Talk:Bernoulli polynomials

Error in integrals?

Definite integrals

$\int _{0}^{1}B_{n}(t)B_{m}(t)\,dt=(-1)^{n-1}{\frac {m!n!}{(m+n)!}}B_{n+m}\quad {\mbox{ for }}m,n\geq 1$

$\int _{0}^{1}E_{n}(t)E_{m}(t)\,dt=(-1)^{n}4(2^{m+n+2}-1){\frac {m!n!}{(m+n+2)!}}B_{n+m+2}$

Should it be (−1)^n + m instead of (−1)^n − 1? Obviously the integrals must depend in a symmetrical way on the two variables n and m. Michael Hardy 19:18, 7 September 2006 (UTC)

Nope, but either m or n will do. If one is odd and the other even, the right-hand side is zero. So the sign only matters when they are both odd or both even, and in that case it is sufficient to look at one of them. Fredrik Johansson 19:51, 7 September 2006 (UTC)

Very nice. Maybe that should be added to the article. Michael Hardy 20:23, 7 September 2006 (UTC)

Relationship between Eulers and Bernoullis

Clearly there is a relationship between B_n and E_n, but it is never stated explicitly. Septentrionalis PMAnderson 01:59, 10 July 2008 (UTC)

$E_{n-1}(x)={\frac {2}{n}}(B_{n}(x)-2^{n}B_{n}(x/2))$. Not worth the fuss, I suspect. Cuzkatzimhut (talk) 20:02, 20 September 2012 (UTC)

The Euler polynomials are also discussed in the article. There is no indication of why. Nor were they mentioned in the introduction. I added a sentence to the intro to tell the reader. Needed: A better introduction that explains why the Euler and Bernouilli polynomials should be in the same article. Also needed: a section stating the relationship (as above). Also, more connection between the two. At this time there is no visible reason to put them together. I'm not knowledgeable about them so I can't do this.

It would also be helpful to make more of the connection between the polynomials and the numbers. My new sentence mentions that the numbers are treated here. There are also separate articles on the numbers. Someone who knows more should improve my introductory sentence with some explanation that there are also Euler and Bernoulli numbers and that they are treated in greater depth elsewhere. Zaslav (talk) 15:28, 23 August 2015 (UTC)

misprint ? suggestions.

I expect the second formula in "differences and derivatives" to be a misprint : replace by En(x+1) + En(x) = 2*x^n  ? [as a consequence of {(e^D + 1)/2} En = x^n ]. One should tell where the Fourier expansion of Euler polynomials is valid : on [0,1] (open interval for n=1). The formula for Bn is more elegant than for En. Write the latter as : ...*[sum over all odd integers of exp(Pi i kx)/k^(n+1)] ? About the 3d formula in "Integrals", one could explain : by Parseval formula applied to the Fourier expansions. Periodic Bernoulli also appear naturally for p-adic l values. I would add : The Fourier expansion given for the Bernoulli polynomial Bn is in fact a Fourier expansion for Pn. If Pn is viewed as a function on the circle group R/Z, the multiplication theorem can be written as Pn(x) = m^(n-1) [sum over y such that my=x of Pn(y)]Kibour (talk) 20:27, 12 December 2008 (UTC).

unfortunate indexing in the basic formula

In the basic defining formula ("explicite formula") the index k is running along the set of bernoulli-numbers and oppositely to the powers of x. Because the binomial coefficients are a symmetric set of numbers for one row of the binomial matrix it seems to be arbitray, in which direction I use the index. So this formula seems to have no problems.

However, later there is the generalization to fractional orders of the bernoulli-polynomials. But for fractional orders the set of nonzero binomial coefficients is infinite and thus not symmetric. Then the order of indexing is relevant - and then the given indexing in the formula is wrong. If we consider fractional orders, then we have to replace the bernoulli-numbers by zeta-values of fractional arguments, and the index for the binomial coefficients and the index for the zeta-arguments must run in the same direction, so it should always run in the opposite direction for the bernoulli-numbers.

The direction of indexes is "correct" in this sense in the section of the alternate formula ("another explicte formula"): there the index of the binomial coefficients run with the exponents of x, which is correct also for the case of fractional orders.

Note: I've just adapted the first formula by changing the index at the binomial coefficients.

--Gotti 11:37, 19 August 2011 (UTC) — Preceding unsigned comment added by Druseltal2005 (talk • contribs)

Values at x=1/4

I noticed that if B_n is the n-th Bernoulli polynomial, then the numerator of ${\frac {-B_{n}({\frac {1}{4}})}{n\cdot (2^{n-1}-1)}}$ is the numerator of the Bernoulli numbers (when n is even) or Euler numbers (when n is odd). The first values are (start at n = 2)

1, -1, -1, 1, 1, -61, -1, 277, 1, -50521, -691, 41581, 1, -199360981, -3617, 228135437, ...

Fourier series

does that fourier series true for any value of x? ISHANBULLS (talk) 13:43, 20 March 2019 (UTC)

Requested move 24 December 2023

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion.

The result of the move request was: not moved. – robertsky (talk) 15:49, 10 January 2024 (UTC)

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Bernoulli polynomials → Bernoulli polynomials and Euler polynomials – The article Bernoulli polynomials is almost as much about the Euler polynomials as it is about the Bernoulli polynomials, so it appears appropriate to rename the article.  --Lambiam 09:14, 24 December 2023 (UTC) — Relisting. BegbertBiggs (talk) 22:17, 2 January 2024 (UTC)

• I would rather split. Bernoulli polynomials are used in practice noticeably more often than Euler polynomials, and it's distracting for readers who are really just interested in Bernoulli polynomials to have properties of Euler polynomials interspersed throughout. (Or it least, that's how it felt to me when I read this article a few years ago.) I would distinguish it from a case like sine and cosine, where ordinarily anyone who learns about one necessarily learns about the other around the same time. Adumbrativus (talk) 21:39, 27 December 2023 (UTC)
• I'd argue against splitting.Talking about Bernoulli without mentioning Euler is like talking about sine and pretending that cosine doesn't exist. Which would be kind of weird.They really do go together. As to renaming... meh. Sounds Victorian. The Bernoulli and Euler polynomials, and the uses thereof, their general properties and relations, and other important and habitual information concerning their structure.. The shorter title is fine, Euler already has almost two dozen WP article titles with his name in it. 67.198.37.16 (talk) 08:43, 5 January 2024 (UTC)
• Personally I'd recommend against changing the name. The proposed replacement is a mouthful. (If it were up to me, we'd also e.g. move sine and cosine -> sine, but I don't care enough to try it right now.) –jacobolus (t) 09:50, 5 January 2024 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Wrong maximum?

In the section "Maximum and minimum", I am pretty sure that $2n!$ has to be read as $2\,n!$ and not as $(2n)!$. However, in that case, setting $n=1$ leads to ${\frac {1}{2}}=M_{1}<{\frac {2\cdot 1!}{(2\pi )^{1}}}={\frac {1}{\pi }}$, i.e., $\pi <2$. (Same problem applies to the bound for $m_{1}$). Could someone please clarify? Hagman (talk) 12:54, 10 March 2024 (UTC)

Notation for the Stirling numbers of the first kind (signed/unsigned)

In the section on Relation to the Falling Factorial, the claim includes the statement that $$\left[{\begin{matrix}n\\k\end{matrix}}\right]=s(n,k)$$ denotes the Stirling number of the first kind. However, the citing link implies that the LHS refers to the unsigned Sitirling number where the RHS is the signed Stirling number. Which is intended? 81.130.237.47 (talk) 00:19, 31 July 2024 (UTC)