Talk:Faulhaber's formula

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

message: 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)
replies: text: I have changed it now. Eric Kvaalen (talk) 06:37, 14 April 2010 (UTC)
replies: text: 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)
thread_title:  First sum fomula: no need for "special case" 

message: 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)
replies: text: 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)
thread_title:  about faulhaber observation in the last of this article 

message: 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)
replies: text: 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)
replies: text:  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)
thread_title:  did or did not know? 

message: 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)
thread_title:  NPOV 

message: 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)
thread_title:  Comments on new sections? ("Proof" and "Alternate expression") 

message: 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)
replies: text: Both are in Bernoulli number.  — Preceding unsigned comment added by Ice ax1940ice pick (talk • contribs) 11:12, 1 July 2014 (UTC)
replies: text: 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)
thread_title:  Doubt 

message: 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)
replies: text: I second your opinion. --46.242.13.183 (talk) 22:26, 6 September 2018 (UTC)
thread_title:  Proof style 

message: 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)
thread_title: A practical use of the Faulhaber’s formulas. 

message: 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)
thread_title: Faulhaber polynomials citation needed

message: 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)
replies: text: 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)
replies: text: 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)
replies: text: 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)
replies: text: Thank you. I have now added text to make things more clear. Bob.v.R (talk) 11:48, 11 November 2017 (UTC)
thread_title:  Unknown matrix variable in "From examples to matrix theorem" section 

message: 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)
thread_title:  Typo in the Summae Potestatum 

message: 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)
thread_title:  A Commons file used on this page or its Wikidata item has been nominated for deletion 

message: 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)
replies: text: Yes, it is wrong. it:somma di potenze di interi successivi note 1. --Bio.oid (talk) 12:10, 29 March 2022 (UTC)
replies: text: oppure in questa stessa discussione: Typo in the Summae Potestatum--Bio.oid (talk) 13:51, 29 March 2022 (UTC)
thread_title:  Error in image taken from Jakob Bernoulli's Summae Potestatum, Ars Conjectandi, 1713 

message: 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)
thread_title:  Generalized Faulhaber formula