Talk:Associated Legendre polynomials

Sign Convention
There are two sign conventions for associated Legendre polynomials; some authors include a factor of $$\left(-1\right)^{m}$$ (Condon-Shortley phase) (e.g. Arfken, Mathematical methods for physicists, p.669). For Geophysicists it is interesting to know that Grant and West (Interpretation Theory in Applied Geophysics, 1965, p.223) omit the factor. See also http://mathworld.wolfram.com/Condon-ShortleyPhase.html —Preceding unsigned comment added by 124.177.116.210 (talk) 00:40, 31 October 2009 (UTC)

orthogonality
I removed the following orthogonality condition recently added by User:Physicistjedi:


 * $$\int_{-1}^{1} \frac{P_\ell ^m P_\ell ^n}{1-x^2}dx = \delta _{m,n} \frac{(\ell+m)!}{m(\ell-m)!}$$

Not only is it broken for m=0, but also it's false for l=1, m=1, n=-1, if one just plugs in the formulas for these from the very next section. linas 00:30, 16 March 2006 (UTC)


 * OK, it needs more explanation. It vanishes for $$m = 0$$ and $$n \neq 0$$ from the kronecker delta and the integral diverges for $$m=n=0$$. So I don't think that there is a problem there. But it is tricky for n=-m, because $$P_\ell ^m$$ and $$P_\ell ^{-m}$$ are the same up to a constant. May be we should put the condition m>0 and write what to do for m<=0.

Formula is correct otherwise. I've checked from MathWorld and Arfken&Weber. What do you think? physicistjedi 03:24, 16 March 2006 (UTC)


 * Division by zero is a dangerous notation for infinity, because it does not indicate the limit in which infinity was approached (which makes a big difference in complex analysis). And if m is positive, and n is negative, I can always exchange m and n to violate the conditions, so you'll have to choose the wording carefully. If the core formula is correct, I have no complaints; but the article should be precise, and avoid things that make a naive reader like me flinch when I see it. linas 04:21, 16 March 2006 (UTC)


 * Also, the statement about m=0 is false: take l=2, m=0, n=2, the integral is zero, not infinite. linas 04:24, 16 March 2006 (UTC)
 * I agree, as I said above $$m = 0$$ and $$n \neq 0$$ case vanishes. physicistjedi 05:05, 16 March 2006 (UTC)

So I propose:
 * $$\int_{-1}^{1} \frac{P_\ell ^m P_\ell ^n}{1-x^2}dx = \begin{cases} \delta _{m,n} \frac{(\ell+m)!}{m(\ell-m)!} & \mbox{if }m>0, n>0\\ 0 & \mbox{if }m=0, n>0\\ \infty & \mbox{if }m=0,n=0\end{cases}$$

Together with:
 * $$P_\ell ^{-m} = (-1)^m \frac{(l-m)!}{(l+m)!} P_\ell ^m$$

Any comments? physicistjedi 05:05, 16 March 2006 (UTC)


 * That formula is not symmetric in m and n (it does not define the integral when m > 0 and n = 0). I'd write it as
 * $$\int_{-1}^{1} \frac{P_\ell ^m P_\ell ^n}{1-x^2}dx = \begin{cases} 0 & \mbox{if } m\neq n \\ \frac{(\ell+m)!}{m(\ell-m)!} & \mbox{if } m=n\neq0 \\ \infty & \mbox{if } m=n=0\end{cases}$$
 * Also, somewhere in the article the range of the parameters should be mentioned (something like $$\ell\in\mathbf{N}$$ and $$m \in \{-\ell, -\ell+1, \ldots, \ell-1, \ell\}$$ for the elementary definition, I guess). -- Jitse Niesen (talk) 06:51, 16 March 2006 (UTC)
 * Wow! I wasn't expecting that this thing turns out to be so complicated. Wikipedia will be more precise then any book! Jitse your formula does not work for n=-m which is not zero. What I wrote may not look symmetric, but m>0,n=0 is obviously same with the second case. But probably we can do better. By the way writing the indices is a good point. How did I skip that? I am adding it immediately. physicistjedi 08:30, 16 March 2006 (UTC)


 * We try to be as precise as possible on WP. We sometimes fail, but not for lack of trying. linas 00:00, 17 March 2006 (UTC)


 * "formula does not work for n=-m" &mdash; yes, you're right. So, either add a fourth case to cover this, or stipulate that n and m should be nonnegative. One more thing that I do not like in "your" formula is that you use both cases and the Kronecker delta, which conceptually do the same. But it seems to be correct, so it is just a matter of presentation. -- Jitse Niesen (talk) 10:18, 16 March 2006 (UTC)


 * So shall we write this:
 * "For $$m \geq 0$$ and $$n \geq 0$$ there is a separate orthogonality relation for the same lower index:
 * $$\int_{-1}^{1} \frac{P_\ell ^m P_\ell ^n}{1-x^2}dx = \begin{cases} 0 & \mbox{if } m\neq n \\ \frac{(\ell+m)!}{m(\ell-m)!} & \mbox{if } m=n\neq0 \\ \infty & \mbox{if } m=n=0\end{cases}$$
 * Negative upper indices are related to positive ones with the following identity:
 * $$P_\ell ^{-m} = (-1)^m \frac{(l-m)!}{(l+m)!} P_\ell ^m$$" physicistjedi 19:39, 16 March 2006 (UTC)
 * Since there are no objections so far I am moving this to the main article. physicistjedi 15:58, 28 March 2006 (UTC)

My recent changes
There are 3 aspects of my recent changes that might warrant discussion or controversy:

(1) I have used the superscript-parameters-in-parentheses convention, the same as was done in the orthogonal polynomials page. This was inspired by the convention used with the Jacobi and Gegenbauer polynomials, but is not "industry standard" beyond that. I'm beginning to wonder whether it's worth it. This isn't "original research", but it might border on "original standardization".

(2) I have taken out the implication that $$P_\ell^m$$ with negative values of m is actually meaningful. It's true that there is a very important notion, in the field of spherical harmonics, that m lies in the range $$-\ell \le m \le \ell$$. (The periodic table of the elements would look very different if this were not so!) But those negative values don't specify things like $$P_4^{-3}$$. They specify a different phase of the polar angle $$\phi$$. Therefore, I believe that, prior to making the jump from Legendre polynomials to spherical harmonics and quantum mechanics, the requirement should be $$0 \le m \le \ell$$.

(3) Some texts (Abramowitz and Stegun, Whittaker and Watson) define things with $$(z^2-1)^{\mu/2}$$ instead of the $$(1-z^2)^{\mu/2}$$ used here. That would be more natural when one is defining analytic functions on the complex plane, but doesn't work for the important case of the real interval [-1, 1]. Accordingly, I have switched things, and changed and checked the formulas. When generalized to the complex plane, things work out the same as in A&S and W&W.

Comments? Complaints? William Ackerman 20:42, 12 May 2006 (UTC)

Not polynomial
Note that the correct name for these functions are Associated Legendre Functions -- they are not always polynomials. A bit of a change just to do without discussion. Thoughts? MFago 23:08, 1 June 2006 (UTC)


 * We should not care much what the correct name is, but we should use the name actually being used. I checked Abramowitz & Stegun and they use Associated Legendre function. That supports changing the name. -- Jitse Niesen (talk) 14:25, 2 June 2006 (UTC)

There is no truly good solution to this. The existing nomenclature is messed up. The problem is that there are 3 levels of generality of the functions $$P_\ell^m$$: I think we have 2 choices (other than the completely correct but unworkable "functions associated with the Legendre polynomials"): I could accept either way. Opinions? William Ackerman 21:38, 12 June 2006 (UTC)
 * The most restricted case occurs when l is a non-negative integer and m is zero. These are the Legendre polynomials.  They are the subject of Abramowitz & Stegun chapter 22.  I don't believe that chapter mentions the case of nonzero m, or uses the word "associated" or "generalized".
 * The most general case is for arbitrary l and m. These are the "Legendre functions", and are the subject of A&S chapter 8.  They are also the subject of a chapter in Whittaker & Watson.  Since they are completely general, there is no need to speak of a "general" version of them, though A&S do use the phrase (in a somewhat confused introduction) "associated Legendre functions".
 * In between, we have the case of l and m integers, with m nonzero. If we call them "generalized Legendre functions", that suggests that they are more general than the "Legendre functions", which they are not.
 * Call them "associated Legendre functions". This isn't a lie, but it suggests that "associated" means something other than "more general", since they are actually less general.
 * Leave them as "associated Legendre polynomials", along with the disclaimers that they aren't really polynomials, without explaining (except to people who read this talk page) why we use such an incorrect term. This term, correct or not, seems to be used in the physics and applied maths literature.  Do people agree?
 * I'm not entirely certain that "associated Legendre polynomials" is more prelevant in the literature, but that, as you suggest, the existing nomenclature is messed up. How about (since Associated Legendre functions redirects here) adding a single sentence to the introduction similar to The associated Legendre polynomials are also referred to as associated Legendre functions which is perhaps more correct as they are not polynomials when $$m\neq0$$. MFago 14:51, 13 June 2006 (UTC)
 * In fact, there is just such a sentence in "Definition" -- I'd suggest simply moving it into the introduction. MFago 20:23, 13 June 2006 (UTC)
 * Of course. When there is no good way of presenting it which conceals the facts, just lay out the facts.  Done.  I also changed a section title to avoid using the phrase "general Legendre functions". William Ackerman 22:17, 13 June 2006 (UTC)

Requested move
Associated Legendre polynomials → Associated Legendre function — As noted by others, these functions are not generally polynomials. The page probably would have been moved already but it can't be, because the target name already exists and redirects to Associated Legendre polynomials. I think we should go with the term function, and let A.L. polynomials redirect to it. EricK 18:01, 21 February 2007 (UTC)

Survey

 * Add  # Support   or   # Oppose   on a new line in the appropriate section followed by a brief explanation, then sign your opinion using ~ .  Please remember that this survey is not a vote, and please provide an explanation for your recommendation.

Survey - in support of the move

 * 1) Support: The functions are simply not always polynomials, and this leads to lots of confusion. As an example, Gauss-Legendre quadrature integration schemes are only exact when the integrand is a polynomial, but if the integrand were to be a Legendre function, all bets are off. Lunokhod 19:59, 21 February 2007 (UTC)


 * 1) Support: I agree: The functions are simply not always polynomials, and this leads to lots of confusion. It's hardly worth citing sources: there are thousands, and the table of Associated Legendre Functions listed in the article clearly shows that they are not all polynomials. 76.93.153.69 (talk) 22:23, 11 December 2016 (UTC)

Discussion

 * Add any additional comments:

"The differential equation is also invariant under a change from ℓ to −ℓ − 1, and the functions for negative ℓ are defined by

P_{-\ell} ^{m} = P_{\ell-1} ^{m},\ (\ell=1,\,2,\, ...). "

This sentence and the formula under it are inconsistent! — Preceding unsigned comment added by 144.122.30.156 (talk) 10:41, 25 September 2013 (UTC)

I have moved it. No I'm nobody special nor am I magic nor some wiz-kid hacker. Check this out. I have removed the tag & the request at Requested moves. What to do with the note at the top ... Jimp 00:51, 22 February 2007 (UTC)

I have rewritten the preface paragraph to fit the new article name. What we call these things, and what the preface needs to say, are quite tricky, because common usage is so messed up. I hope this is OK. William Ackerman 02:07, 22 February 2007 (UTC)

Integer variables?
Why do I get automaticly redirected to the site with the Associated Legendre functions when i search for the page on the Legendre Functions? They are obviously not identical. It would be enlightening to have a page on the general case also. Neltah 15:45, 11 May 2007 (UTC)

Errors
It appears that the list of example legendre polynomials is incorrect for negative m. For example, P(m=-1,l=1) is said to be equal to -P(m=1,l=1), but according to the formula, which I believe is correct as it matches Gradshteyn et al, it should be -1/2 P(m=1,l=1)! Further errors seem to occur for other values of negative m on the list!

Definition/proof
The following relation in the article defines associated Legendre functions for positive and negative m:


 * $$P_\ell^{(m)}(x) = \frac{(-1)^m}{2^\ell \ell!} (1-x^2)^{m/2}\ \frac{d^{\ell+m}}{dx^{\ell+m}}(x^2-1)^\ell.$$

This means that the following relation must be proved (and cannot be taken as a definition as is now stated):


 * $$P_\ell ^{(-m)} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!} P_\ell ^{(m)}$$

I tried to find a simple proof (in the literature and by my own wits), but could only come up with a very tedious proof (and did not find anything in the literature). Does anybody know a simple proof? It would be good to either have it, or to refer to it.--P.wormer 15:49, 7 June 2007 (UTC)


 * If we can assume that both sides of the following equation are proportional (with proportionality constant $$C_{lm}$$), then the proof is simple. Because both sides are solutions of the same associated Legendre equation this proportionality is a plausible conjecture.
 * $$ (1-x^2)^{-m/2}\ \frac{d^{\ell-m}}{dx^{\ell-m}}(x^2-1)^\ell

= C_{lm} (1-x^2)^{m/2}\ \frac{d^{\ell+m}}{dx^{\ell+m}}(x^2-1)^\ell $$
 * Use the well-known equation

\frac{d^k}{dx^k} x^n = \frac{n!}{(n-k)!} x^{n-k}, \qquad k \le n. $$
 * Comparing the highest power of x in both sides of the following:
 * $$  \frac{d^{\ell-m}}{dx^{\ell-m}}(x^2-1)^\ell

= C_{lm} (-1)^m (x^2-1)^{m}\ \frac{d^{\ell+m}}{dx^{\ell+m}}(x^2-1)^\ell $$
 * gives

\frac{(2\ell)!}{(\ell+m)!} x^{\ell+m} = (-1)^m C_{lm} \frac{2\ell!}{(\ell-m)!} x^{\ell+m} $$
 * so that

C_{lm} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!}, $$
 * which is the desired result. It remains to prove the proportionality that was conjectured. --P.wormer 08:50, 8 June 2007 (UTC)

Extremely bad notation
Nobody writes brackets (parentheses) around the order. They need to be removed. Brackets (parentheses) are reserved for the Jacobi polynomials and they are not used for associated Legendre functions in the literature I have seen. HowiAuckland 21:59, 16 September 2007 (UTC)
 * The parentheses surprised me; I've not seen them elsewhere and a quick trawl through Google books and my own bookshelves threw up zero instances of this notation. Unless they are supported by the references, the paranthesis should go. --catslash (talk) 00:49, 10 August 2009 (UTC)
 * ...and I shall remove them if nobody leaps to their defence within the week --catslash (talk) 16:12, 11 August 2009 (UTC)
 * (Done)--catslash (talk) 09:59, 22 August 2009 (UTC)

Complex Generality
The beginning of this article should refer to the most general form of associated Legendre function, i.e. those with complex order, degree and argument. These are important special functions and are used to specifically describe spherical harmonics, oblate and prolate spheroidal harmonics, conical functions, toroidal harmonics, etc. All of these require different choices (and sometimes complex valued) order, degree and argument. Also, the argument can be real valued but simply greater than unity. Furthermore there are many formulae for associated Legendre functions which can be written most easily and in the most generality using general complex valued order and degree. There are many examples of these and those of you who are familiar with Abramowitz and Stegun are well aware. In order for this article to be the most general, this should be done. If somebody wants to start a separate article which just works for those associated Legendre functions which are used in spherical harmonics, i.e. those with integer degree and order, and with real argument between plus and minus one, this should be performed. However, this article should be applied to the most general case, otherwise this article should be renamed. HowiAuckland 02:10, 24 September 2007 (UTC)

Proposal to add proof of orthogonality for fixed m
I spent 2 weeks searching the web, trying to find a proof of the orthogonality of Associated Legendre Functions for fixed m without success. So, working together with a theoretical physicst (retired) we developed one. Some of the proof relies on logic I found on the web and some we developed on our own. We would like to contribute this proof to the Associated Legendre Function wiki page. It was suggested to me by RHaworth (who seems to be a Wikipedia administrator) that I work with an established editor on this. I am happy to do so. Please contact me if you are interested in working on this. Dnessett (talk) 17:23, 11 April 2009 (UTC)

Bkocsis (talk) 14:29, 6 April 2010 (UTC)
 * Doesn't this follow directly from the defining differential equation and Sturm-Liouville theorem?

Gaunt formula
In the definition of p, note that n-m-u is always nonpositive, so p=0. Therefore something is wrong with the formula. —Preceding unsigned comment added by 129.88.33.185 (talk) 09:40, 1 October 2009 (UTC)

I noticed the same thing. Not corrected after 3+ years. Steven J Haker (talk) 17:12, 1 March 2013 (UTC)

Bias of the original article
The by far most important and basic Associated Legendre functions are those for the classical case with integer $$m$$ and $$n$$ with $$1 \le m \le n $$. The reader has the right to expect that the article is focused on this case! That some mathematicians have generalize the concept is another matter! This is then a very special field that could be mentioned afterwards (if at all!). The proof I inserted puts the focus to the classical case!

The Legendre polynomials and the associated Legendre functions are needed for the spherical harmonics in 3 dimensional physical space and there is no need nor use for anything then integer $$m$$ and $$n$$ with $$1 \le m \le n $$ in this context!

Stamcose (talk) 15:32, 31 December 2009 (UTC)


 * But about 90% of the article already is focused on this case. I do not think however that it should be exclusively focused on that case (also, I think you mean $$0\le m\le n$$).  Finally, as I already indicated at Talk:Legendre polynomials, we don't usually put detailed proofs consisting entirely of routine manipulations into encyclopedia articles.  We certainly don't put them at the very top of the article (in this case before the relevant Rodrigues formula is even stated.)  I find that in both cases, the entire proof can be summarized fairly succinctly, and I would recommend doing that instead in the Definition... section.   Sławomir Biały  (talk) 17:23, 31 December 2009 (UTC)

No, it is genial mathematics! That only can be understood by carrying out the proof! Without the hints in Courant-Hilbert we would have had no chance to find the proof but even with the hints it requires concentration and precision to get it right! Very useful to have the proof in Wikipedia!

Stamcose (talk) 18:11, 31 December 2009 (UTC)


 * I have no idea what "genial mathematics" means, nor why it necessitates an exclamation point in this case. However, I really must disagree that "we would have had no chance to find the proof" without any hint.  At any rate, this misses the point.  Giving detailed proofs simply is not part of the mandate of an encyclopedia: there are textbooks for such things.  Indeed, a conscientious reader wishing to find out more about a subject is expected to look in the references for proofs (among other things).  Some context on what the principal aims of Wikipedia are can be found by examining what Wikipedia is not, and in particular Wikipedia is not a textbook.     Sławomir Biały  (talk) 19:57, 31 December 2009 (UTC)

(also, I think you mean $$0\le m\le n$$).

No, I do not mean that! The first real associated Legandre function is $$P_n^1 (x) = \sqrt {1 - x^2} {d \over dx } P_n(x) $$.

The convention $$P_n^0 (x) = P_n(x) $$ is a pure formality that can be convenient for writing some formulas but without any real content!

Stamcose (talk) 13:47, 1 January 2010 (UTC)
 * A short simple proof adds useful information for the reader; what about creating a separate page for that proof ? Ulner (talk) 15:02, 1 January 2010 (UTC)


 * (after edit conflict) I disagree with the assertion that $$P_n^0 (x) = P_n(x) $$ has no content.  By itself this might be true, but there are properties that link up the various Legendre functions of different orders (e.g., recursion formulas, differentiation formulas, etc.)  The fact that the Legendre polynomials fit in with this natural hierarchy of functions seems to be quite significant indeed.  Moreover, I doubt that this absolutest point of view would be supported by most sources (Arfken and Weber, for instance, explicitly allow m to be zero).  The point does not seem to be worth arguing about, nor of making a big issue out of in the text.  That most sources allow m to be zero, and negative too, is what we should go with (per WP:WEIGHT).


 * But this is already somewhat offtopic. I am beginning to think that, contrary to the original post, the article emphasizes the integral case too much.  Most serious treatments of associated Legendre functions also discuss associated Legendre functions of the second kind (see, for instance, MacRobert's treatise on spherical harmonics).  As far as I know, there is no compelling reason that these need to have integral arguments, and they are more important in the region |x| > 1.  Their omission seems to be a more serious flaw with the article than any (real or imagined) over-emphasis on the not-necessarily-positive-integer case.  Someone should start compiling sources for the article.   Sławomir Biały  (talk) 15:30, 1 January 2010 (UTC)

You removed again writing:

''' The issue of whether to include routine proofs in these two articles was discussed on the talk page here, at Talk:Legendre polynomials, and at WT:WPM.) I see no discussion there or here, only a statement of yourself.

The only "third view point" anywhere was:

'''A short simple proof adds useful information for the reader; what about creating a separate page for that proof? Ulner (talk) 15:02, 1 January 2010 (UTC)''' Rule of conduct in Wikipedia:

Do not remove what others have been writing!

If I had not respected this rule I would have completely changed your text about spherical harmonics that I do not consider up to standard!

Shall we play ping-pong undoing each others correction?

You wrote yourself on your page:

'''I have retired again because of personal attacks against me that were not refactored. I may be back at some later time. Me... If you don't love me then, I'm a bit shy. Please be gentle, I really am a good kitty. If I'm not, then please slap me with a trout.'''

There could be some good reasons for those "personal attacks"! What about that you really and definitely "retire" instead of preventing other people from improving the articles.

Stamcose (talk) 23:01, 3 January 2010 (UTC)


 * Actually, I initially commented on the inappropriateness of the first proof at Talk:Legendre polynomials. Instead of eliciting a response, however, instead you added another tedious and routine proof (this one to the Associated Legendre function article).  At this point, I commented here about the inappropriateness of including these kinds of proofs, and I also initiated a discussion at WT:WPM to solicit broader input on what to do with the recently-added proofs.  The consensus that emerged there (including the opinion of User:Ulner) was that the proofs were inappropriate for an encyclopedia article, and that a reader was better served by a "thumbnail" version.


 * Finally, you are of course free to continue our discussion at Talk:Spherical harmonics. The current version agrees substantially with the Courant and Hilbert reference, whereas you have not yet supplied any references for your own preferred version in response to my request.  At that article, I have also added a new subsection which discusses spherical harmonics in the context of the orbital angular momentum.  Outside input might also be helpful there in determining the best way forward.   Sławomir Biały  (talk) 23:28, 3 January 2010 (UTC)

To say that a paragraph should be removed because it is "tedius" is conceptually very strange! A reader does not need to read every paragraph, he is allowed to skip those he finds "tedious" or un-iteresting. But for somebody looking for a proof this information could be of great value! Are you worried about the cost of file space? We vaste more file space on these discussions!

There is another recursion formula the proof of which I want to find myself.

This is
 * $$\frac{d^m P_n}{dx^m}=\frac{2 \cdot n - 1}{n - m} \cdot x \cdot \frac{d^m P_{n-1}}{dx^m} - \frac{n + m - 1}{n - m} \cdot \frac{d^m P_{n-2}}{dx^m}

$$

Would be of great value if also such a proof was included!

Stamcose (talk) 18:35, 4 January 2010 (UTC)


 * As I've already said, there is a longstanding consensus on Wikipedia that detailed step-by-step proofs are not appropriate in encyclopedia articles. I certainly didn't make this up (contrary to what your edit summary would appear to suggest), but in this case I rather strongly agree that the proof doesn't belong.  If you will again read what I wrote above and in my own edit summary, the proof is not just tedious, but also quite a routine calculus exercise: it follows just by differentiating the Legendre equation m times and using the Leibniz rule for repeated differentiation.  Finally, while I certainly don't object to including this last formula (provided, that is, you can find sources for it), I would also oppose including a proof on much the same grounds.  Sławomir Biały  (talk) 19:15, 4 January 2010 (UTC)
 * I agree with Sławomir Biały, there seems to be a consensus on Wikipedia that detailed step-by-step proofs are not appropriate in encyclopedia articles; if Stamcose don't agree please discuss this at this talk page or here: []. Removing the detailed proofs are not vandalism (as Stamcose said in a recent comment); the reason is that removing the proofs follow the consensus among Wikipedia editors. Ulner (talk) 23:20, 4 January 2010 (UTC)


 * See WikiProject_Mathematics/Proofs. linas (talk) 21:56, 13 November 2010 (UTC)

Generalizations
I found the following posted on my personal talk page; I think it's better to reply here. linas (talk) 21:32, 13 November 2010 (UTC)


 * I do not understand you idea : . As such, Legendre polynomials can be generalized (In what way?)  to express the symmetries of semi-simple Lie groups (not SO(3)?)  and Riemannian symmetric spaces. (not euclidic ?) Gvozdet (talk) 12:02, 11 October 2010 (UTC)


 * Hi, Sorry for the late response, I do not get on Wikipedia very often. This is not "my idea"; rather, there are papers, books and conferences on the topic. Simply-put, whenever one has a space with a continuous symmetry, one also has a symmetry group describing the symmetry; these are essentially the Lie groups. One may define a Laplacian on such spaces, and then study the eigenfunctions of the Laplacian on these spaces. These solutions can be considered to result in "generalizations" of the associated Legendre polynomials. The bottom of the article on spherical harmonics provides more detail, and at least four references. linas (talk) 21:32, 13 November 2010 (UTC)

Recurrence relations
I believe at least some of the recurrence formulae were copied from a source that excludes the Condon-Shortley phase (which the rest of this article includes). For example I tested the second relation on Wolfram Alpha here and got the negative of what the article says it should be.

I will correct the ones I think are wrong and add another recurrence relation I derived from the others and found useful. However since I don't have a reference (e.g. Abramowitz and Stegun write about Legendre functions of a complex variable and it's subtly different) and I haven't derived them myself, I can't be 100% sure of their accuracy.

Edit: This book has recurrence relations equivalent to some in the article but defines the polynomials without the Condon-Shortley factor.

202.8.37.86 (talk) 05:57, 27 January 2012 (UTC)

Reference needed
"This equation has nonzero solutions that are nonsingular on [−1, 1] only if ℓ and m are integers with 0 ≤ m ≤ ℓ, or with trivially equivalent negative values." Where is this statement taken from? I have not been able to find it in the literature. 5.61.176.43 (talk) 00:07, 9 April 2016 (UTC)

I looked at the reference above and I have found no statement in it that supports this claim. 5.61.176.43 (talk) 19:48, 19 April 2016 (UTC)

Assessment comment
Substituted at 21:34, 26 June 2016 (UTC)