Editing Bernoulli number (section)

=== Connection with Worpitzky numbers ===
The definition to proceed with was developed by Julius Worpitzky in 1883. Besides elementary arithmetic only the factorial function {{math|''n''!}} and the power function {{math|''k<sup>m</sup>''}} is employed. The signless Worpitzky numbers are defined as

: <math> W_{n,k}=\sum_{v=0}^k (-1)^{v+k} (v+1)^n \frac{k!}{v!(k-v)!} . </math>

They can also be expressed through the [[Stirling numbers of the second kind]]

: <math> W_{n,k}=k! \left\{ {n+1\atop k+1} \right\}.</math>

A Bernoulli number is then introduced as an inclusion–exclusion sum of Worpitzky numbers weighted by the [[Harmonic progression (mathematics)|harmonic sequence]] 1,&nbsp;{{sfrac|1|2}},&nbsp;{{sfrac|1|3}},&nbsp;...

: <math> B_{n}=\sum_{k=0}^n (-1)^k \frac{W_{n,k}}{k+1}\ =\ \sum_{k=0}^n \frac{1}{k+1} \sum_{v=0}^k (-1)^v (v+1)^n {k \choose v}\ . </math>

:{{math|1=''B''<sub>0</sub> = 1}}
:{{math|1=''B''<sub>1</sub> = 1 − {{sfrac|1|2}}}}
:{{math|1=''B''<sub>2</sub> = 1 − {{sfrac|3|2}} + {{sfrac|2|3}}}}
:{{math|1=''B''<sub>3</sub> = 1 − {{sfrac|7|2}} + {{sfrac|12|3}} − {{sfrac|6|4}}}}
:{{math|1=''B''<sub>4</sub> = 1 − {{sfrac|15|2}} + {{sfrac|50|3}} − {{sfrac|60|4}} + {{sfrac|24|5}}}}
:{{math|1=''B''<sub>5</sub> = 1 − {{sfrac|31|2}} + {{sfrac|180|3}} − {{sfrac|390|4}} + {{sfrac|360|5}} − {{sfrac|120|6}}}}
:{{math|1=''B''<sub>6</sub> = 1 − {{sfrac|63|2}} + {{sfrac|602|3}} − {{sfrac|2100|4}} + {{sfrac|3360|5}} − {{sfrac|2520|6}} + {{sfrac|720|7}}}}

This representation has {{math|''B''{{su|p=+|b=1}} {{=}} +{{sfrac|1|2}}}}.

Consider the sequence {{math|''s<sub>n</sub>''}}, {{math|''n'' ≥ 0}}. From Worpitzky's numbers {{OEIS2C|id=A028246}}, {{OEIS2C|id=A163626}} applied to {{math|''s''<sub>0</sub>, ''s''<sub>0</sub>, ''s''<sub>1</sub>, ''s''<sub>0</sub>, ''s''<sub>1</sub>, ''s''<sub>2</sub>, ''s''<sub>0</sub>, ''s''<sub>1</sub>, ''s''<sub>2</sub>, ''s''<sub>3</sub>, ...}} is identical to the Akiyama–Tanigawa transform applied to {{math|''s<sub>n</sub>''}} (see [[#Connection with Stirling numbers of the first kind|Connection with Stirling numbers of the first kind]]). This can be seen via the table:

:{| style="text-align:center"
|+ '''Identity of<br/>Worpitzky's representation and Akiyama–Tanigawa transform'''
|-
|1|| || || || || ||0||1|| || || || ||0||0||1|| || || ||0||0||0||1|| || ||0||0||0||0||1|| 
|-
|1||−1|| || || || ||0||2||−2|| || || ||0||0||3||−3|| || ||0||0||0||4||−4|| || || || || || || 
|-
|1||−3||2|| || || ||0||4||−10||6|| || ||0||0||9||−21||12|| || || || || || || || || || || || || 
|-
|1||−7||12||−6|| || ||0||8||−38||54||−24|| || || || || || || || || || || || || || || || || || || 
|-
|1||−15||50||−60||24|| || || || || || || || || || || || || || || || || || || || || || || || || 
|-
|}

The first row represents {{math|''s''<sub>0</sub>, ''s''<sub>1</sub>, ''s''<sub>2</sub>, ''s''<sub>3</sub>, ''s''<sub>4</sub>}}.

Hence for the second fractional Euler numbers {{OEIS2C|id=A198631}} ({{math|''n''}}) / {{OEIS2C|id=A006519}} ({{math|''n'' + 1}}):

:{{math|1= ''E''<sub>0</sub> = 1}}
:{{math|1= ''E''<sub>1</sub> = 1 − {{sfrac|1|2}}}}
:{{math|1= ''E''<sub>2</sub> = 1 − {{sfrac|3|2}} + {{sfrac|2|4}}}}
:{{math|1= ''E''<sub>3</sub> = 1 − {{sfrac|7|2}} + {{sfrac|12|4}} − {{sfrac|6|8}}}}
:{{math|1= ''E''<sub>4</sub> = 1 − {{sfrac|15|2}} + {{sfrac|50|4}} − {{sfrac|60|8}} + {{sfrac|24|16}}}}
:{{math|1= ''E''<sub>5</sub> = 1 − {{sfrac|31|2}} + {{sfrac|180|4}} − {{sfrac|390|8}} + {{sfrac|360|16}} − {{sfrac|120|32}}}}
:{{math|1= ''E''<sub>6</sub> = 1 − {{sfrac|63|2}} + {{sfrac|602|4}} − {{sfrac|2100|8}} + {{sfrac|3360|16}} − {{sfrac|2520|32}} + {{sfrac|720|64}}}}

A second formula representing the Bernoulli numbers by the Worpitzky numbers is for {{math|''n'' ≥ 1}}

: <math> B_n=\frac n {2^{n+1}-2}\sum_{k=0}^{n-1} (-2)^{-k}\, W_{n-1,k} . </math>

The simplified second Worpitzky's representation of the second Bernoulli numbers is:

{{OEIS2C|id=A164555}} ({{math|''n'' + 1}}) / {{OEIS2C|id=A027642}}({{math|''n'' + 1}}) = {{math|{{sfrac|''n'' + 1|2<sup>''n'' + 2</sup> − 2}}}} × {{OEIS2C|id=A198631}}({{math|''n''}}) / {{OEIS2C|id=A006519}}({{math|''n'' + 1}})

which links the second Bernoulli numbers to the second fractional Euler numbers. The beginning is:

:{{math|{{sfrac|1|2}}, {{sfrac|1|6}}, 0, −{{sfrac|1|30}}, 0, {{sfrac|1|42}}, ... {{=}} ({{sfrac|1|2}}, {{sfrac|1|3}}, {{sfrac|3|14}}, {{sfrac|2|15}}, {{sfrac|5|62}}, {{sfrac|1|21}}, ...) × (1, {{sfrac|1|2}}, 0, −{{sfrac|1|4}}, 0, {{sfrac|1|2}}, ...)}}

The numerators of the first parentheses are {{OEIS2C|id=A111701}} (see [[#Connection with Stirling numbers of the first kind|Connection with Stirling numbers of the first kind]]).