Talk:Bernoulli number/Archive 2

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Archive 1

Archive 2

Explicit definition

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);

> 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.

@Maxal: The *unsigned* Stirling numbers of the first kind are called Stirling cycle numbers, see DLMF 26.13.3 [1] (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

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

Contour Integral Definition of Bernoulii numbers

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

Integral representation.

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

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 the 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)

Bernoul/Bernoulli number/numbers - Eh? What?

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)

"no simple formulas for B_n exist"

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 B_n 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 B_n exist". ) --pm a 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)

The relation to the Euler numbers and π

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 E_2n are in magnitude approximately (2/π)(4²ⁿ − 2²ⁿ) times larger than the Bernoulli numbers B_2n. 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"?

I dislike that this section doesn't actually give an explicit generating function (and nowhere is one given in the article) in the form B_n or B(n) = ("The nth Bernouilli number equals..."). All we get is a summation with a B_m(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 (talk • contribs)

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)

Oh, thank you!

This page is so beautiful-- I just have to say thank you to everyone for keeping all of this together in a living and "comprehensible format". Thank you and thank you. --Rednblu (talk) 00:25, 28 June 2016 (UTC)

Please use different symbols for the two sign conventions

As it is, reading the article is a headache because the sign convention is not at all obvious. Sometimes it's mentioned in an adjacent paragraph, and sometimes it's not mentioned at all and you basically have to figure it out yourself.

My suggestion would be to consistently use $B n$ for one of the sign conventions, and $B' n$ for the other sign convention, similar to how the Hermite polynomial conventions are disambiguated. --Fylwind (talk) 09:26, 25 November 2016 (UTC)

Or maybe $B - n$ and $B + n$ , which might be even clearer. <--Fylwind (talk) 09:41, 25 November 2016 (UTC)

Let's try to make it less technical

Hi everybody. At present the info box at the beginning of the article says, the article is too technical. I agree.

One point which I find is not so happily solved (but it's a matter of opinion) is:

One of the authors connected the two types of Bernoulli numbers $B_{m}^{-{}}$ and $B_{m}^{+{}}$ with the Bernoulli polynomials. In doing so he introduced the polynomial argument $n$ and then specified it to 0 and 1.

Well, like this he had to introduce several times formulas that really belong to the article Bernoulli polynomials. I suggest that in the sections where this applies we just list the two formulas with $B_{m}^{-{}}$ and $B_{m}^{+{}}$ , and leave $n$ and the polynomials out. I already did this in the sections Recursive definitions and Generating functions. If you are interested in this article (or wrote the sections I have edited), I would welcome your feedback: do you favour further changes in this direction, or would you rather revert to the previous way? Thanks. --Herbmuell (talk) 09:25, 2 July 2017 (UTC)

I'm neutral on this. I'm fine as long as the removed material is still accessible from the other article and a brief description of the relationship to Bernoulli polynomials is kept along with a {{Main}} link. --Fylwind (talk) 23:57, 3 July 2017 (UTC)

I'll keep that in mind.

The changes to make the article less technical while retaining more or less the same mathematical content look like improvements to me. —David Eppstein (talk) 00:12, 4 July 2017 (UTC)

Okay.

The next point I changed:

The designation "First and Second Bernoulli numbers" was not referenced. I don't believe it's recognized, and doesn't make much sense (First and Second Stirling number makes sense, they are completely different). --Herbmuell (talk) 07:20, 9 July 2017 (UTC)

The next point I want to address is redundancy:

The section Values of the Bernoulli numbers was largely redundant. I copied some material to other, suitable places and then deleted the entire section. --Herbmuell (talk) 11:45, 9 July 2017 (UTC)
The section Different viewpoints and conventions is largely an essay and therefore unsuitable for Wikipedia. Personally I found only the graphics and the last part about the zeta-function worthy of being kept. I deleted all the rest. Comments and criticism are welcome. --Herbmuell (talk) 12:19, 9 July 2017 (UTC)

The next point I want to address is overlap with other articles:

Eg. this article with the articles Faulhaber's formula and Euler–Maclaurin formula. To be continued. --Herbmuell (talk) 05:40, 9 July 2017 (UTC)

I deleted several subsections:

The subsection Definitions - Algorythmic description didn't fit in and looked like original research. I deleted it. --Herbmuell (talk) 11:54, 10 July 2017 (UTC)
The subsection Combinatorial definitions - Representation of the Bernoulli numbers didn't fit in and lacked an explanation. I deleted it. --Herbmuell (talk) 11:54, 10 July 2017 (UTC)
The subsection Combinatorial definitions - Connection with Balmer series lacked an explanation. I deleted it. --Herbmuell (talk) 12:46, 10 July 2017 (UTC)

not a dead link, but the bot might put the template back there...

I removed "{ { dead link|date=July 2017 |bot=InternetArchiveBot |fix-attempted=yes } }" in Saalschütz ref, but how avoid that the bot puts it back? — MFH:Talk 04:14, 14 August 2018 (UTC)

PS: Tried to report a false positive using the IAB bug reporting form at https://tools.wmflabs.org/iabot/index.php?page=reportfalsepositive but could not submit it. It said: "These URLs are already alive or have since been whitelisted and will not be reported". But then the bot should have repaired the damage he did...!?! — MFH:Talk 04:24, 14 August 2018 (UTC)

A companion to the Bernoulli numbers

(moved from my talk page, Sapphorain (talk) 13:58, 31 August 2019 (UTC))

I suggest removing Section 13 "A companion to the Bernoulli numbers" from the Bernoulli number page. The section has little to do with Bernoulli numbers and a lot to do with OEIS sequences authored by Paul Curtz. I bet that he also introduced his "companion to the Bernoulli numbers" on Wikipedia. Another reason to delete his section is that the French Wikipedia page [[2]] does not contain anything related. Thanks. Jsondow (talk) 12:39, 31 August 2019 (UTC)

I agree that this section should be deleted. It has indeed very little to do with the Bernoulli numbers. And it is appears to be no more than a badly presented list of various unrelated sequences of numbers, with no clear purpose. It should definitely not be part of this article, and I don’t see where it would be appropriate to move it. Sapphorain (talk) 13:58, 31 August 2019 (UTC)

Since no opposition has been voiced for 6 months, I have removed this section. Sapphorain (talk) 20:43, 22 February 2020 (UTC)