Talk:Bernoulli number

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 Field:  Number theory

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."

"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 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 , but that is otherwise correct and should be added to the article. Pokajanje|Talk 21:52, 1 August 2013 (UTC)

Oh, thank you![edit]

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[edit]

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 Bn for one of the sign conventions, and Bn for the other sign convention, similar to how the Hermite polynomial conventions are disambiguated. --Fylwind (talk) 09:26, 25 November 2016 (UTC)

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

Let's try to make it less technical[edit]

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 and 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 and , 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)

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)