Talk:Legendre polynomials

Completeness Relation
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)

Applications of the Legendre polynomials
Can anyone provide more concrete examples of applications of the Legendre polynomials, or when the Legendre differential equation arises? 171.64.133.56 22:48, 24 February 2006 (UTC)


 * In fact, they form a basis for the space of piecewise continuous functions defined on this interval, so any such function can be written as a linear combination of Legendre polynomials.

These can't be bases and linear combinations in the sense of linear algebra. Are we talking Hilbert space basis here?

Also, is it correct that P_n solves the differential equation with parameter n, or is n used in two different senses here? AxelBoldt 23:22 Feb 12, 2003 (UTC)

The Legendre polynomials form an orthogonal basis for the Hilbert space of square-integrable functions on the interval from -1 to 1, so we're talking about a "linear combination" in the Hilbert-space sense, i.e., not a finite linear combination. So it's easy to see that if sn is the nth partial sum, then
 * $$\lim_{n\rightarrow\infty}\int_{-1}^1\left|s_n(x)-f(x)\right|^2\,dx=0.$$

But if you want pointwise convergence to a piecewise continuous function, that takes more work. The person who wrote those words doesn't seem aware of the difference kinds of convergence or of the fact that mathematicians may often construe "linear combination" to mean finite linear combination. Michael Hardy 22:11 Mar 13, 2003 (UTC)


 * Yes, well, Hilbert spaces should be discussed in the article on orthogonal polynomials anyway, so as to cover the whole kit-n-kaboodle. linas 05:32, 28 Mar 2005 (UTC)


 * Strange that in the same section you would talk about pointwise convergence towards piecewise continuous functions, and state the completeness relation in a distributional form, but no one dares to talk about the (far easier formulated) convergence in the Hilbert spaces sense (i.e., with respect to the norm in L_2([-1,1])) — Preceding unsigned comment added by 178.3.54.188 (talk) 13:29, 3 April 2022 (UTC)

--

Usually the singular makes more sense that the plural in the title of an article, but in this case, the title "Legendre polynomial" makes the same amount of sense as "Beattle" as the title of an article about John, Paul, George, and Ringo. The whole sequence Legendre polynomials is to be thought of as a unit. Michael Hardy 22:15 Mar 13, 2003 (UTC)

Despite the style pronouncements at , Legendre polynomial is not appropriate as a title for an article about the

Legendre polynomials

(which see!). An article about John, Paul, George, and Ringo would not be titled Beatle, but Beatles' or The Beatles''. So it is here. One does speak of a "Legendre polynomial" in the singular in some contexts, but generally those are of interest only because this polynomial sequence, like others, is thought of as a unit.

Associated Legendre functions?
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)


 * 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)

Standardization
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)


 * 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)


 * 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)


 * 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)

Need section on roots
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)


 * 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)

Rodrigues' formula
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)
 * 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)

Generating Function
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)

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

Additional properties of Legendre polynomials
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)

Sign error in series expansion ?
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)

Explicit representations
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)

Now done. — Preceding unsigned comment added by Uhap027 (talk • contribs) 20:51, 6 December 2017 (UTC)

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Chebyshev polynomial table
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)

missing definition of [t^n]
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)