Talk:Legendre polynomials

message: Is it appropriate to add the completeness relation to this article? 
: \sum_{n=0}^{\infty} \frac{2 n + 1}{2} P_n(x) P_n(y) = \delta(x - y) for -1 < x < 1 and -1 < y < 1.
I feel that they are as important as orthogonality relations yet receive little mention. 75.42.225.88 (talk) 15:28, 3 July 2011 (UTC)
thread_title: Completeness Relation

message: My quantum mechanics text says speaks of something called the associated Legendre functions, which appears to be distinct from the associated Legendre polynomials.  They have the form:
:f_{lm}(\theta) = \frac{(\sin\theta)^{|m|}}{2^l l!} \left [ \frac{d}{d(\cos\theta)} \right ]^{l+|m|}(\cos^2(\theta)-1)^l
Is this something that somehow missed having an article? --Smack (talk) 05:46, 7 May 2005 (UTC)
replies: text: No, you just didn't read the article carefully enough. Associated Legendre polynomials clearly mentions x=\cos\theta. linas 16:55, 8 May 2005 (UTC)
thread_title:  Associated Legendre functions? 

message: I have reverted the word "normalize" back to "standardize",
since the polynomials are not normal, and the standardization
is just a convention.  I'm still not happy with this -- it's
a conflict between saying exactly the right thing vs. belaboring
a point that isn't really important.  Improvements?
William Ackerman 23:38, 14 March 2006 (UTC)
replies: text: Works for me. Nothing wrong with adding a sentance that says: "Sometimes this is called a "normalization", although the correct meaning of normalization is that the integral of the square of the function should be unity." linas 00:14, 15 March 2006 (UTC)
replies: text: I still think normalized is the correct word here. Normalization does not necessarily mean to transform something to norm one, it is used for any transformation to get something in standard form. However, it is not that important. I do wonder though why the word "standardize" is written in bold. -- Jitse Niesen (talk) 09:16, 15 March 2006 (UTC)
replies: text: You are using the term normalization in a too narrow sense. Requiring P_n(1)=1 is a true normalization, just not w.r.t. the Hilbert space norm, but the sup-norm (L_\infty-normalization). In the same spirit, the property that Legendre polynomials take the maximal value in x=1 is of independent interest and deserves to be mentioned explicitly  — Preceding unsigned comment added by 178.3.54.188 (talk) 13:23, 3 April 2022 (UTC)
thread_title:  Standardization 

message: The roots of the Legendre polynomials are very important, e.g. for Gaussian quadrature.  There are a number of beautiful proofs that P_n(x) has exactly n roots in [-1,1].  It would be nice to mention this property (perhaps with a proof or at least with a reference) in the article.
—Steven G. Johnson (talk) 04:54, 16 March 2009 (UTC)
replies: text: If you define the Legendre polynomials by Rodrigues' formula, then it is just Rolle's theorem, so a proof is not out of the question (but see my post below).  That the roots are simple follows then from the fact that the polynomials satisfy a second order linear homogeneous differential equation, and so cannot vanish to higher order at any point of their domain, by uniqueness of solutions.  Much more can be said about the zeros of the Legendre polynomials, and there should be a section about them. Sławomir Biały (talk) 15:06, 29 December 2009 (UTC)
thread_title:  Need section on roots 

message: I have added to the lead paragraph a summary of the proof of Rodrigues' formula.  However, I now wonder whether such a detailed proof benefits the article.  We don't usually put detailed proofs in Wikipedia articles, particularly when these proofs are fairly routine manipulations; I think taking n+1 derivatives probably qualifies as routine.  On a related note, a more useful structure might be to have a section on the various equivalence characterizations of the Legendre polynomials: differential equation, Rodrigues' formula, generating function, recursive definition.  It can state briefly the reasons underlying the equivalences (hopefully without too much detail).  Sławomir Biały (talk) 15:02, 29 December 2009 (UTC)
replies: text: I think that for special functions explanations of some easy (quick) proofs in the article makes sense. Ulner (talk) 23:24, 4 January 2010 (UTC)
thread_title:  Rodrigues' formula 

message: I am changing "generating function" to "ordinary generating function".
I also think the word "Taylor" is incorrect and should be removed; which I will do in about a week unless somebody objects.  — Preceding unsigned comment added by Rrogers314 (talk • contribs) 19:07, 6 April 2014 (UTC)
thread_title:  Generating Function 

message: The current article indicates:
* "Legendre functions of the fractional order of \nu"
* "Legendre polynomial of the degree n"
I'm afraid that the words order and degree would be contradictory to each other. --Kkddkkdd (talk) 14:56, 16 April 2015 (UTC)
replies: text: I've changed order to degree in this section. --catslash (talk) 16:22, 16 April 2015 (UTC)
thread_title:  Legendre functions of fractional order 

message: The last two identities here are wrong. The correct limit is 
lim_{\ell\rightarrow\infty} P_{\ell}(\cos \theta/\ell)
=
J_0(\theta) + \mathcal{O}(\ell^{-1})
One cannot just take finite angles an expect the identity to work.  — Preceding unsigned comment added by 130.225.212.8 (talk) 14:01, 6 May 2016 (UTC)
thread_title: Additional properties of Legendre polynomials

message: It seems that the series expansion in powers of x^k that is given in the article yields the wrong sign for Legendre polynomials of odd order. This can be checked by using this formula to calculate the lowest order polynomials by hand or with Mathematica. To obtain the correct formula, a factor (-1)^l would have to be included.
There is no reference given for this formula, and it does not appear in either Abramowitz or Arfken.  — Preceding unsigned comment added by 160.103.208.6 (talk) 11:09, 8 December 2016 (UTC)
thread_title:  Sign error in series expansion ? 

message: In the section "Recursive definition", paragraph starting "Explicit representations include", the second formula involves binomial coefficients with negative (upper) argument. While the formula is not incorrect, it would be more elegant to express in terms of binomial coefficients with positive arguments. This is easy to do. I propose to re-express this formula in this way, unless there are objections. Uhap027 (talk) 12:43, 3 December 2017 (UTC)
thread_title:  Explicit representations 

message: Hello fellow Wikipedians,
I have just modified 2 external links on Legendre polynomials. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
*Added archive https://web.archive.org/web/20090920070434/http://edocs.ub.uni-frankfurt.de/volltexte/2007/3757/pdf/A009566090.pdf to http://edocs.ub.uni-frankfurt.de/volltexte/2007/3757/pdf/A009566090.pdf
*Added archive https://web.archive.org/web/20060427014500/http://www.du.edu/~jcalvert/math/legendre.htm to http://www.du.edu/~jcalvert/math/legendre.htm
When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
Cheers.—InternetArchiveBot (Report bug) 03:58, 20 December 2017 (UTC)
thread_title:  External links modified 

message: The section Legendre polynomials in trigonometry consists mostly of a big table, which as far as I can tell is just giving the coefficients for the Chebyshev polynomials (of the first kind) with respect to the basis of Legendre polynomials—there is no noticeable pattern in these coefficients, nor any hint of an application that would make this interesting. Not everything verifiable is worth including in an article.
The very last formula in that section is however different—it seems to say that the Chebyshev polynomials of the second kind U_n are convolutions of the Legendre polynomials, which is an interesting and unexpected relation—although it could use a source. 130.243.68.155 (talk) 14:37, 11 June 2019 (UTC)
thread_title:  Chebyshev polynomial table 

message: in section Rodrigues' formula and other explicit formula,  second equation, [t^n] is undefined.  I would guess it means the Taylor coefficient of t^n, but to my knowledge, this is not universal notation. Chris2crawford (talk) 01:04, 27 February 2024 (UTC)
thread_title:  missing definition of [t^n]