Talk:Bernoulli number

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Bernoulli's vs. Faulhaber's formula[edit]

I do not understand your last changes. For example it now reads: "Bernoulli's formula was generalized by V. Guo and J. Zeng to a q-analog (Guo & Zeng 2005)." Before it read: "Faulhaber's formula was generalized by V. Guo and J. Zeng to a q-analog (Guo & Zeng 2005)." The paper which is referenced (did you look at it?) is: "Guo, Victor J. W.; Zeng, Jiang (2005), "A q-Analogue of Faulhaber's Formula for Sums of Powers". For me your changes make little sense. Please, can you comment on it? Wirkstoff (talk) 21:26, 3 September 2009 (UTC)

First of all, article says that "Bernoulli's formula is sometimes called Faulhaber's formula", i.e., "Bernoulli's formula" and "Faulhaber's formula" are synonymous. However, for consistency the article should use one of them while mentioning the other as equivalent. Currently, "Bernoulli's formula" is used as primary one. Second, my changes were not only about replacing "Faulhaber's formula" with "Bernoulli's formula" in one sentence, I've also split historical and factual information (that were messed up) between "History" and "Sum of powers" sections. Hence, I'm restoring my changes. Maxal (talk) 00:43, 4 September 2009 (UTC)
"First of all, article says that "Bernoulli's formula is sometimes called Faulhaber's formula" Yes, but to call 'Bernoulli's formula' 'Faulhaber's formula' is an error! i.e., "Bernoulli's formula" and "Faulhaber's formula" are synonymous. "... the other as equivalent." This is not the case! Please read the formulas and compare. These are very different formulae. Please read what Donald Knuth writes. "Faulhaber never discovered the Bernoulli numbers" so he can not have stated Bernoulli's formula which are based on the Bernoulli numbers. Hence, I'm restoring my changes. Please reconsider. For better reference I save the section here.

Faulhaber's formula

Bernoulli's formula is sometimes called Faulhaber's formula. There is no evidence which justifies this nomenclature. Johann Faulhaber found remarkable ways to calculate sum of powers but he never stated Bernoulli's formula.

Faulhaber realized that for odd m, Sm(n) is not just a polynomial in n but a polynomial in the triangular number N = n(n + 1)/2. For example Faulhaber's formulas read as follows:

 1 + 2 + \cdots + n = N;
 1^2 + 2^2 + \cdots + n^2 = N \left(2n+1\right) /3 ;
 1^3 + 2^3 + \cdots + n^3 = N^2.

To call Bernoulli's formula Faulhaber's formula does injustice to Bernoulli and simultaneously hides the genius of Faulhaber as Faulhaber's formula is in fact more efficient than Bernoulli's formula. According to Knuth (Knuth 1993) a rigorous proof of Faulhaber’s formula was first published by Carl Jacobi in 1834 (Jacobi 1834). Donald E. Knuth's in-depth study of Faulhaber's formula concludes:

“Faulhaber never discovered the Bernoulli numbers; i.e., he never realized that a single sequence of constants B0, B1, B2, … would provide a uniform

 \quad \sum n^m = \frac 1{m+1}\left( B_0n^{m+1}-\binom{m+1}1B_1n^m+\binom{m+1} 2B_2n^{m-1}-\cdots +(-1)^m\binom{m+1}mB_mn\right)

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  \sum n^m from polynomials in N to polynomials in n.” (Knuth 1993, p. 14)

Faulhaber's formula was generalized by V. Guo and J. Zeng to a q-analog (Guo & Zeng 2005). Wirkstoff (talk) 10:55, 4 September 2009 (UTC)

While I agree that the name "Faulhaber's formula" may not be historically justified enough to replace the name "Bernoulli's formula", Wikipedia is not a place to correct this kind of historical unfairness. What is currently called "Faulhaber's formula" is the same as "Bernoulli's formula" (see Faulhaber's formula and Faulhaber's formula at MathWorld), and from a factual point of view it does not matter what Faulhaber actually discovered. We cannot drop "Faulhaber's formula" name based on Knuth's investigation, because many research papers are using the name "Faulhaber's formula" in place of "Bernoulli's formula" (Guo and Zeng paper is an example). Maxal (talk) 11:24, 4 September 2009 (UTC)
Moreover, the current article is about Bernoulli numbers, not Bernoulli's formula. The piece of information that you quoted above is more appropriate for the article about Bernoulli's formula which is Faulhaber's formula. I'm copying it there to form "History" section. Maxal (talk) 11:48, 4 September 2009 (UTC)
You write: "We cannot drop "Faulhaber's formula" name based on Knuth's investigation, because many research papers are using the name "Faulhaber's formula" in place of "Bernoulli's formula" (Guo and Zeng paper is an example)." Yes, of course we can, even more, we have to! Knuth is an worldwide authority, MathWorld an advertising private website with no authority. You write: "Guo and Zeng paper is an example". So I will not argue longer with you as you obviously do not care about even a minimal amount of accuracy even if it could be achieved with two mouse clicks only. The link to the paper is in the article, download and READ before you tell us your fantasy. Guo and Zeng are writing about what they say they write about in the title of their refereed paper: Faulhaber's formula. It is important to differ between the formulas of Faulhaber and Bernoulli and their is a simple way to do it: to call Faulhaber's formula Faulhaber's formula and Bernoulli's formula Bernoulli's formula. Knuth does so, Guo and Zeng do so. MathWorld and you not. Why do you think we should follow you and Mathworld and not Knuth and (besides others) Guo and Zeng? Wirkstoff (talk) 14:19, 4 September 2009 (UTC)
I'm not saying that everybody should ultimately follow MathWorld (or any other source), I'm just stating that naming of Bernoulli's formula as "Faulhaber's formula" exists and we cannot disregard that fact. If you want to distinguish Bernoulli's formula from Faulhaber's one - please do. But the best place to start mentioning the difference is the Faulhaber's formula article. Having speculations about the difference between the two formulas won't make sense while the Faulhaber's formula article (in Wikipedia!) is exposing Bernoulli's formula. Actually, I don't mind reverting the statement about Guo and Zeng generalizing the Faulhaber's formula — my major point is that this statement should be in the "Sum of powers" section not the "History" one. Maxal (talk) 02:49, 5 September 2009 (UTC)

Kowa Seki, again, and why I reverted[edit]

AMorozov edited the lead. It says now: "They [the Bernoulli numbers] were first studied by the Swiss mathematician Jakob Bernoulli and the Japanese mathematician Seki Kōwa at around the same time." This needs some citation.

The introduction says: "At approximately the same time in Japan an equivalent method for calculating sums of powers was discovered by Seki Kōwa. However, Seki did not present his method as a formula based on a sequence of constants." Note the difference: 'method for calculating sums of powers' versus 'a single sequence of constants B0, B1, B2, … which provide a uniform formula for all sums of powers'. This article is about these constants, not about 'methods for calculating sums of powers'.

AMorozov, please give a reliable reference which affirms that the constants in Bernoulli's formula were known to Seki Kōwa. This is absolutely necessary for such a claim. Wirkstoff (talk) —Preceding undated comment added 18:59, 16 August 2009 (UTC).

Actually, I don't really have any problems with your reversion. To be frank I am no mathematician myself, and was only trying to add what the Seki Kōwa page seemed to claim. The details of how Kōwa's work relates to Bernoulli's is lost on me, and I trust your better judgment on whatever history has transpired there. ~ AMorozov 〈talk〉 23:05, 17 August 2009 (UTC)
My knowledge of old Japanese mathematics is very limited. I just want to see a reliable reference. If there exists one this clearly should be added to the Bernoulli number page. If there exists none to give reference to a Wikipedia page is of course absurd as long as there is no better founded information on this page. The Seki Kōwa page says: "For example, discovery of Bernoulli numbers (published in 1712) ... are attributed to him." No citation given. It would be great to add to the Bernoulli number page a facsimile of the page of Kōwa's work where he demonstrates the 'Seki' numbers -- if it exists. What I know is that Kōwa used the binomial numbers to calculate the sums of powers (and there is indeed a figure of Pascal's triangle in his work). At this time other methods to calculate sums of powers were known. For example Faulhaber's formula, which predates Bernoulli's work by 82 years (and which Bernoulli knew). If it were true that Faulhaber invented the Bernoulli numbers, as the Wikipedia page on Faulhaber's formula (wrongly) claims, then we need not speculate about a possible anticipation of the Bernoulli numbers by Seki. Don't use Wikipedia as a primary source when you write for Wikipedia. Wirkstoff (talk) 11:15, 18 August 2009 (UTC)
"In 1712, one year before Bernoulli's Ars Conjectandi (1713), Seki obtained Bernoulli numbers by Ruisai shosa-ho." Selin, Helaine (1997), Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer, p. 891, ISBN 0792340663, retrieved 2009-08-18 Dominus (talk) 17:11, 18 August 2009 (UTC)
"...[Seki] wrote about diophantine equations, magic squares, and Bernoulli numbers (before Bernoulli)..." Poole, David (2005), Linear algebra: a modern introduction, Cengage Learning, p. 279, ISBN 0534998453, retrieved 2009-08-18 Dominus (talk) 17:18, 18 August 2009 (UTC)
"...the Japanese mathematician Takakazu Seki Kôwa (1642–1708) discovered [the Bernoulli] numbers before Jacob Bernoulli I." Styan, George P. H.; Trenkler, Götz, Journal of Applied Mathematics and Decision Sciences 2007, doi:10.1155/2007/13749 http://hindawi.com/GetPDF.aspx?doi=10.1155/2007/13749 |url= missing title (help), retrieved 2009-08-18 Dominus (talk) 17:28, 18 August 2009 (UTC)
I undid my revision. I will try to look into these references. Thanks for your help to clarify this point. Nevertheless, I have my doubts. These references are certainly no primary sources. A facsimile would be the best proof. Anyone out there who is in the position to provide such a facsimile? Wirkstoff (talk) 20:38, 18 August 2009 (UTC)
Wikipedia policy prefers secondary sources to primary sources. A facsimile of the original page would not be a reliable source, since it would depend on too much interpretation by Wikipedia editors: is the page authentic? Is it from the work that is claimed to be from? Does it really say what it is claimed is said? Wikipedia editors are not competent to make these sorts of decisions. In contrast, when an article in an encyclopedia of non-Western science, published by a reliable technical publisher, and presumably written by a credentialed scholar of the subject, asserts flat out that Seki had priority, that is quite sufficient to meet Wikipedia's standards, unless there is some specific reason to doubt this particular source. Please familiarize yourself with Wikipedia's policies on verifiability. Thanks. —Dominus (talk) 22:33, 18 August 2009 (UTC)
The reference to the "Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer, p. 891" meets no doubt not only Wikipedia's standards. So why was it not included to the Bernoulli page when the reference to Seki was made? This would have made this discussion unnecessary. To add a facsimile of Seki's calculation like the one which shows Bernoulli's calculation would nicely illustrate the article in my opinion. I don't see that this would touch "Wikipedia's policies on verifiability" in any way. It would be similar to the article Pascal's triangle #History which presents the arithmetic triangle in different historic forms. Wirkstoff (talk) 13:57, 19 August 2009 (UTC)
This is all very Greek to me, but it might be a good idea to add them as inline citations, now that someone took the pain to look it up. ~ AMorozov 〈talk〉 15:26, 19 August 2009 (UTC)
Well, I've gone ahead and put in the citations, and reworded things a little in the lead. I hope this is alright with you guys. ~ AMorozov 〈talk〉 17:27, 20 August 2009 (UTC)
Thanks very much. I agree with Wirkstoff that it would be excellent to be able to display a page from Seki's work that includes the Bernoulli numbers. I did not mean to suggest that displaying it would be a policy violation, only that depending on it as a reference to prove Seki's discovery would be. —Dominus (talk) 18:20, 20 August 2009 (UTC)
Perhaps the wording could be optimized? "They were discovered in 1712 by the Japanese mathematician Seki Kowa.[1][2][3] This was followed immediately by the Swiss mathematician Jakob Bernoulli who published his findings in the Ars Conjectandi in 1713..." According to Wikipedia Jakob Bernoulli died on 16 August 1705. Wirkstoff (talk) 17:42, 21 August 2009 (UTC)
I took a shot at it. According to Wikipedia, Seki's publication was also posthumous; he died in 1708. —Dominus (talk) 17:53, 21 August 2009 (UTC)
Seki's tabulation of Bernoulli numbers
The facsimile that Wirkstoff wants seems to be here. (Attached at right.) The main body of the table is evidently Pascal's triangle, written in rod numerals. The caption claims that the bottom row of values is the Bernoulli numbers, but I haven't been able to understand it yet. —Dominus (talk) 18:06, 21 August 2009 (UTC)
I can read it now, more or less; it is certainly the Bernoulli numbers. The rightmost value in the bottom row is B_0, and is a special case, 全 ("everything"). Each other entry is either 空 (zero, literally, "empty") or a fraction in the form d分之n, where d is the denominator and n is the numerator. There is a small double mark 二 in the margin before d and a small single mark 一 in the margin after n. So for example the fraction in the third column from the right (B_2) is written as 六分之一, which means 1/6; the fraction in the seventh column (B_6) is written 四十二分之一, which means 1/42. The sign is given by the characters under the marginal 一 mark, with 加 ("add") indicating positive, and 減 ("reduce") indicating negative.
So the complete reading of the table, right to left is: "everything", 1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0.
For a table of Japanese numeral symbols, see Japanese numerals. —Dominus (talk) 18:50, 21 August 2009 (UTC)
I have just discovered that the fractional notation is explained at Chinese_numerals#Fractional_values. —Dominus (talk) 19:39, 21 August 2009 (UTC)
These are very valuable contributions. Thank you Dominus! Wirkstoff (talk) 22:49, 21 August 2009 (UTC)
Good work indeed, Dominus. I hope you guys don't mind me adding the pic as a thumbnail in the article. ~ AMorozov 〈talk〉 05:34, 22 August 2009 (UTC)
Thank you both for your kind words. I enjoyed doing the research. I have removed the references to Pool and to Styan, which I felt were not especially trustworthy. (For example, Poole is a linear algebra textbook, repeating the claim about Seki in passing; I have no reason to believe that Poole is an authority on Japanese mathematics.) in place of those, I added a reference to Smith and Mikami History of Japanese Mathematics (1914), which addresses the matter in more detail. —Dominus (talk) 06:32, 22 August 2009 (UTC)
Looks good to me (: ~ AMorozov 〈talk〉 06:37, 22 August 2009 (UTC)

Inconsistency[edit]

Inconsistency here: Bernoulli numbers as defined on Bernouilli number page are alternately negative and positive.

But Taylor series for Tan(x) and Cot(x) use Benouilli numbers that are all positive. Formula for Taylor series should use absolute value |Bn| and not Bn.

Or Bernouilli numbers should be defined as all positive.

There is no inconsistency at all. The article says:
The Bernoulli numbers also appear in the Taylor series expansion of the tangent...
and that is exactly correct; they appear in the Taylor series expansion. The Taylor series for tan(x) is:
\tan x = \sum^{\infin}_{n=1} \frac{B_{2n} (-4)^n (1-4^n)}{(2n)!} x^{2n-1}
in which B_{2n} denotes the 2nth Bernoulli number; so the Bernoulli numbers do indeed appear in the Taylor series expansion. The multiplication by (-4)n ensures that all the coefficients are positive. I hope this clears things up for you. -- Dominus 12:40, 29 July 2005 (UTC)

Yes it does, thanks.

Another question:

\sum_{k=0}^{m-1} k^n = 0^n + 1^n + 2^n + \cdots + {(m-1)}^n

for various fixed values of n. The closed forms are always polynomials in m of degree n + 1

But term of highest degree appears to be (m-1)^n which has a degree of 'n'.

there is a sum of m terms involved here. If you take in the sum of powers only those summands say with k   >   m/2− 1 then the cut-off sum will be already bigger than
 (m/2) (m/2)^n ={1\over {2^{n+1}}} m^{n+1} \ .
Therefore the order of magnitude (for fixed n) is asymptotically  O(m^{n+1}) and not  O(m^{n}) (the exact order is  {1\over{n+1}}\,m^{n+1}+O(m^{n})).  .

Big O notation[edit]

Thanks for the timely revert, Dmharvey. I mistakenly thought you were using the omega notation in its historical sense, equivalent to big O. Elroch 22:21, 10 February 2006 (UTC)

Relationship of Bernouilli numbers to Riemann zeta function[edit]

I decided to change the language used to describe the relationship of the Bernouilli numbers to the Riemann zeta function, which grated with me as it stood. As I understand it, two sequences are the same "up to a factor" if one is a constant multiple of the other, and describing one sequence as "essentially" another sequence was wooly language at best. Elroch 22:29, 10 February 2006 (UTC)

I would say that the Bernoulli numbers are related to the Riemann zeta function for positive integers, and not for negative integers, as stated here. The usual convention of the zeta function is to write fractions. (Gio74 (talk) 12:21, 12 September 2009 (UTC))

Kowa Seki[edit]

A recent edit removed the assertion that the Bernoulli numbers were first studied by Bernoulli, and instead attributed them to the great Japanese mathematician Seki "in 1683", and asserting that Bernoulli did not study them until "the 18th Century". The implication here is that Bernoulli was greatly anticipated by Seki. But Bernoulli (1654-1705) and Seki (1642-1708) were contemporaries, and without two dates, I am reluctant to believe any claims of priority.

If anyone has any real information, I would be glad to hear it. Meantime, I am going to change the article again to note Seki's discovery. -- Dominus 01:15, 28 March 2006 (UTC)

the great JAPANESE mathematician did it FIRST! KOWASHITA! Washowasho! 128.226.162.163 (talk)japniggerfromhell —Preceding comment was added at 17:39, 27 November 2007 (UTC)

Recursive Definition[edit]

I could be mistaken, but I don't see the recursive definition as being recursive. Maybe the 0 on the right hand side should be Bm+1? Psellus 23:19, 7 July 2006

It's recursive but not phrased in an explicitly recursive manner. For example, try substituting m = 3 and then solve for B3. If you like you can rearrange the equation to show the recursion more explicitly, but it's quite elegant the way it's written currently. Dmharvey 23:48, 7 July 2006 (UTC)

I was afraid it would turn out to be something like this. OK, I will look at it harder. Thanks very much. Psellus 23:54, 7 July 2006 (UTC)

Asymptotic expansion[edit]

I made a check about that formula, it is ok but not especially good. The ordinary formula which is B(n) = 2*n!/Pi^n/2^n (shown just above) is much better : for n=1000 the approximation is good for the first 300 digits(!) and that one is good to something like 20 digits. I don't see the point of showing that formula. Maybe is it new but by far less efficient than the usual one. In my opinion that formula should be removed and we could maybe put a reference to that guy that found it. The litterature on bernoulli numbers is quite large and I am sure many authors looked at these pages.

 Plouffe 08:29, 18 February 2007 (UTC)

It is apparently the ordinary asymptotic formula with a Stirling-like approximation for the factorial. Perhaps it could be useful for computing Bernoulli numbers with double (16-digit) precision? Nevertheless, it seems redundant to mention in this article since anyone who'd be interested in an n-digit approximation for large Bernoulli numbers could simply take the original asymptotic formula and substitute an n-digit factorial approximation of his own choice. Maybe the section could be reworded to say something to this effect? By the way, the article should also mention how to compute Bernoulli numbers exactly with the zeta function and the von Staudt-Clausen theorem. Fredrik Johansson 19:50, 18 February 2007 (UTC)
An asymptotic expansion using the factorial n! is in many cases not so useful as an expansion without n!. This is perceived for example on Mathworld [Bernoulli Number, formula (35)] which displays the 'standard' asymptotic form. However, the formula on Mathworld is by far not as efficient as the formula given here. Moreover, not every user of the Bernoulli numbers is ready to substitute first an approximation of the factorial in some other approximation of the Bernoulli numbers. Overall the formula looks like a good balance between simplicity and efficiency in a great many practical cases. —The preceding unsigned comment was added by 84.136.133.103 (talk) 23:41, 3 April 2007 (UTC).

Another set of identities[edit]

Hi -

I've got another set of identities based on an recursive definition, which I found recently, and which is not much known.

However, scanning some internet resources, I found the basic idea also mentioned in Zhi Wei Sun's article about "courious results concerning Bernoulli and Euler-polynomials", where he cites this relation according to von Ettinghaus in the early 19'th century.

This recursive definition relates some basic number-theoretic sequences in a very simple scheme. Would it be appropriate to link to this article of mine?

GeneralizedBernoulliRecursion.pdf

--Gotti 21:08, 18 February 2007 (UTC)

No neutral point of view - removed paragraphs[edit]

There is no place for such remarks:

So Pavlyk could have saved his employer Wolfram a lot of resources had he used free and open source software. Pavlyk wrote: "The numerator begins with 571642756... and ends with ...597039303." This result is false as a quick check with the asymptotic formula in the next section reveals. Interestingly, none of the math-professors in the newsgroup seem to have noticed.

The following removed paragraphs are personal opinions of Peter Luschny, see an essay of Luschny The Bernoulli Confusion. This guy (is he an expert on number theory or a graduate mathematician at all? I don't think so!) rigidly claims, against the consensus of experts, to change the definition of Bernoulli numbers so that B(1) = 1/2. For both cases, B(1) = 1/2 or B(1) = -1/2, there are advantages and disadvantages. There is no favored viewpoint! One can start with the summation of powers up to n-1 (simple) or the Riemann zeta function on the negative x-axis (advanced).

This convention is looked at as an unhappy choice by many mathematicians as it is not in accordance with the viewpoint suggested by the Riemann zeta function. Therefore more and more modern writers depart from it. John H. Conway for example uses the convention Bn = Bn(1) in his books. He remarked that this convention also makes "certain formulas more aesthetically pleasing to our eyes".
and is often preferred by number theorists
In conclusion: The Bernoulli numbers Bn admit a variety of different representations. Since not all of these agree with one another conventions are introduced. Which among them should be adopted may depend not only on mathematical insight. But mathematical insight, simplicity and aesthetics suggest that the convention supporting the concordance is the best one in almost all of the cases.
Indeed for many mathematicians there is no valid alternative to the above concordance as they agree with G. H. Hardy: "The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics."

Cryptic passage[edit]

After the words

The combinatorics of this representation can be seen from:

There follows a sequence of expressions in which the pattern is clear EXCEPT for the pattern in the plus and minus signs. I left a message on the talk page of the user who wrote that passage inquiring about it, but he's been away from us for six days now, and left no email address. Can anyone explain that pattern? Michael Hardy (talk) 02:26, 8 July 2008 (UTC)

My understanding is that the sign is (−1)n+k multiplying the term containing the number of words of length n on k distinct symbols, as in the summation formula which precedes the passage in question. I don't know if there is an easy way to make this more visually apparent in the formula. siℓℓy rabbit (talk) 03:00, 8 July 2008 (UTC)

No, that is inconsistent with what appears on the page. The cases of B1 and B3 should have a minus sign in the term with the empty set. Either you're wrong or what user:Wirkstoff wrote there is wrong. Michael Hardy (talk) 04:01, 8 July 2008 (UTC)

...OK, maybe he's just neglecting the sign in the case where the term is zero. Not the clearest way of writing. Michael Hardy (talk) 04:16, 8 July 2008 (UTC)

Sign[edit]

I remove this:

The von Staudt-Clausen representation of the Bernoulli numbers gives in a natural way B1 = 1/2.This confirms the observation, that the deeper arithmetical properties of the Bernoulli numbers do not support the convention B1 = −1/2. (See also the section on conventions above.)

I hold this to be (unfounded) opinion. If all the v. Staudt sums were 1, it would be differenct, but they are, including all the odd values, infinity, (0 or 1), 1, .5, 1, .5, 1, .5, 1, .5, 1, ,5, 1, .5, 2, .5, -6... Who shall say that 0 or 1 is more natural for the second member of this series? Septentrionalis PMAnderson 00:50, 9 July 2008 (UTC)

Silly Rabbit, I think the change you reverted was correct[edit]

Look at this formula:

 B_{n}=\sum_{k=0}^{n}(-1)^{k+n}\frac{k!}{k+1}
\left\{\begin{matrix} n \\ k \end{matrix}\right\} \ .

Take n=1.

For k=0, the summand is (-1)^1 0!/1 (1 Choose 0), which is -1.
For k=1, the summand is (-1)^2 1!/2 (1 Choose 1), which is +1/2.

Hence, B_1 is -1/2.

However, in the change you reverted, you now have B_1 = 1/2.

I.e., (1 Choose 0) is |{\emptyset}|=1, not |\emptyset|=0.

Loisel (talk) 15:50, 9 July 2008 (UTC)

Except that \left\{\begin{matrix} n \\ k \end{matrix}\right\} is not a binomial coefficient, but a Stirling number of the second kind. It measures the number of distinct words of length n that can be formed out of a set of k symbols. Thus in particular
\left\{\begin{matrix} 1 \\ 0 \end{matrix}\right\} = 0
since exactly zero words of length one can be formed out of a set of zero symbols. siℓℓy rabbit (talk) 16:38, 9 July 2008 (UTC)

All right. I stand corrected. Loisel (talk) 16:47, 9 July 2008 (UTC)

Verifiability and fair attribution.[edit]

I affirm that I published the approximation formula and the inclusion formulas cited in the article. To the best of my knowledge they were new when published in January 2007. I think they should be published by Wikipedia if due attribution to the author is made. I think they should not be published without attribution.

Silly Rabbit removed the attribution to the author of the formulas but not the formulas nor did he gave any reason why he did so.

Why? Notability? Worthy of notice? I think they are worthy of notice.

Verifiability? It might very well be the case that this requirement is not met. Not in the sense of correctness, even Simon Plouffe says on this page that the formula is correct (in fact every freshman in mathematics should be able to verify them) but in the sense of 'source for quotations'.

But in this case the formulas are to be deleted, not the attribution only! The editorial practice which Silly Rabbit shows is more than questionable.

I respect the requirement of verifiability put forward by the rules of Wikipedia. But I also think that their is room for common sense. Is their a quote for each mathematical formula displayed on Wikipedia? I doubt that. I think it would be absurd to require this in such a narrow sense.

However, I can not agree with the arbitrariness shown in the editing practice of Silly Rabbit. Publish a result and give due credit or don't publish the result for whatever reason. Therefore I revert the paragraphs containing the formulas to something which is similar to what it was before they were included.

(*) I replace in the section 'Asymptotic approximation'

Substituting an asymptotic approximation for the factorial function in this formula gives an asymptotic approximation for the Bernoulli numbers. For example

 |B_{2 n}| \sim 4 \sqrt{\pi n} \left(\frac{n}{ \pi e} \cdot 
\frac{480 n^2 + 9}{480 n^2 -1}\right)^{2n}.

This formula (Peter Luschny, 2007) is based on an approximation of the factorial function given by Gergo Nemes in 2007. For example this approximation gives

 |B(1000)| \approx 5.318704469415522033\ldots\times 10^{1769} \,

which is off only by three units in the least significant digit displayed.

(*) by

 |B_{2 n}| \sim 4 \sqrt{\pi n} \left(\frac{n}{ \pi e} \right)^{2n}.

(**) And I replace in the section 'Inequalities'

The following two inequalities (Peter Luschny, 2007) hold for n > 8 and the arithmetic mean of the two bounds is an approximation of order n−3 to the Bernoulli numbers B2n.

 4 \sqrt{ \pi n} \left(\frac{n}{\pi e} \right)^{2n}  
\left[1 + \frac{1}{24n}\right] < |B_{2 n}| < 4 \sqrt{ \pi n} 
\left( \frac{n}{ \pi e} \right)^{2n} \left[1+\frac{1}{24n}
\left(1+\frac{1}{24n}\right)\right]

Deleting the squared brackets on both sides and replacing on the right hand side the factor 4 by 5 gives simple inequalities valid for n > 1. These inequalities can be compared to related inequalities for the Euler numbers.

For example the low bound for 2n = 1000 is 5.31870445... × 101769, the high bound is 5.31870448... × 101769 and the mean is 5.31870446942... × 101769.

(**) by

 4 \sqrt{ \pi n} \left(\frac{n}{\pi e} \right)^{2n}  < |B_{2 n}| 
<  5 \sqrt{ \pi n} \left(\frac{n}{\pi e} \right)^{2n}

Peter Luschny

P.S. http://www.luschny.de/math/primes/bernincl.html
http://www.luschny.de/math/factorial/approx/SimpleCases.html —Preceding unsigned comment added by 85.179.164.224 (talk) 16:30, 22 December 2008 (UTC)

The reason I removed the pseudo-referenced material was that there was no reference in the References section, so that the needs of WP:V were clearly not met. As a reader, I was unable to verify the material. If you are able to provide a proper reference to the material, including a properly peer-reviewed journal, then it can stay in the article. Otherwise, Wikipedia is not the place to publish original research, even if it appears elsewhere in a self-published resource. Is this published in a peer-reviewed source or isn't it? The links you have provided, while interesting and probably suitable for an external links section of the article, do not seem to me suitable to base sections of the article on, per Wikipedia policy. At any rate, if you give a Harvard reference like (Luschny, 2007) then you really ought also to say where the material was published, just like if you were to try to write an article in a scientific journal. siℓℓy rabbit (talk) 22:32, 22 December 2008 (UTC)

-
>> The reason I removed the pseudo-referenced material was that there was no reference
>> in the References section, so that the needs of WP:V were clearly not met. As a reader,
>> I was unable to verify the material. If you are able to provide a proper reference to
>> the material, including a properly peer-reviewed journal, then it can stay in the article.
>> Otherwise, Wikipedia is not the place to publish original research, even if it appears
>> elsewhere in a self-published resource. Is this published in a peer-reviewed source or isn't it?
What are telling me? I wrote: "It might very well be the case that this
requirement is not met." And I wrote "I respect the requirement of verifiability
put forward by the rules of Wikipedia." So the type of your answer is absolutely
misplaced. You do not need to tell me what I can read and obviously did read on
the respective pages of Wikipedia policy. You are ignoring what I wrote but give
the impression that there is a need to tell me about. This is quite impertinent.
And at the same time you do not answer my main question: Why did you leave the
formulas on the page and deleted the attribution only? This is what I really want
to know. This is why I object your editorial practice. This does not meet the needs
of Wikipedia in any way. Your answer is still missing.
>> The links you have provided, while interesting and probably suitable
>> for an external links section of the article, do not seem to me suitable to base
>> sections of the article on, per Wikipedia policy.
>> At any rate, if you give a Harvard reference like (Luschny, 2007) then you really
>> ought also to say where the material was published, just like if you were to try
>> to write an article in a scientific journal.
How can you say that? I did not give "a Harvard reference like (Luschny, 2007)"
nor any other kind of reference. There was a discussion in the newsgroup
de.sci.mathematik where I spoke about this formula.
http://groups.google.com/group/de.sci.mathematik/browse_frm/thread/4a980ba9ca71f0df/
I looked up the editing history. From what I see I conclude that I published the
formula on my website and that someone put this formula to Wikipedia.
I also often wrote about integer sequences in newsgroups and there are quite
a few of these sequences to be found on OEIS published by people who read these
threads and care more to publish these sequences then I do.
I reiterate the question:
As a reader, are you able to verify the material which is attributed to Ramanujan?
Why did you not delete the attribution to Ramanujan and left only his formulas?
"The reason I removed the pseudo-referenced material was that there was no reference
in the References section, so that the needs of WP:V were clearly not met."
Ok. Where are references to Ramanujan's work? Why did you not remove it?
As a reader, are you able to verify the material which is "pseudo-referenced"
to von Ettingshausen?
Why did you not delete the attribution to von Ettingshausen and left only his formulas?
"The reason I removed the pseudo-referenced material was that there was no reference
in the References section, so that the needs of WP:V were clearly not met."
Ok. Where are references to von Ettingshausen's work? Why did you not remove it?
And so on. It is not difficult to extend this list. As far as I can judge your
editing does not work on the standards of Wikipedia but on your own ones, and
these are quite arbitrary.
Peter Luschny
I deleted the attribution and replaced it with a {{citation needed}} tag. The material was not properly cited, and Google scholar did not turn up any hits for Luschny 2007. I'm sorry that you were unable to publish your original research, but Wikipedia is not the place to do it. I also acknowledge that there are other substantial problems with the referencing of the article: this was just the first one that I noticed, and so attempted to fix. I am now in the process of adding {{citation}} templates, so that other bogus references can be worked out appropriately. siℓℓy rabbit (talk) 01:23, 23 December 2008 (UTC)
I think there are two issues.
  1. Mr. Luschny wants fair attribution. This is easy; just replace "(Peter Luschny, 2007)" with "due to Peter Luschny".
  2. Wikipedia needs verifiable sources. This is easy and already done; just replace the broken citation with {{citation needed}}. If a citation is not provided eventually, then the fact will be removed, but there is no need to do this immediately if the fact itself is not disputed (it may be published later with attribution to Luschny by some other mathematician who read the estimate on the usenet newsgroup).
If there are no objections, I suggest adding back the sharper asymptotics and inequalities by Peter Luschny with the corresponding attribution and citation needed cleanup tags. Finding specific publications of the estimates is actually fairly hard, and often a decade goes by after the community knows of the sharper estimate before someone actually records it in a reliable source (for instance, how many subgroups does the symmetric group on n points have? I think there is 7 year gap between announcement of the sharp result in the community and the first reliable source stating that it is true, only the nearly sharp estimate is proven that I have seen). JackSchmidt (talk) 01:46, 23 December 2008 (UTC)
This seems to be a reasonable compromise. At any rate, this at least provides the desired attribution without masquerading as a citation, as the "(Luschny, 2007)" format certainly suggested. siℓℓy rabbit (talk) 02:05, 23 December 2008 (UTC)
Fair enough, but I am still quite skeptical. The previous version of this article said it relied on work of Nemes. Tracking down this reference I found it was published in a place called Stan's Library which boasts on its front page "Get published for free with the unique warranty that your work will receive immediate attention!" (the web sites emphasis not mine). This seems pretty far from from peer reviewed, and I am skeptical if any reasonable source will show up. If they don't appear in some reasonable time frame we should remove the material. If it turns out there is a 7 year gap, we should wait 7 years then include it. How many times was it announced that the Poincare conjecture had been proven? Thenub314 (talk) 14:40, 23 December 2008 (UTC)
I think the material should be removed. I tend towards including things which others would delete but I do think that the wiki notability guideline should be adhered to, especially in the case of an author with their own original research. Dmcq (talk) 21:26, 23 December 2008 (UTC)

(unindent) The material at stake is a trivial application of the Stirling formula that anyone can derive on the back of an enveloppe. That doesn't warrant either publication in a peer-reviewed journal (try it and you'll be politely directed to a community college mathematical gazette at best) or formal citation. Bikasuishin (talk) 13:23, 31 December 2008 (UTC)

True, then again I don't think WP need mention things if nobody else does. Does the approximation bounds section need to be in at all? - people will just wonder if they are missing something and wonder why it is so wide with that multiply by 5 in there. Dmcq (talk) 14:41, 31 December 2008 (UTC)
Actually I already cut the "inequality" section and some of the blatter in the "asymptotic" section. It might be OK to keep the asymptotic itself, though; some mathematical software packages like PARI can compute approximations of the Bernoulli numbers for large indices, so it's possible that they make use of a formula such as that one. One would need to check the relevant papers. Bikasuishin (talk) 15:09, 31 December 2008 (UTC)

Are these values correct?[edit]

I tried to check the table below with the asymptotic formula now given on the content page. Some digits are true, some not. I think the table has to be put under quarantine here until someone has checked it. Hopefully there is a better asymptotic expansion so that we can decide this question. Don't forget to tell us how you did! And please give a checkable reference.

n N D Bn = N / D
102 -9.4598037819... × 1082 33330 -2.8382249570... × 1078
103 -1.8243104738... × 101778 342999030 -5.3187044694... × 101769
104 -2.1159583804... × 1027690 2338224387510 -9.0494239636... × 1027677
105 -5.4468936061... × 10376771 9355235774427510 -5.8222943146... × 10376755
106 -2.0950366959... x 104767553 936123257411127577818510 -2.2379923576... × 104767529
107 -4.7845869850... × 1057675291 9601480183016524970884020224910 -4.9831764414... × 1057675260
108 -1.8637053533... × 10676752608 394815332706046542049668428841497001870 -4.7204482677... × 10676752569

The displayed values for n = 107 and n = 108 were computed in less than one second with the von Staudt-Clausen formula and the asymptotic formula given below.

Peter Luschny —Preceding unsigned comment added by 85.179.164.224 (talk) 18:32, 22 December 2008 (UTC)

Who found the Bernoulli numbers first[edit]

(I read in J.H. Conway's "The Book of Numbers", page 108:) Those constants in Faulhaber's formula are known as Bernoulli numbers because they are discussed at length in Ars Conjectandi (1713), Jacob Bernoulli's posthumous masterpiece, in which the latter points out that they were originally discovered by Johann Faulhaber.

(This is not quite in line with what the Wikipedia section says.) —Preceding unsigned comment added by Puddington (talkcontribs) 21:46, 19 April 2009 (UTC)

This page says "Faulhaber did not discover the Bernoulli numbers but Jacob Bernoulli refers to Faulhaber in Ars Conjectandi published in Basel in 1713, eight years after Jacob Bernoulli died, where the Bernoulli numbers (so named by De Moivre) appear." McKay (talk) 04:04, 20 April 2009 (UTC)
No, Johann Faulhaber did not know these coefficients. He had wonderful and efficient formulas for summing the powers but not Bernoulli's coefficients. It is absurd and erroneous to call the second formula in the introduction "Faulhaber's formula". Wirkstoff (talk) 21:21, 29 May 2009 (UTC)

Mistake in Ars Conjectandi[edit]

Either there's a mistake in the png, or there's a mistake in the Ars Conjectandi itself. The last term of the sum(n9) should be -3/20 instead of -1/12. An easy way to verify this is to put in 1 for n. All the coefficients in that row, when added, should equal 1. Epte (talk) 06:18, 27 May 2009 (UTC)

Your observation is correct. No, it is not a mistake of the 'png'. However, this fact well known. Go ahead and add a comment to the section. Unfortunately I do not know to which printing of Ars Conjectandi this pseudo-facsimile refers to. You find one for example in the "Rare Book Collection" of "The Caltech Archives". There might be other printings without this error. Wirkstoff (talk) 21:13, 29 May 2009 (UTC)

Not new formula[edit]

There have been a few news reports of a 16 y.o Iraqi boy in Sweden with a "formula to explain the calculation of Bernoulli numbers." References [1] or Google translation of swedish news site. The second reference includes the formula

 \forall n,\ B_n = \sum_{i=0}^n \left( \frac{1}{i+1} \sum_{p=0}^i  (-1)^p C^p p^n\right)

—Preceding unsigned comment added by Zeimusu (talkcontribs) 09:21, 28 May 2009

Actually that C^p is C^p_i (i.e. the binomial coefficient). Nice formula, though I don't know how novel it is. A straightforward implementation would seem to lead to a summation of O(n2) terms, and each term involves the factor pn (whose size is on the order of O(n log n) bits). At a quick glance this would seem to be way slower than Harvey's O(n2 log(n)2+eps) method (mentioned in this article) but I could be wrong. --Jmk (talk) 12:29, 28 May 2009 (UTC)
In fact the formula seems very similar to Worpitzky's formula developed in 1883 (according to this very article). Is this the new formula, or what is it? --Jmk (talk) 14:14, 28 May 2009 (UTC)
Ok, in fact Dalarnas Tidningar clearly says (quoting Lars-Åke Lindahl from Uppsala university): Men det han har kommit fram till är välbekant och han har inte löst något som tidigare har varit okänt, säger Lindahl till TT. ("What he has developed is well known, and he has not found something that was unknown before.") So it would appear that the "new formula" is simply a re-discovery of Worpitzky's 1883 formula. Nice for a high-school kid, but nothing really new. --Jmk (talk) 15:27, 28 May 2009 (UTC)
Dalarnas Tidningar has now posted a clarification, basically stating that Mohammed has independently discovered a previously known formula. (This is also part of the online article, but seems to have been cut from the paper edition; perhaps that is the source of the confusion.) -- magetoo 18:09, 28 May 2009 (UTC)
If "Dalarnas Tidningar" was not able to give a journalistic clean account in the first place how were they able to verify by hindsight that the young genius did not read Wikipedia? I know a lot of young students who do so. Wirkstoff (talk) 21:38, 29 May 2009 (UTC)

Iraq-born teen cracks maths puzzle[edit]

I have redirected Mohamed Altoumaimi to this page, due to extensive media coverage (primarily in Swedish language media), eg:

--Mais oui! (talk) 20:18, 28 May 2009 (UTC)

Well, the popular press has been exaggerating their pants off. See the previous section here (Not new formula). The formula is already known since 1883 so there was no "puzzle" to be "cracked". --Jmk (talk) 20:53, 28 May 2009 (UTC)
I can't see it being more than some news hype, it might go somewhere else in the article but certainly not in the leader. Just not notable enough in this context. Dmcq (talk) 14:12, 29 May 2009 (UTC)
Here's a citation from Uppsala University to saying No new mathematical solution by Swedish Teen, so I'll remove the bit in the leader. Dmcq (talk) 14:25, 29 May 2009 (UTC)

I know, I can't believe it either. Teens nowadays are really starting to upstage us adults.

(here is the full article)

STOCKHOLM (AFP) – A 16-year-old Iraqi immigrant living in Sweden has cracked a maths puzzle that has stumped experts for more than 300 years, Swedish media reported on Thursday.

In just four months, Mohamed Altoumaimi has found a formula to explain and simplify the so-called Bernoulli numbers, a sequence of calculations named after the 17th century Swiss mathematician Jacob Bernoulli, the Dagens Nyheter daily said.

Altoumaimi, who came to Sweden six years ago, said teachers at his high school in Falun, central Sweden were not convinced about his work at first.

"When I first showed it to my teachers, none of them thought the formula I had written down really worked," Altoumaimi told the Falu Kuriren newspaper.

He then got in touch with professors at Uppsala University, one of Sweden's top institutions, to ask them to check his work.

After going through his notebooks, the professors found his work was indeed correct and offered him a place in Uppsala.

But for now, Altoumaimi is focusing on his school studies and plans to take summer classes in advanced mathematics and physics this year.

"I wanted to be a researcher in physics or mathematics; I really like those subjects. But I have to improve in English and social sciences," he told the Falu Kuriren. —Preceding unsigned comment added by Philemmons (talkcontribs) 06:29, 30 May 2009 (UTC)

I saw the article as well. However redirecting may not be the best way to go. At the very least, he should have a short blurb in this article, and may even be notable enough for his own article. He made the front page of yahoo...Drew Smith What I've done 07:14, 30 May 2009 (UTC)

What was his contribution? The yahoo blurb doesn't actually include or explain the formula. Also, I second the view that a simple redirect is insufficient; I actually think it's worse than nothing. Without reference to the kids name anywhere in the article, it seems more mocking than helpful. -(anonymous)

Well I think the formula is probably a little to complex to post on wikiedia. Also, formulas are a pain in the ass to do in wikicode, so that further complicates the problem of posting it on wikipedia. I have posted my views on the subject at the helpdesk and the village pump, so if I can get confirmation from a few more editors I will begin work on his article.Drew Smith What I've done 08:40, 30 May 2009 (UTC)
If you will read the section of talk that you are writing in you will see that it is not a new formula and Uppsala has had to issue a statement refuting it. Secondly Wikipedia already did have a close variant of the formula as you can see in the previous talk section. In fact probably the only thing that might be noteworthy is that there was a media circus about it and Uppsala had to issue that statement. Dmcq (talk) 08:47, 30 May 2009 (UTC)
What statement are you reffering to? The only thing I've seen on this kid says that Uppsala has verified his work.Drew Smith What I've done 10:43, 30 May 2009 (UTC)
This from Uppsala University saying No new mathematical solution by Swedish Teen. Dmcq (talk) 14:25, 29 May 2009 (UTC)
This isn't refuting his "claim to fame" it is just re-iterating that he isn't the first to do it. The article I read said that very few experts have been able to come up with a solution, and that the only thing unique about the boy is that he did it so young. Which brings me to the part about being offered a place in Uppsala. Yahoo claims he was offered a place, Uppsala claims he was not given a place. Both are "reliable" sources. This sounds to me like Uppsala is "saving face" at having been turned down, by using some fancy wording. Either way, I think the kid is notable enough for his own article, but only barely. Many "child prodegies" get there own article, so I think he should as well. Again, I have posted this elsewhere, and will wait for other editors opinions before starting work on it,Drew Smith What I've done 11:13, 30 May 2009 (UTC)
Not a new formula. Bit about very few experts have been able to come up with a solution is unmeasurable nonsense, they wouldn't be able to publish that they had come up with the same solution. Conspiracy theory about Uppsala unfounded. Yahoo reliable source compared to university laughable. 16 not a child as far as maths prodigies concerned, need to start earlier, hmm maybe I'm a prodigy I figured how to do cube roots by the time I was nine - where's my wikipedia page? Dmcq (talk) 11:31, 30 May 2009 (UTC)
I am simply trying to get other editors opinions on whether this is notable or not. And I don't know what the standards are where you come from, but in my school every one was doing cube roots by age nine.Drew Smith What I've done 11:41, 30 May 2009 (UTC)
Pictogram voting info.svg Administrator note: Given the edit-warring about whether this kid should be in the article, I have fully protected the article for a week. Please gain consensus about this and request unprotection once consensus exists. Regards SoWhy 17:22, 30 May 2009 (UTC)

Reference 3 links to an article on Naharnet. The link is temporary and no longer goes to the correct story. This link to the same story may be permanent: http://www.naharnet.com/domino/tn/NewsDesk.nsf/AwayPolitics/5DAFD8132D1DFCC5C22575C4002BAD8F?OpenDocument Xot (talk) 20:02, 30 May 2009 (UTC)

I'm just here to say that Jmk and Dcmq are right. Melchoir (talk) 22:43, 30 May 2009 (UTC)

No, he should not be mentioned here or have a separate article. He rediscovered the formula, which is amazing for a 16-year-old, but not notable. I proposed the redirect for deletion. --Apoc2400 (talk) 21:03, 31 May 2009 (UTC)
Thank you for your opinion, and for not associating me with the vandals that constantly add the info without consensus. As it appears consensus is that he isn't notable, I won't bother writing an article.Drew Smith What I've done 21:32, 31 May 2009 (UTC)
I have removed {{prod}} from Mohamed Altoumaimi because WP:PROD does not apply to redirects. You can nominate it at Wikipedia:Redirects for discussion instead. PrimeHunter (talk) 21:41, 31 May 2009 (UTC)

One of the references for this is incorrect. Currently citation #3, to [2] should go to [3]. Atonix (talk) 13:12, 1 June 2009 (UTC)

I have removed the reference in [4] as a duplicate of the preceding reference [5]. PrimeHunter (talk) 14:29, 1 June 2009 (UTC)

Well, the current (frozen) article text has things backwards: it states the "correctly simplifying the computation" as a fact, while demoting the "discovered before and well known" to something that mathematicians "claim". According to the sources, it should be other way round.

  • No news source has given any precise description of what new was allegedly invented: all exclamations of "solution to a 300-year-old problem" were vague and empty of content.
  • The only precise formula that has been shown (in Falu Kuriren's photo) is essentially Worpitzky's formula from 1883.
  • The university has clearly and unambiguously stated that the formula "was well known and readily available in several databases".

Based on these sources, a more faithful representation of the events is: A high-school student re-discovered a well-known formula. There was a short media hype based on the misunderstanding that a 300-year-old problem has now been solved. But such a short-lived media hype is hardly notable enough for Wikipedia (Wikinews might be a better place), so I would say it does not belong to this article. If someone disagrees, please show a respectable source for the novelty, otherwise it is just gossip. --Jmk (talk) 08:03, 2 June 2009 (UTC)

The page says (at this moment): ".. submitted a proof to Uppsala University, correctly simplifying the computation of Bernoulli numbers." What a nonsense. The formula which is (or is equivalent to) Worpitzky's formula does by *no* means simplify the computation of the Bernoulli numbers. Please compare with the Akiyama-Tanigawa algorithm which is a hundred times more efficient than the Worpitzky formula. Will this balderdash never end? —Preceding unsigned comment added by 85.183.213.3 (talk) 14:30, 2 June 2009 (UTC)

The story has now been removed from the article as non-notable. However, eager contributors keep [6] [7] bringing it back. Presumably these contributors are in good faith, and sincerely believe that a 300-year-old problem was now solved for the first time. After all, the original (false) story got much better media coverage than its meager correction. Should we give in and include a sentence or two about the media circus, just to make things clear? --Jmk (talk) 07:52, 4 June 2009 (UTC)

I have no particular objection to it being further down the article somewhere. My guess is it would eventually be removed as not notable but it is notable enough for the moment and I'm no stickler about wikipedia's rule about notability being eternal. Dmcq (talk) 08:30, 4 June 2009 (UTC)
Actually, given that the target audience for this material is general-interest readers who are following up on a news story they've just read, I would suggest the opposite: add a couple sentences to the end of the lead section, where they can't be missed. Mention Worpitzky, Akiyama-Tanigawa, and Altoumaimi. Then, when traffic dies back down under 300/day, remove the latter. Melchoir (talk) 09:50, 4 June 2009 (UTC)
A temporary mention is OK by me when it's made clear he didn't discover anything new. PrimeHunter (talk) 11:29, 4 June 2009 (UTC)

Integer sequence?[edit]

This page is currently in the "Integer sequence" category. Since Bernoulli numbers are not integers, maybe it should be moved to the parent category, "Sequences and series". (I didn't see a "rational number sequences" category.) --Spiffy sperry (talk) 19:27, 31 May 2009 (UTC)

It is correctly categorized as an "Integer sequence". Consider the first formula in the introduction. It should be clear by that formula that B_n is an integer for all n. Bender2k14 (talk) 04:40, 1 June 2009 (UTC)
The first formula in the introduction is not Bernoulli numbers. Bernoulli number#Values of the Bernoulli numbers and http://mathworld.wolfram.com/BernoulliNumber.html show non-integer numbers. PrimeHunter (talk) 10:32, 1 June 2009 (UTC)
Sloane gets round the problem by having an integer sequence for the numerator and another for the denominator in lowest terms so they can be considered as two integer sequences. On-Line Encyclopedia of Integer Sequences describes a number of other tricks they use, I hadn't realized they did things like the digits of pi too! Dmcq (talk) 11:20, 1 June 2009 (UTC)

Why is this important?[edit]

I just read about these numbers in the news (mentioned above) and came here hoping to learn why they were important, but other than a mention of number theory my question remains unanswered. To all the smart people who write the math wikis, could you please remember us simple folk who don't already know about these subjects and what to learn the basics of "what" and "why?." Thank you. —Preceding unsigned comment added by Skintigh (talkcontribs) 14:44, 1 June 2009 (UTC)

Yes that is a continuing problem with the maths articles. The Manual of style for Mathematics has that up front in the introduction but even where some editors make the leader or an introduction of a maths article a bit friendlier there are more of the others who just seem to want to remove any shred of informality. It is a difficult and ongoing task and the formalists are winning. For instance have a look at Talk:Topology and the start of the Topology article itself to see the problem. That's a top level major area of mathematics - read the start and see if you have the foggiest what any of it is about. Dmcq (talk) 09:25, 2 June 2009 (UTC)

I am the strongest of all advocates of informality, and also a strong advocate of formality. They both need to be there. Michael Hardy (talk) 17:59, 2 June 2009 (UTC)

I'm almost ashamed to admit I came across them mainly first in the context of the old version of the umbral calculus. Now that was something that lacked formalism even if it wasn't informal! Conway and Guy's Book of Numbers which I can thoroughly recommend as a popular mathematics book has a nod towards all that when dealing with them. Dmcq (talk) 19:02, 2 June 2009 (UTC)

Agreed, that topology article is basically useless for 99% of people reading it. I already know a bit about topology but that first paragraph could have been describing anything. In fact, I first learned about topology in 5th grade, and the Boston Museum of Science has an exhibit on it. I wonder how many excited young minds come home from there, look up the subject on Wikipedia, and instantly lose interest and go back to the Wii. (Not that I'm saying we should write at a 5th grade level...) As for formality, I'm all for it. But "formal" != "only elites and PhDs are allowed to understand what we are talking about and our article should be useless to all others." It reminds me of some undergrad texts I've read that can't be understood until you have a PhD. Not helpful. All I ask is just a brief paragraph on "what" and "why" before launching into the gobbledygook.--Skintigh (talk) 18:29, 3 June 2009 (UTC)

Values of the Bernoulli numbers[edit]

I was dismayed to see that the actual values of the Bernoulli numbers are not mentioned anywhere in the article before section 17. Is there some reason why there is no tabulation of the numbers earlier, perhaps in the lede section, or in a sidebar adjacent to the lede section? This seems very strange to me. —Dominus (talk) 14:11, 31 August 2009 (UTC)

I put an enumeration into the lede. —Dominus (talk) 15:30, 1 September 2009 (UTC)
Wirkstoff: Thanks, that looks much better. —Dominus (talk) 20:03, 1 September 2009 (UTC)

Moved history section downwards[edit]

The article began with an extremely long historical discussion, including the section "Reconstruction of 'Summae Potestatum'", before getting into any of the mathematical issues. I moved the history down a long way. The first section in the article now discusses what I think is the most salient fact about the Bernoulli numbers, which is that they appear in the formulas for the sums of the first nth powers. This is the context in which they were first discovered by both Bernoulli and Seki, and the context in which they are most likely to be of interest to the general reader. —Dominus (talk) 14:32, 31 August 2009 (UTC)

The contribution by Yahord[edit]

I undid the following contribution by Yahord:

Rodrigues-like formula. This formula was invented by V.M. Kalinin.

B_{2k}=(-1)^{k-1}\frac{2k}{ 2^{2k}(2^{2k}-1) } \lim_{t\rightarrow 0}\left\{\left[(1+t^2)\frac{d}{dt}\right]^{2k-2}(1+t^2)\right\}

This is why: The identity says:

B_{2k}=(-1)^{k-1}\frac{2k}{ 2^{2k}(2^{2k}-1) } T_{k}

where Tk is (up to sign) the (2k+1)th coefficient of the exponential expansion of tan, (the tangent (or "Zag") numbers). This is explained in detail in the article (section "An algorithmic view: the Seidel triangle"). Obviously the tangent numbers can be computed in various ways; if this is of interest this can be stated in an article on the tangent numbers. Wirkstoff (talk) 23:11, 31 August 2009 (UTC)

Generalization of the sum-of-powers formula[edit]

The formula for sum-of-powers can be generalized to a real power:

S_{a}\left( n \right)=\sum\limits_{k=1}^{n}{k^{a}}=\frac{1}{a+1}\sum\limits_{k=0}^{\infty }
{\left( \begin{matrix}    a+1  \\   k  \\ \end{matrix} \right)
B_{k}\left( 0 \right)\left( \left( 1+n \right)^{a+1-k}-1 \right)}


B_{k}\left( 0 \right) is the bernoulli numbers with B_{1}\left( 0 \right)=-\frac{1}{2}.

It follows from the Euler–Maclaurin formula with f\left( n \right)=\left( 1+n \right)^{a} and p=\infty

--77.127.51.123 (talk) 08:29, 7 October 2009 (UTC)

Though your remark is correct, is it not more natural to consider the function f\left( n \right)=n^{a} instead of f\left( n \right)=\left( 1+n \right)^{a} when speaking of the power function? In this case the Euler–Maclaurin formula gives for n ≥ 1
S_{a}\left( n \right)=\sum\limits_{k=1}^{n}{k^{a}}= 1 + \frac{1}{a+1}\sum\limits_{k=0}^{\infty }
{\left( \begin{matrix}   a+1  \\   k  \\ \end{matrix} \right)
B_{k}\left( 1 \right)\left(  n^{a+1-k}-1 \right)}
B_{k}\left( 1 \right) are the Bernoulli numbers with B_{1}\left( 1 \right)=\frac{1}{2}.

Explicit definition[edit]

Have the explicit formulas been checked ? Something looks very wrong : Maple gives g:=n->sum(sum((-1)^(j+1)*binomial(k-1,j-1)*j^n/(k),j=1..k+1),k=1..n+1);

             n + 1 /k + 1                                      \
             ----- |-----     (j + 1)                         n|
              \    | \    (-1)        binomial(k - 1, j - 1) j |
   g := n ->   )   |  )   -------------------------------------|
              /    | /                      k                  |
             ----- |-----                                      |
             k = 1 \j = 1                                      /

> binomial(6,3);

                                 20

> seq(g(n),n=0..10);

                  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Where is my mistake? Dfeldmann (talk) 13:07, 13 November 2009 (UTC)

Well, it does not really matter what Maple gives. Bernoulli was a mathematician, not a maplematician ;) Please learn the difference between 'sum' and 'add' in Maple. [F1] helps. In any case g := proc(n) local k, j; add(add((-1)^(j+1)*binomial(k-1,j-1)*j^n/(k),j=1..k+1),k=1..n+1) end; gives the announced results. Wirkstoff (talk) —Preceding undated comment added 22:10, 11 December 2009 (UTC).

The Erster Art and Zweiter Art terminology is a disservice.[edit]

@Maxal: The *unsigned* Stirling numbers of the first kind are called Stirling cycle numbers, see DLMF 26.13.3 [8] (and also OEIS A008275, the very first comment.) You also changed 'Stirling set numbers' to 'Stirling numbers of the second kind'. Note what D. E. Knuth said: "Nielsen [..] did the world a disservice by originating the Erster Art and Zweiter Art terminology, because that terminology has no mnemonic value and is historically inaccurate." (Two notes on notation.) So I wonder why you did Wikipedia your disservice. Wirkstoff (talk) 21:58, 11 August 2010 (UTC)

  • Why do you complain here? If you think that the notions of Stirling numbers of first/second kind are inadequate, argue that in the discussions of the corresponding articles on the Stirling numbers (i.e., in Talk:Stirling numbers of the first kind and Talk:Stirling numbers of the second kind). However, I would notice that despite of historical inaccuracy, these notions are most widely accepted and used. Maxal (talk) 18:44, 12 August 2010 (UTC)

Assorted Identities[edit]

Please give a reference to the last two identies. Laosinchai (talk) 09:49, 21 April 2011 (UTC)

Contour Integral Definition of Bernoulii numbers[edit]

I note an alternative definition for the Bernoulii numbers, Bn, at http://mathworld.wolfram.com/BernoulliNumber.html (equation 2), involving a contour integral. I note that this integral can, in addition, produce Bn for non-integer n.
Can any advise:
(i) if this definition should be included in the article?
(ii) of name(s) / application(s) for Bn when n is not an integer? Mhallwiki (talk) 22:59, 10 August 2011 (UTC)

Integral representation.[edit]

The integral representation "suggested by Peter... In 2004" is in fact just Euler's formula for the zeta and Bernoulli function/numbers of even argument, but where the zeta function is replaced by it's very well known integral representation. A quick change of variable shows this ! Hence, there is nothing new here. — Preceding unsigned comment added by 2.28.226.145 (talk) 07:27, 17 January 2012 (UTC)

Removed one section[edit]

I removed the following material from the article since it is hard to understand and does not seem important enough to me:


A representation of the second Bernoulli numbers

This is an unreduced version of fractions A192456/A191302 in OEIS. The columns have the same denominators 2,6,15,105,105,... = A190339 in OEIS. Hence the following array with an ambiguous first term:

A representation of the second Bernoulli numbers
B0  =  1 = 2/2
B1  =  1/2
B2  =  1/2 − 2/6
B3  =  1/2 − 3/6
B4  =  1/2 − 4/6 +  2/15
B5  =  1/2 − 5/6 +  5/15
B6  =  1/2 − 6/6 +  9/15  −   8/105
B7  =  1/1 − 7/6 + 14/15 −  28/105
B8  =  1/1 − 8/6 + 20/15 −  64/105  +  8/105
B9  =  1/2 − 9/6 + 27/15 − 120/105 + 36/105

The numerators of the columns in the preceding array from the second: A000027, A000096, 4*A005581, 4*A005582, 16*A005583, 16*A005584 in OEIS (sequence A191532 in OEIS).

Another array could be written with the same denominators for every row: 1,2,6,6,30,30,... = A080326 in OEIS (see sequence A123536).


AxelBoldt (talk) 01:16, 2 May 2012 (UTC)

"no simple formulas for Bn exist"[edit]

I think this sentence should be adjusted :

There is a widespread misinformation that no simple closed formulas for the Bernoulli numbers exist (...) The last two equations show that this is not true.

The quoted (classical, well-known) double summation formulas are exactly what one would call not simple, and possibly, what generated the slogan no simple formulas for Bn exist. One could argue whether those formulas are really that complicated (it depends on what one wants to do with them, after all). But, as it is, the sentence is somehow misleading (the quoted formulas are not unknown to those who believe that "no simple formulas for Bn exist". ) --pma 09:41, 7 December 2012 (UTC)

"Simple" is indeed in the eye of the beholder. However, a more objective statement would be that no closed-form expression is known. In fact, I wonder if it is proved that no closed-form exists. McKay (talk) 02:49, 22 May 2013 (UTC)

Bernoul/Bernoulli number/numbers - Eh? What?[edit]

What are Bernoulli numbers? To find out, you have to already know what they are - that's my conclusion from the article's introduction. It gives no real help to the "outsider" coming relatively fresh to the subject. But putting the actual Bernouille maths aside, plus their purpose, relevance, usefulness or importance (which aren't sufficiently clear upfront), let me just illustrate how confusingly an outsider is "thrown" into immediate difficulty trying to understand the topic. Here goes: A)- One often hears of "Bernoulli numbers" via tv, newspapers etc. B)- Yet Wikipedia redirects to this article titled in the singular, ie. "Bernoulli number". C)- The article's very first sentence is "In mathematics, the Bernoul numbers Bn are a sequence ...". Talk about upfront confusion! If they are a sequence, why is this a "single number" article?? And what are Bernoul numbers?? (as opposed to Bernoulli)?? Surely not a typing error - parts of the article are regularly argued over by dozens of mathematical angels dancing on pinheads. They wouldn't let "Bernoul" stand as an error for months and years, would they? My point is this: the article's great for those who already know a lot about number theory, but it doesn't help ousiders whose understanding is just limited to arithmetic (yes, it's the same thing and yet not the same thing, I know...). It's probably been said before but worth restating - the Introduction should give a brief but clear summary of the nature, purpose and usefulness of the number(s), sufficient to not throw up immediate questions! Perhaps a "maths teacher" might review/improve the Intro accordingly? Pete Hobbs (talk) 16:08, 6 April 2013 (UTC)

Are you more comfortable with the article on binomial coefficients? There is some similarity between the two contexts, so if you find that article better please explain how and those features can be brought across to this article. Haklo (talk) 02:41, 8 April 2013 (UTC)
More comfortable? Hardly! I came to read about Bernoulli etc while chasing some history info. My point was (I thought) more to do with efficiency than coefficiency! Although I don't mean that as a put-down - you might be right about some similarity in contexts, I'm just unable (really not qualified) to judge that. But I will say the Binomial coefficients Intro was clearer than the Intro para of this Bernoul(li) Number(s) article. By which I mean it gave a clear definition, instead of throwing up immediate questions. Let me try putting my query another way: the article starts by defining "Bernoul numbers bn" but does NOT define a "Bernoulli number" - a more important definition that should surely be the first thing on the page. Pete Hobbs (talk) 13:49, 10 April 2013 (UTC)

The relation to the Euler numbers and π[edit]

The first portion of this section, namely

The Euler numbers 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 E2n are in magnitude approximately (2/π)(42n − 22n) times larger than the Bernoulli numbers B2n. In consequence:
 \pi \  \sim \  2 \left(2^{2n} - 4^{2n} \right) \frac{B_{2n}}{E_{2n}}.
This asymptotic equation reveals that π lies in the common root of both the Bernoulli and the Euler numbers. In fact π could be computed from these rational approximations.

seems more like numerology than mathematics. Very many asymptotic expressions involve π so getting π by dividing two of them is uninteresting without a purpose in mind. Also, what does "lies in the common root" mean? McKay (talk) 02:45, 22 May 2013 (UTC)

First note that the formula has an exact and well defined mathematical meaning. So it is math and not numerology. Also you can (I hope) see that it is a true formula. The formula relates the Bernoulli numbers and Pi so it is on topic. Moreover it relates the Bernoulli numbers and Pi in an interesting way (via the Euler numbers). Thus the proposition is true and interesting; thus it should be kept. If you can formulate the exposition of this theorem in a better way than go on and make a proposal. Ed Tawny (talk) 21:28, 31 May 2013 (UTC)

No, it isn't interesting. Anyway, what is the source for this section? You aren't allowed to add your own analysis. McKay (talk) 03:24, 31 July 2013 (UTC)

"Generating Function"?[edit]

I dislike that this section doesn't actually give an explicit generating function (and nowhere is one given in the article) in the form Bn or B(n) =  ("The nth Bernouilli number equals..."). All we get is a summation with a Bm(n) thrown in there. Pokajanje|Talk 15:30, 25 July 2013 (UTC)

That's what a generating function is. McKay (talk) 04:39, 26 July 2013 (UTC)
Is there a function that gives the nth Bernouilli number with argument n? The algorithmic description doesn't define it with mathematical notation. Pokajanje|Talk 15:49, 27 July 2013 (UTC)
Is somethink like  \zeta(1-n) \cdot n \cdot (-1)^n enough? (please check the n and n-1 coefficients, I'm not completely sure about them)--Gotti 08:25, 30 July 2013 (UTC) — Preceding unsigned comment added by Druseltal2005 (talkcontribs)
It would seem to be B(n) = -n\zeta(1 - n)(-1)^n, but that is otherwise correct and should be added to the article. Pokajanje|Talk 21:52, 1 August 2013 (UTC)