{math|''n'' ≥ 1}} as<ref>{{citation
 | last = Stanley | first = Richard P. | author-link = Richard P. Stanley
 | arxiv = 0912.4240
 | contribution = A survey of alternating permutations
 | doi = 10.1090/conm/531/10466
 | mr = 2757798
 | pages = 165–196
 | publisher = American Mathematical Society | location = Providence, RI
 | series = Contemporary Mathematics
 | title = Combinatorics and graphs
 | volume = 531
 | year = 2010| isbn = 978-0-8218-4865-4 | s2cid = 14619581 }}</ref>{{r|Elkies2003}}

:<math> S_n  = 2 \left(\frac{2}{\pi}\right)^n \sum_{k = 0}^\infty \frac{ (-1)^{kn} }{(2k+1)^n} = 2 \left(\frac{2}{\pi}\right)^n \lim_{K\to \infty} \sum_{k = -K}^K (4k+1)^{-n}. </math>

The magic of these numbers lies in the fact that they turn out to be rational numbers.  This was first proved by [[Leonhard Euler]] in a landmark paper ''De summis serierum reciprocarum'' (On the sums of series of reciprocals) and has fascinated mathematicians ever since.{{r|Euler1735}} The first few of these numbers are

: <math> S_n = 1,1,\frac{1}{2},\frac{1}{3},\frac{5}{24}, \frac{2}{15},\frac{61}{720},\frac{17}{315},\frac{277}{8064},\frac{62}{2835},\ldots  </math> ({{OEIS2C|id=A099612}} / {{OEIS2C|id=A099617}})

These are the coefficients in the expansion of {{math|sec ''x'' + tan ''x''}}.

The Bernoulli numbers and Euler numbers can be understood as ''special views'' of these numbers, selected from the sequence {{math|''S''<sub>''n''</sub>}} and scaled for use in special applications.

: <math>\begin{align}
  B_{n} &= (-1)^{\left\lfloor \frac{n}{2}\right\rfloor} [ n \text{ even}] \frac{n! }{2^n - 4^n}\, S_{n}\ , & n&= 2, 3, \ldots \\
  E_n &= (-1)^{\left\lfloor \frac{n}{2}\right\rfloor} [ n \text{ even}] n! \, S_{n+1}  & n &= 0, 1, \ldots
\end{align}</math>

The expression [{{math|''n''}} even] has the value 1 if {{math|''n''}} is even and 0 otherwise ([[Iverson bracket]]).

These identities show that the quotient of Bernoulli and Euler numbers at the beginning of this section is just the special case of {{math|''R''<sub>''n''</sub> {{=}} {{sfrac|2''S''<sub>''n''</sub>|''S''<sub>''n'' + 1</sub>}}}} when {{mvar|n}} is even. The {{math|''R''<sub>''n''</sub>}} are rational approximations to {{pi}} and two successive terms always enclose the true value of {{pi}}. Beginning with {{math|''n'' {{=}} 1}} the sequence starts ({{OEIS2C|id=A132049}} / {{OEIS2C|id=A132050}}):

: <math> 2, 4, 3, \frac{16}{5}, \frac{25}{8}, \frac{192}{61}, \frac{427}{136}, \frac{4352}{1385}, \frac{12465}{3968}, \frac{158720}{50521},\ldots \quad \longrightarrow \pi. </math>

These rational numbers also appear in the last paragraph of Euler's paper cited above.

Consider the Akiyama–Tanigawa transform for the sequence {{OEIS2C|id=A046978}} ({{math|''n'' + 2}}) / {{OEIS2C|id=A016116}} ({{math|''n'' + 1}}):

:{| class="wikitable" style="text-align:right;"
! 0 
|1||{{sfrac|1|2}}||0||−{{sfrac|1|4}}||−{{sfrac|1|4}}||−{{sfrac|1|8}}||0
|-
! 1 
| {{sfrac|1|2}}|| 1|| {{sfrac|3|4}}|| 0|| −{{sfrac|5|8}}|| −{{sfrac|3|4}}||
|-
! 2 
| −{{sfrac|1|2}}|| {{sfrac|1|2}}|| {{sfrac|9|4}}|| {{sfrac|5|2}}|| {{sfrac|5|8}}|| ||
|-
! 3 
| −1|| −{{sfrac|7|2}}|| −{{sfrac|3|4}}|| {{sfrac|15|2}}|| || ||
|-
! 4 
| {{sfrac|5|2}}|| −{{sfrac|11|2}}|| −{{sfrac|99|4}}|| || || ||
|-
! 5 
| 8|| {{sfrac|77|2}}|| || || || ||
|-
! 6 
| −{{sfrac|61|2}}|| || || || || ||
|}

From the second, the numerators of the first column are the denominators of Euler's formula. The first column is −{{sfrac|1|2}} × {{OEIS2C|id=A163982}}.