Talk:Faulhaber's formula

Umbral section
In the 'Umbral' section it looks as if the factor 1 / (p+1) has been omitted twice.(Crackling 22:55, 13 March 2007 (UTC))

First sum fomula: no need for "special case"
In this article, in the first formula, there is no need for a special case for p=1: because if $${(-1)^p}$$ be inserted, all is well - since all odd Bernoulli numbers (excepting B_1) vanish. (There should be no other consequences - as far as I can see.) Hair Commodore 16:32, 10 April 2007 (UTC)
 * I have changed it now. Eric Kvaalen (talk) 06:37, 14 April 2010 (UTC)
 * A special case is still needed for p=0 though, because the formula right now gives 1+1+...1 = n+1/2, which is obviously wrong. 10:33, 21 February 2019 (UTC)

about faulhaber observation in the last of this article
There is an error in faulhaber's polynomials. This article says that : 1^{11}+ 2^{11} + 3^{11} + \cdots + n^{11} = {32a^6 - 64a^5 + 68a^4 -40a^3 + 5a^2 \over 6} But It's : 1^{11}+ 2^{11} + 3^{11} + \cdots + n^{11} = {32a^6 - 64a^5 + 68a^4 -40a^3 + 10a^2 \over 6} Thank you. —Preceding unsigned comment added by 63.243.163.199 (talk • contribs)
 * Setting n=1 makes it clear that there was an error, and I have changed it according to what you say. However, Knuth's paper does say 5n^2. I don't know whose mistake it is. Eric Kvaalen (talk) 06:37, 14 April 2010 (UTC)
 * A letter from Knuth informs me that in the paper version of his article the mistake is not there. So probably Faulhaber had it right and it was just a misprint in the preprint that was later corrected. Eric Kvaalen (talk) 09:16, 14 May 2010 (UTC)

Upper limit of sum
Where is the mistake? Here on in the Bernoulli_number article? The upper limits differ by 1.--MathFacts (talk) 19:58, 24 April 2009 (UTC)


 * Both articles are right. This one says:
 * The formula says
 * $$\sum_{k=1}^n k^p = {1 \over p+1} \sum_{j=0}^p {p+1 \choose j} B_j n^{p+1-j}\qquad \left(\mbox{with } B_1 = {1 \over 2} \mbox{ rather than }-{1 \over 2}\right)$$
 * (the index j runs only up to p, not up to p + 1).
 * The other one says:
 * $$\sum_{k=0}^{m-1} k^n = 0^n + 1^n + 2^n + \cdots + {(m-1)}^n $$
 * This article takes B1 to be +1/2; the other has &minus;1/2. That's why they don't contradict each other.  Look at the polynomials listed in this article, and you see that in each case if you change the coefficient of the second-highest-degree term from plus to minus, the effect is exactly the same as that of deleting the last term on the left side of the identity. Michael Hardy (talk) 21:27, 24 April 2009 (UTC)
 * The other one says:
 * $$\sum_{k=0}^{m-1} k^n = 0^n + 1^n + 2^n + \cdots + {(m-1)}^n $$
 * This article takes B1 to be +1/2; the other has &minus;1/2. That's why they don't contradict each other.  Look at the polynomials listed in this article, and you see that in each case if you change the coefficient of the second-highest-degree term from plus to minus, the effect is exactly the same as that of deleting the last term on the left side of the identity. Michael Hardy (talk) 21:27, 24 April 2009 (UTC)
 * $$\sum_{k=0}^{m-1} k^n = 0^n + 1^n + 2^n + \cdots + {(m-1)}^n $$
 * This article takes B1 to be +1/2; the other has &minus;1/2. That's why they don't contradict each other.  Look at the polynomials listed in this article, and you see that in each case if you change the coefficient of the second-highest-degree term from plus to minus, the effect is exactly the same as that of deleting the last term on the left side of the identity. Michael Hardy (talk) 21:27, 24 April 2009 (UTC)
 * This article takes B1 to be +1/2; the other has &minus;1/2. That's why they don't contradict each other.  Look at the polynomials listed in this article, and you see that in each case if you change the coefficient of the second-highest-degree term from plus to minus, the effect is exactly the same as that of deleting the last term on the left side of the identity. Michael Hardy (talk) 21:27, 24 April 2009 (UTC)

did or did not know?
The article says Faulhaber did not know the formula in this form while MathWorld claims the opposite:
 * In a 1631 edition of Academiae Algebrae, J. Faulhaber published the general formula for...

What information is correct? Did Faulhaber know (and publish) the general formula? Maxal (talk) 04:10, 30 April 2009 (UTC)


 * I seem to recall that he knew the first couple of dozen or so cases. I'll see if I can find that. Michael Hardy (talk) 20:12, 30 April 2009 (UTC)


 * No, Faulhaber did definitely not know this formula. It is absurd to call this formula Faulhaber's formula. Mathworld is, again, unreliable and cannnot be used a primary source. Wirkstoff (talk) 12:26, 4 July 2009 (UTC)

Move?
Since the formula is not actually due to Faulhaber, should we move the article to "Bernoulli's formula"? Eric Kvaalen (talk) 06:37, 14 April 2010 (UTC)
 * The Wikipedia article title policy says that articles should be listed under their common names: "[Use] names and terms commonly used in reliable sources, and so likely to be recognized...Articles are normally titled using the most common English-language name of the subject of the article."  This holds even when that name is erroneous or misattributed.  For example, Pell's equation is under that title because the equation is called "Pell's equation", even though Pell had nothing to do with it, the equation was solved by Brouncker, and the solution was attributed to Pell entirely by mistake.   Similarly Zorn's lemma is not listed as "Kuratowski's lemma", even though Kuratowski unquestionably published it first.
 * Unless it appears that the term "Bernoulli's formula" is more common than "Faulhaber's formula", the article should be titled "Faulhaber's formula", perhaps with "Bernoulli's formula" as a redirect to it. —Dominus (talk)

NPOV
Lots of things in math aren't named after the person to actually discover them. We don't need to remind the reader at every opportunity, and certainly don't need the commentary. Twin Bird (talk) 22:05, 21 October 2010 (UTC)

Comments on new sections? ("Proof" and "Alternate expression")
I have more material I would like to add to this page, but I don't want to do it without gauging whether it is in the spirit of Wikipedia, since I'm aware that proof-like material is not quite encyclopaedic.

I believe what I've written so far is explicative rather than merely demonstrative, since the proof is in a historic style, illuminating the relation between coefficients that led to the formulation of the Bernoulli numbers, and the transformation between the two expressions explains why there are two conventions for B_1.

What I would like to add next is a generalisation to non-natural powers which shows the relationship between Faulhauber's formula and the Hurwitz (and Riemann) zeta function, and further demonstrates why the alternate expression is more natural (since the more conventional one is not a Taylor series when p is non-natural). — Preceding unsigned comment added by Haklo (talk • contribs) 21:25, 13 June 2012 (UTC)

A new Theorem?
Please see at point 2 of Discussion in the following article: https://commons.wikimedia.org/wiki/File:Pubblicazione_english_01.pdf

--Ancora Luciano (talk) 15:47, 11 August 2013 (UTC)


 * That observation is a special case of the Stolz–Cesàro_theorem, for $$a_{n+1}-a_n=(n+1)^m$$, and $$b_{n+1}-b_n=(n+1)^{m+1}-n^{m+1}$$. It is interesting though, isn't it?
 * Haklo (talk) 02:06, 12 August 2013 (UTC)


 * Performing calculations with Excel, I encounter these other amazing results:

n lim (Σn n3)/(Σn n). n2 = 1/2 n→∞ 1 n lim (Σn n5)/(Σn n). n4 = 1/3 n→∞ 1 n lim (Σn n7)/(Σn n). n6 = 1/4 n→∞ 1 n lim (Σn n9)/(Σn n). n8 = 1/5 n→∞ 1 which, by induction, can be generalized in a formula. Note that the denominators of the results are the positions of the exponents in the numerator in the sequence of odd numbers. Also this is a special case? To see the genesis of this, go to: http://en.wikipedia.org/wiki/Talk:Summation#Sum_of_the_first_.22n.22_cubes_-_even_cubes_-_odd_cubes_.28geometrical_proofs.29

--Ancora Luciano (talk) 15:49, 12 August 2013 (UTC)


 * Yes; it might be fair to say that most limits involving summations are a special case of Stolz–Cesàro theorem, because that theorem relates the differences of terms in a sequence to the terms, and the total of a summation is itself a sequence indexed by the upper limit of the summation, where the difference between one total and the "next" is the next term of summation.
 * Haklo (talk) 00:34, 13 August 2013 (UTC)

The formula's above don't make sense, but it looks as if Ancora Luciano means the following: n           n   lim ( Σ  k3) / { ( Σ  k). n2 } = 1/2 n→∞ k=1          k=1 n           n    lim ( Σ  k5) / { ( Σ  k). n4 } = 1/3 n→∞ k=1          k=1 n           n  lim ( Σ  k7) / { ( Σ  k). n6 } = 1/4 n→∞ k=1          k=1 n           n  lim ( Σ  k9) / { ( Σ  k). n8 } = 1/5 n→∞ k=1          k=1

Bob.v.R (talk) 00:58, 18 October 2017 (UTC)

Doubt
The article seems to use 1/2 and - 1/2 as B 1. — Preceding unsigned comment added by Ice ax1940ice pick (talk • contribs) 11:08, 1 July 2014 (UTC)
 * Both are in Bernoulli number. — Preceding unsigned comment added by Ice ax1940ice pick (talk • contribs) 11:12, 1 July 2014 (UTC)
 * Different expressions require different conventions; is there any part of the page where it is ambiguous which convention is needed? Haklo (talk) 03:04, 2 July 2014 (UTC)

Proof style
Is a proof based on an exponential generating function for Bernoulli polynomials appropriate for this page?

I deliberately chose a very elementary proof as the first version, partly because Faulhaber's formula seems to be the historical origin of the Bernoulli numbers; i.e., formula first, Bernoulli numbers later.

The proof I had written avoids calculus, generating functions, and even glosses over an induction step, in the hope that it would be an easier "entry point" for those wishing to get an idea of the significance of the Bernoulli numbers.

Haklo (talk) 09:00, 5 November 2014 (UTC)


 * I second your opinion. --46.242.13.183 (talk) 22:26, 6 September 2018 (UTC)

A practical use of the Faulhaber’s formulas.
The Faulhaber’s formulas F(m) can be used to calculate the second partial sums of m-th powers, with the following general formula: a(n,m) = (n+1) * F(m) - F(m+1). This formula generated seven new contributions to the OEIS database: A250212, A253636,  A253637,  A253710,  A253711,  A253712,  A253713. His proof is linked in each contribution. The previous sequences, obtained in a different manner, were already present in the database. Later ones were blocked shots by editors, for practical reasons. — Preceding unsigned comment added by Ancora Luciano (talk • contribs) 16:29, 21 January 2015 (UTC)

Faulhaber polynomials citation needed
What is a reference for the formula which follows "More generally," in the Faulhaber polynomials section of the article? Jsondow (talk) 17:07, 28 January 2017 (UTC) Jsondow (talk) 22:20, 30 January 2017 (UTC)

Unknown matrix variable in "From examples to matrix theorem" section
The variable $$\overline{A}_m$$ appears without prior use or definition. Perhaps it was introduced in an earlier, now-deleted section. Previous comment was coming from Ntropyman at 16 October 2017, time 04:27h.
 * No, the formula where it appears is its definition. The meaning of the overline is somehow explained in the following text: one pass from $$A_m$$ to $$\overline{A}_m$$ by changing the signs of the elements such that the sum of the column index and the row index is odd ($$\textstyle a_{i,j} \to (-1)^{i+j}a_{i,j}$$). I agree that this section would require some cleanup. D.Lazard (talk) 08:05, 16 October 2017 (UTC)
 * Both $$A_m$$ and $$\overline A_m$$ are undefined in the current version of the article; cleanup is indeed needed. Bob.v.R (talk) 00:23, 18 October 2017 (UTC)
 * If D.Lazard's definition of the overline is correct, this should be more clearly mentioned in the article. However, this would still not answer the first question which needs to be answered: what is the definition of $$A_m$$? Bob.v.R (talk) 06:36, 11 November 2017 (UTC)
 * Both $$A_m$$ and $$\overline A_m$$ are already defined at the first point they appear: $$\overline A_m$$ as the inverse of $$G_m$$, and $$A_m$$ as "like $$\overline A_m$$ but with all positive elements". —David Eppstein (talk) 06:48, 11 November 2017 (UTC)
 * Thank you. I have now added text to make things more clear. Bob.v.R (talk) 11:48, 11 November 2017 (UTC)

Typo in the Summae Potestatum
For $$p = 9$$, it says that the coefficient of the term in $$n^2$$ is $$-1/12$$. It should say $$-3/20$$. — Preceding unsigned comment added by 2602:306:CC10:5D70:B857:4E4E:215A:FE59 (talk) 04:01, 11 June 2018 (UTC)

clarification not needed
There is a complaint about a notation not explained, which is explained in the first few lines of the article. There is a complaint about an index name change, which is futile. Nobody able to read this article would be confused by this and it draws attention to the change of the lower limit. Of course now the formula is more in agreement what is customary in the textbooks.

A meta remark. It requires courage and expertise to remove that remark, both of which I lack. Why are those remarks anonymous? I would like to file a complaint of vandalism against such author. — Preceding unsigned comment added by 80.100.243.19 (talk) 12:51, 30 January 2020 (UTC)
 * The complaints in the tag make sense, as the notation for failing factorial is not of a common use. So, not recalling it suppose that the reader of this section has memorized the lead. As one cannot impose such an effort to the reader, the tag is justified. For the change of notation, this is the same: It is easy to verify it, but it is a task for the editor, not for the reader. Nevertheless, I have modified the formulation, and removed the tag.


 * By the way, this tag is not anonymous, you can find its its author by consulting the history of the article. D.Lazard (talk) 14:54, 30 January 2020 (UTC)

A Commons file used on this page or its Wikidata item has been nominated for deletion
The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for deletion: Participate in the deletion discussion at the nomination page. —Community Tech bot (talk) 17:37, 11 April 2020 (UTC)
 * Pascal’s and Bernoulli’s triangle.jpg

Proposed merge of Polynomials calculating sums of powers of arithmetic progressions into Faulhaber's formula
This new article covers substantially the same substantive material, from a slightly different perspective. There is no meaningfully separate study of the sums of powers of arithmetic progressions in general and the study of the sum $$1^m + 2^m + \ldots + n^m$$. -- JBL (talk) 21:10, 25 March 2022 (UTC)
 * Support. I agree that this does not seem sufficiently distinct for a separate article. —David Eppstein (talk) 05:07, 26 March 2022 (UTC)
 * Support Makes sense to me. XOR&#39;easter (talk) 16:07, 26 March 2022 (UTC)
 * Support When it was a draft, I commented that this was essentially a content fork of Faulhaber's formula. D.Lazard (talk) 17:05, 26 March 2022 (UTC)
 * Not merge Even if from a historical point of view it seems the opposite, the traditional Faulhaber formula identifies a countable infinity of polynomials calculating sums of powers but this is only a point of the four-dimensional complex space $$\mathbb{C}^2$$ where each point does the same.
 * $$(1,1) \in \mathbb{C}^2 \quad\text{but not vice versa}$$
 * The subject of the newly created voice is the sets of particular polynomials that populate this vast space, not the historical formula.
 * In my opinion, this is a possible in-depth study that must be well connected but remain separate. Also because the same insights must be available coming, for example, from arithmetic progressions.--Bio.oid (talk) 20:05, 26 March 2022 (UTC)
 * Everyone in this discussion understand the (essentially trivial) way in which the topic of the new article generalizes the topic of this article. Given the relative importance of the generalization, it is best treated in a brief section of this article. --JBL (talk) 17:12, 28 March 2022 (UTC)
 * $$ \mathbb{C}^2\notin (1,1) $$ is trivial but the reflections on the downsizing of the claims of the point-formula in a much vaster four-dimensional complex universe are not trivial. Also in the wikipedia universe a similar reflection is necessary.
 * The will to incorporate "Polynomials calculating sums of powers of arithmetic progressions" conflicts with other articles that would be deprived of it. This is because those who are interested in articles such as Pascal matrix, arithmetic progressions, Bernoulli polynomials without being interested in the famous formula would lose the opportunity to deepen unless they also incorporate the same article, appropriately modified, in them. It seems to me a useless waste of energy that could be used much better for example by explaining with adequate sources that strange Bernoulli polynomial with + exponent that is encountered in the cannibal article :-).
 * In short, it will be my limit, but I can only see negative aspects in this merger project but it seems to me that the positive ones have not yet been explained. With wiki love --Bio.oid (talk) 09:19, 29 March 2022 (UTC)


 * Don't merge. I am a random visitor from the Internet, with other words, one of your users. I find the idea of the merging simply hilarious - I googled for Faulhabers' formula and I got exactly what I wanted. You want now eliminate this and merge into a cloudy, similar thing. Do not do that! 37.76.55.242 (talk) 14:37, 23 August 2023 (UTC)


 * Don't merge; maybe delete?. It is now a year and a half later, and no merge has occurred. Both articles are long, almost eye-wateringly long; making an even larger mega-article seems wrong. The tradition in WP is that grand generalizations of some concept get their own article. FWIW, currently, neither article hints at what the generalization is. The commentary above suggests that its some algebraic variety, some "...four-dimensional complex space $$\mathbb{C}^2$$ where each point does the same." But neither article mentions the complex numbers or use the word "variety". All coefficients appear to be integers or rationals, not complex numbers. I imagine this theory generalizes to integral domains or ring of integers or something like that, but nothing to that effect is mentioned in either article. What is this variety that User:bio.oid speaks of? Where are these "points in general position"? ... Hmm. I keep looking at Polynomials calculating sums of powers of arithmetic progressions and frankly, I do not understand why that topic is notable and worthy of an article on WP. It seems to describe dull, obvious generalizations that you can cook up in an afternoon or two of scratching around; why is the topic notable? Am I missing something? There don't seem to be any theorems or claims being made. Is there a reason not to prod or AfD the thing? 67.198.37.16 (talk) 20:28, 20 February 2024 (UTC)

Error in image taken from Jakob Bernoulli's Summae Potestatum, Ars Conjectandi, 1713
Hello - there seems to be an error in the formula for 9th powers in the image "Jakob Bernoulli's Summae Potestatum, Ars Conjectandi, 1713". The formula for 9th powers should be

$$\frac{1}{10}n^{10}+\frac{1}{2}n^9+\frac{3}{4}n^8-\frac{7}{10}n^6+\frac{1}{2}n^4-\frac{3}{20}n^2.$$

The last term seems to be wrong. — Preceding unsigned comment added by HarryToby (talk • contribs) 11:17, 29 March 2022 (UTC)
 * Yes, it is wrong. somma di potenze di interi successivi note 1. --Bio.oid (talk) 12:10, 29 March 2022 (UTC)
 * oppure in questa stessa discussione: Typo in the Summae Potestatum--Bio.oid (talk) 13:51, 29 March 2022 (UTC)

Generalized Faulhaber formula
A generalization Faulhaber's formula for two types of iterated sums is presented in the following two papers:

1. El Haddad, R. (2022). A generalization of multiple zeta value. Part 1: Recurrent sums. Notes on Number Theory and Discrete Mathematics, 28(2), 167-199, DOI: 10.7546/nntdm.2022.28.2.167-199.

2. El Haddad, R. (2022). A generalization of multiple zeta values. Part 2: Multiple sums. Notes on Number Theory and Discrete Mathematics, 28(2), 200-233, DOI: 10.7546/nntdm.2022.28.2.200-233.

Examples of particular cases of the generalization Faulhaber's formulas are present in the OEIS (some of which are cited in the two papers).

I suggest adding these formulas to the Faulhaber's formula page as it is very relevant to the topic.

The papers present two types of iterated sums called recurrent sums and multiple sums. Then present the generalizations for recurrent and multiple sums (See Theorem 4.8 of [1] and Theorem 5.1 of [2]).

Kaizen Grey (talk) 19:50, 14 May 2022 (UTC)

Implementation of the merger project
This does not appear a merge operation but a crude attempt at transplantation. What is your opinion?. --37.161.247.176 (talk) 06:54, 10 July 2024 (UTC)