Editing Bernoulli number (section)

==History==
=== Early history ===
The Bernoulli numbers are rooted in the early history of the computation of sums of integer powers, which have been of interest to mathematicians since antiquity.

[[File:Seki Kowa Katsuyo Sampo Bernoulli numbers.png|thumb|right|A page from [[Seki Takakazu]]'s ''Katsuyō Sanpō'' (1712), tabulating binomial coefficients and Bernoulli numbers]]

Methods to calculate the sum of the first {{mvar|n}} positive integers, the sum of the squares and of the cubes of the first {{mvar|n}} positive integers were known, but there were no real 'formulas', only descriptions given entirely in words. Among the great mathematicians of antiquity to consider this problem were [[Pythagoras]] (c. 572–497&nbsp;BCE, Greece), [[Archimedes]] (287–212&nbsp;BCE, Italy), [[Aryabhata]] (b. 476, India), [[Abu Bakr al-Karaji]] (d. 1019, Persia) and Abu Ali al-Hasan ibn al-Hasan ibn [[al-Haytham]] (965–1039, Iraq).

During the late sixteenth and early seventeenth centuries mathematicians made significant progress.  In the West [[Thomas Harriot]] (1560–1621) of England, [[Johann Faulhaber]] (1580–1635) of Germany, [[Pierre de Fermat]] (1601–1665) and fellow French mathematician [[Blaise Pascal]] (1623–1662) all played important roles.

Thomas Harriot seems to have been the first to derive and write formulas for sums of powers using symbolic notation, but even he calculated only up to the sum of the fourth powers.  Johann Faulhaber gave formulas for sums of powers up to the 17th power in his 1631 ''Academia Algebrae'', far higher than anyone before him, but he did not give a general formula.

Blaise Pascal in 1654 proved [[Faulhaber's formula|''Pascal's identity'']] relating {{math|(''n''+1)<sup>''k''+1</sup>}} to the sums of the {{math|''p''}}th powers of the first {{math|''n''}} positive integers for {{math|''p'' {{=}} 0, 1, 2, ..., ''k''}}.

The Swiss mathematician Jakob Bernoulli (1654–1705) was the first to realize the existence of a single sequence of constants {{math|''B''<sub>0</sub>, ''B''<sub>1</sub>, ''B''<sub>2</sub>,...}} which provide a uniform formula for all sums of powers.{{sfnp|Knuth|1993}}

The joy Bernoulli experienced when he hit upon the pattern needed to compute quickly and easily the coefficients of his formula for the sum of the {{mvar|c}}th powers for any positive integer {{math|''c''}} can be seen from his comment. He wrote:

:"With the help of this table, it took me less than half of a quarter of an hour to find that the tenth powers of the first 1000 numbers being added together will yield the sum 91,409,924,241,424,243,424,241,924,242,500."

Bernoulli's result was published posthumously in ''[[Ars Conjectandi]]'' in 1713.  [[Seki Takakazu]] independently discovered the Bernoulli numbers and his result was published a year earlier, also posthumously, in 1712.{{r|Selin1997_891}} However, Seki did not present his method as a formula based on a sequence of constants.

Bernoulli's formula for sums of powers is the most useful and generalizable formulation to date. The coefficients in Bernoulli's formula are now called Bernoulli numbers, following a suggestion of [[Abraham de Moivre]].

Bernoulli's formula is sometimes called [[Faulhaber's formula]] after Johann Faulhaber who found remarkable ways to calculate sum of powers but never stated Bernoulli's formula. According to Knuth{{sfnp|Knuth|1993}} a rigorous proof of Faulhaber's formula was first published by [[Carl Gustav Jacob Jacobi|Carl Jacobi]] in 1834.{{r|Jacobi1834}} Knuth's in-depth study of Faulhaber's formula concludes (the nonstandard notation on the LHS is explained further on):

:''"Faulhaber never discovered the Bernoulli numbers; i.e., he never realized that a single sequence of constants'' {{math|''B''<sub>0</sub>, ''B''<sub>1</sub>, ''B''<sub>2</sub>,}} ''... would provide a uniform''
::<math display=inline>\sum n^m = \frac 1{m+1}\left( B_0n^{m+1}-\binom{m+1} 1 B_1 n^m+\binom{m+1} 2B_2n^{m-1}-\cdots +(-1)^m\binom{m+1}mB_mn\right) </math>
:''for all sums of powers. He never mentioned, for example, the fact that almost half of the coefficients turned out to be zero after he had converted his formulas for'' {{math|Σ ''n<sup>m</sup>''}} ''from polynomials in {{mvar|N}} to polynomials in {{mvar|n}}."{{sfnp|Knuth|1993|p=14}}''

In the above Knuth meant <math>B_1^-</math>; instead using <math>B_1^+</math> the formula avoids subtraction:
:<math display=inline> \sum n^m = \frac 1{m+1}\left( B_0n^{m+1}+\binom{m+1} 1 B^+_1 n^m+\binom{m+1} 2B_2n^{m-1}+\cdots+\binom{m+1}mB_mn\right). </math>

=== Reconstruction of "Summae Potestatum" ===
[[File:JakobBernoulliSummaePotestatum.png|thumb|right|upright=1.5|Jakob Bernoulli's "Summae Potestatum", 1713{{efn|Translation of the text: 
" ... And if [one were] to proceed onward step by step to higher powers, one may furnish, with little difficulty, the following list:<br>
Sums of powers<br>
<math>\textstyle \int n = \sum_{k=1}^n k = \frac {1}{2} n^2 + \frac {1}{2} n </math><br>
::::⋮
<math>\textstyle \int n^{10} = \sum_{k=1}^n k^{10} = \frac {1}{11} n^{11} + \frac {1}{2} n^{10} + \frac {5}{6} n^9 - 1 n^7 + 1 n^5 - \frac {1}{2} n^3 + \frac {5}{66} n </math><br>
Indeed [if] one will have examined diligently the law of arithmetic progression there, one will also be able to continue the same without these circuitous computations:  For [if] <math>\textstyle c </math> is taken as the exponent of any power, the sum of all <math>\textstyle n^c </math> is produced or<br>
<math>\textstyle \int n^c = \sum_{k=1}^n k^c = \frac {1}{c+1} n^{c+1} + \frac {1}{2} n^c + \frac {c}{2} An^{c-1} + \frac {c(c-1)(c-2)}{2\cdot 3\cdot4} Bn^{c-3} + \frac {c(c-1)(c-2)(c-3)(c-4)}{2\cdot 3\cdot 4 \cdot 5 \cdot 6} Cn^{c-5} + \frac {c(c-1)(c-2)(c-3)(c-4)(c-5)(c-6)}{2\cdot 3\cdot 4 \cdot 5 \cdot 6 \cdot 7  \cdot 8} Dn^{c-7} + \cdots </math><br>
and so forth, the exponent of its power <math>n</math> continually diminishing by 2 until it arrives at <math>n</math> or <math>n^2</math>.  The capital letters <math>\textstyle  A, B, C, D, </math> etc. denote in order the coefficients of the last terms for <math>\textstyle \int n^2 , \int n^4 , \int n^6 , \int n^8 </math>, etc. namely<br>
<math>\textstyle  A = \frac {1}{6} , B = - \frac {1}{30} , C = \frac {1}{42} , D = - \frac {1}{30} </math>."<br>
[Note:  The text of the illustration contains some typos:  ''ensperexit'' should read ''inspexerit'', ''ambabimus'' should read ''ambagibus'', ''quosque'' should read ''quousque'', and in Bernoulli's original text ''Sumtâ'' should read ''Sumptâ'' or ''Sumptam''.]
* {{citation |last=Smith |first=David Eugene |date= 1929 |chapter=Jacques (I) Bernoulli: On the 'Bernoulli Numbers' |title= A Source Book in Mathematics |chapter-url=https://archive.org/details/sourcebookinmath00smit/page/85 |location=New York |publisher=McGraw-Hill Book Co. |pages=85–90 }}
* {{citation |last=Bernoulli |first=Jacob |date=1713 |title=Ars Conjectandi |url=https://archive.org/details/jacobibernoulli00bern/page/97 |location=Basel |publisher=Impensis Thurnisiorum, Fratrum |pages=97–98 |language=la |doi=10.5479/sil.262971.39088000323931}}
}}]]

The Bernoulli numbers {{OEIS2C|id=A164555}}(n)/{{OEIS2C|id=A027642}}(n) were introduced by Jakob Bernoulli in the book ''[[Ars Conjectandi]]'' published posthumously in 1713 page 97. The main formula can be seen in the second half of the corresponding facsimile. The constant coefficients denoted {{math|''A''}}, {{math|''B''}}, {{math|''C''}} and {{math|''D''}} by Bernoulli are mapped to the notation which is now prevalent as {{math|''A'' {{=}} ''B''<sub>2</sub>}}, {{math|''B'' {{=}} ''B''<sub>4</sub>}}, {{math|''C'' {{=}} ''B''<sub>6</sub>}}, {{math|''D'' {{=}} ''B''<sub>8</sub>}}. The expression {{math|''c''·''c''−1·''c''−2·''c''−3}} means {{math|''c''·(''c''−1)·(''c''−2)·(''c''−3)}} – the small dots are used as grouping symbols. Using today's terminology these expressions are [[Pochhammer symbol|falling factorial powers]] {{math|''c''<sup>{{underline|''k''}}</sup>}}. The factorial notation {{math|''k''!}} as a shortcut for {{math|1 × 2 × ... × ''k''}} was not introduced until 100 years later. The integral symbol on the left hand side goes back to [[Gottfried Wilhelm Leibniz]] in 1675 who used it as a long letter {{math|''S''}} for "summa" (sum).{{efn|The {{harvp|''Mathematics Genealogy Project''|n.d.}} shows Leibniz as the academic<!--not doctoral--> advisor of Jakob Bernoulli. See also {{harvp|Miller|2017}}.}} The letter {{math|''n''}} on the left hand side is not an index of [[summation]] but gives the upper limit of the range of summation which is to be understood as {{math|1, 2, ..., ''n''}}. Putting things together, for positive {{math|''c''}}, today a mathematician is likely to write Bernoulli's formula as:

: <math> \sum_{k=1}^n k^c = \frac{n^{c+1}}{c+1}+\frac 1 2 n^c+\sum_{k=2}^c \frac{B_k}{k!} c^{\underline{k-1}}n^{c-k+1}.</math>

This formula suggests setting {{math|''B''<sub>1</sub> {{=}} {{sfrac|1|2}}}} when switching from the so-called 'archaic' enumeration which uses only the even indices 2, 4, 6... to the modern form (more on different conventions in the next paragraph). Most striking in this context is the fact that the [[falling factorial#Real_numbers_and_negative_n|falling factorial]] {{math|''c''<sup>{{underline|''k''−1}}</sup>}} has for {{math|''k'' {{=}} 0}} the value {{math|{{sfrac|1|''c'' + 1}}}}.{{sfnp|Graham|Knuth|Patashnik|1989|loc=Section 2.51}} Thus Bernoulli's formula can be written
: <math> \sum_{k=1}^n k^c = \sum_{k=0}^c \frac{B_k}{k!}c^{\underline{k-1}} n^{c-k+1}</math>

if {{math|''B''<sub>1</sub> {{=}} 1/2}}, recapturing the value Bernoulli gave to the coefficient at that position.

The formula for <math>\textstyle \sum_{k=1}^n k^9</math> in the first half of the quotation by Bernoulli above contains an error at the last term; it should be <math>-\tfrac {3}{20}n^2</math> instead of <math>-\tfrac {1}{12}n^2</math>.