Talk:Laguerre polynomials

Applications of Laguerre polynomials
Can anyone provide more concrete examples of when the Laguerre polynomials arise in real-life science or engineering applications? 171.64.133.56 22:49, 24 February 2006 (UTC)

Well, I fear it's not exactly what you mean, anyway, an application in combinatorics is the following:

How many anagrams with no fixed letters of a given word are there?

It turns out that the answer is:
 * $$\int_0^\infty L_{n_1} (x) L_{n_2}(x)..L_{n_r}(x) e^{-x} dx$$,

for a word with n1 letters X1, n2 letters X2,... nr letters Xr.

PMajer 14:00, 3 May 2007 (UTC)

We use Laguerres to calculate molecules vibrational properties with the Morse oscillator approximation, the wavefunction of a Morse oscillator is (unnormalized): $$ \psi_\nu= \exp\left(-\exp (-\beta x\right) \times \frac{k}{2}) \times \exp(\frac{-\beta x(k-2\times n-1)}{2}) \times L(n,k-2\times n-1,k \exp(-\beta x)) $$ Where k is the force constant of the bond and n is the vibrational quantum number and beta is a function of the anharmonic parameters. — Preceding unsigned comment added by 82.3.202.166 (talk) 18:46, 3 December 2011 (UTC)


 * Real life? They are the cornerstone of the Hydrogen_atom in Quantum mechanics, and likewise the cornerstone of the quantum Phase_space_formulation. Laguerres are as basic as life itself, as a result.  Cuzkatzimhut (talk) 23:51, 18 September 2013 (UTC)

Definition of Laguerre polynomials
Mathias, 20th of June 2011: I added in the article that physicist do not always use the same definition of the associated Laguerre Polynomials and gave some references. I think this is useful information, since any quantum mechanics student will need this and nowadays will look here and to avoid confusing this should be state here. Could someone please put the reference in the buttom of the text. Thank you. — Preceding unsigned comment added by 150.244.101.168 (talk) 11:21, 20 June 2011 (UTC)

There are two accepted definitions of the Laguerre Polynomials, that differ in a $$n!$$ factor. Since these polynomials are referenced in some articles (such as hydrogen atom), we should be careful about which definition to use.

To be coherent with the rest of the article, I have changed the few examples of laguerre polynomials to the standard previously used in the article.

John C PI 17:48, 19 December 2005 (UTC) I came across a couple of (online) articles, where Laguerre polynomials were connected with the analysis of particles, oscillation, resonance-frequences and such. However I'm not able to give a true overview Gotti

I came across a couple of (online) articles, where Laguerre polynomials were connected with the analysis of particles, oscillation, resonance-frequences and such. However I'm not able to give a true overview Another application is in the theory of summation of divergent series. One finds it in G.H. Hardy "divergent series" in connection with a more generalized concept of Hausdorff means. A special consideration was done by Kurt Endl; two articles (german language) are online available at Goettingen Digitizing Centre (GDZ).

Gottfried Helms

--Gotti 07:59, 13 October 2006 (UTC)

I added in a section with explicit comparison with the physicist convention, including some introductory physics references that use the so-called physicist convention. This will hopefully be helpful for some articles such as hydrogen-like atom and hydrogen atom which need to use Laguerre polynomials to define atomic wavefunctions. Twistar48 (talk) 10:36, 16 November 2021 (UTC)

Bibliography: Self-references
Here, as in many other math-related articles, User:Rea5, and other anonymous IPs (probably a dynamic IP) have been adding references to a book authored by Refaat El Ataar. This is not a notable math book (specially because it was edited in 2006!), so many users have been reverting those reference inclusions. Probably, it's a self-reference.

If you are the user who includes this references, please discuss it here first and explain why you think that book should be listed here. Otherwise, references to Refaat El Ataar books in this article will keep being removed.

--John C PI 14:37, 31 January 2006 (UTC)

Usage
It crops up in quantum mechanics for the solution of the spherically symmetric (Coulomb) potential.

While I'm here, would it not be better to express the ODE as

$$ \left(x \frac{d^2}{dx^2} + (1 - x) \frac{d}{dx} + n\right) \, y(x) = 0 $$

for consistency of notation?

Cdyson37 (T) 17:55, 27 May 2006 (UTC)


 * Done. See if this is OK. William Ackerman 16:34, 30 May 2006 (UTC)

For non integer n does the ODE have a solution ?..and if so then:

$$ \int_{0}^{\infty}\int_{0}^{\infty}dudxL_{u}(x)L_{2}(x)exp(-x)(\Gamma (u+1))^{-2} =1 $$

where you 'integrate' over the index 'u' inside the Laguerre function.


 * Well, someone said this section was supposed to be about "usage" when apparently they actually meant "use". But here's a comment about usage:


 * $$ \int_{0}^{\infty}\int_{0}^{\infty} du \, dx \, L_{u}(x)L_{2}(x)\exp(-x)(\Gamma (u+1))^{-2} =1 $$


 * Note that in the TeX display above:


 * * I've put proper spacing between du and dx and after dx;
 * * I've indented the TeX display so that its left edge is to the right of the left edge of the text;
 * * I've put a backslash in \exp. This not only prevent italicization but in some cases provides proper spacing.


 * That's the way to do it. See Manual of Style and Manual of Style (mathematics).


 * I've always found the practice of writing "dx f(x)" instead of "f(x) dx" to be horribly obnoxious. Apparently it's standard among physicists.  They have my condolences (don't read this sentence). Michael Hardy 20:19, 20 June 2007 (UTC)


 * Hi Michael - in defense of physicists, they don't always do it, usually only when its helpful, e.g. in multiple integrals:


 * $$\int_1^3dx\int_4^6 dy\int _0^1 dz\,f(x,y,z)$$


 * Using this notation, its clear which limits go with which variable, rather than writing the (obnoxious imho) x=1, y=4, or z=0 under the integrals. PAR (talk) 21:51, 24 March 2008 (UTC)

Error
Guys, the paragraph about Relation to hypergeometric functions, had a mistake. The Pochhamer symbol should have a superscript n, not a subscript. Check Abramovitz ans Stegan. I have edited this.Spastas (talk) 00:54, 26 March 2008 (UTC)


 * Maybe you're missing the fact that there are differing and conflicting conventions on the use of the Pochhammer symbol. See Pochhammer symbol. Michael Hardy (talk) 15:49, 25 March 2008 (UTC)


 * Michael, you are right. I was missing a lot of facts yesterday. I checked Abramowitz and Stegun again. In chapter 22, there is only a relation between the confluent hypergeometric function and the Laguerre polynomials. I have verified both the relations given in this page and they are correct. The Pochhammer symbol is defined with a subscript and not a superscript as I claimed earlier. I think the proper thing to do here is write the Pochhammer symbol with a subscript as is customary in special functions. As the Wolfram site says, this is an unfortunate notation because it is confusing and that is what confused me. I think it would be better if you changed rather than me. I am not too good at this. Spastas (talk) 00:54, 26 March 2008 (UTC)

Error in the reference "The polynomials' asymptotic behaviour for large n, but fixed α and x > 0, is given by"
Hi. The Abramowitz's reference for the asymptotic behaviour for large $$n$$ is wrong. If you see Abramowitz formula 13.3.8, has nothing to do with this asymptotics. I've searched in Abramowitz, Szego and Askey books looking for this asymptotics expressions, but I have not found them. Are they correct? Thank you very much.


 * The polynomials' asymptotic behaviour for large $$n$$, but fixed $$\alpha$$ and $$x>0$$, is given by
 * $$L_n^{(\alpha)}(x) \approx \frac{n^{\frac{\alpha}{2}-\frac{1}{4}}}{\sqrt{\pi}} \frac{e^{\frac{x}{2}}}{x^{\frac{\alpha}{2}+\frac{1}{4}}} \cos\left(2 \sqrt{x \left(n+\frac{\alpha+1}{2}\right)}- \frac{\pi}{2}\left(\alpha+\frac{1}{2} \right) \right)$$, and


 * $$L_n^{(\alpha)}(-x) \approx \frac{n^{\frac{\alpha}{2}-\frac{1}{4}}}{2\sqrt{\pi}} \frac{e^{-\frac{x}{2}}}{x^{\frac{\alpha}{2}+\frac{1}{4}}} \exp\left(2 \sqrt{x \left(n+\frac{\alpha+1}{2}\right)} \right)$$. .  —Preceding unsigned comment added by Ehuertasce (talk • contribs) 18:06, 23 May 2009 (UTC)

This formula must come from Hypergeometric form of the polynomial. However, it looks more like 13.5.14. --Taweetham (talk) 12:39, 17 September 2009 (UTC)

Or it is Th 8.22.3 and formula 1.71.7 in G. Szegö. In: (fourth edition.),Orthogonal Polynomials Amer. Math. Soc. Colloq. Publ. 23, Amer. Math. Soc., Providence, RI (1975).--Taweetham (talk) 13:08, 17 September 2009 (UTC)

Algorithm source
An algorithm is given for calculating the generalized Laguerre polynomials. Where does it come from? Nicolas Bigaouette (talk) 23:29, 14 December 2009 (UTC)

Applications
I think this article would be improved by a section detailing some applications of (generalized) Laguerre Polynomials, for example in the treatment of the hydrogen atom or in the "isotonic oscillator". Perhaps with a short example if such a thing is possible. 5colourmap(talk) 07 April 2010

Restructure
I don't much like the structure of the section on Generalized Laguerre Polynomials because you have to go looking for the equation that they satisfy and I think that should have a more prominent position in the article. What does everyone else think? 5colourmap(talk) 07 April 2010

Expressed as a summation of binomial coefficients/falling factorials
From observing the first few derivatives of $$e^{-x}x^n$$, it is clear that Laguerre polynomials can be expressed as (ignoring the 1/n! normalization)

$$e^x \frac{d^n}{dx^n} (e^{-x}x^n) = \sum_{k=0}^n \left[ \binom{n}{k} (-x)^{n-k} \frac{n!}{(n-k)!} \right]$$

This is helpful in expressing polynomials of a higher degree than $$L_6$$. Should it be added in? I can't find it in a source anywhere, but it works.. I don't know how WP:OR ties in with math articles..--Dudemanfellabra (talk) 15:30, 24 April 2011 (UTC)

WP:OR cleanup
As part of an ongoing effort to clean our special functions articles of WP:OR, I plan to be removing uncited information from this article in the next few weeks. I have tagged some sections that need citations. When removing the content, I will of course try to reference it myself, but it seems unlikely that I will come up with anything for the vast majority (different authors use different conventions, etc.) I tagged two sections a few weeks ago. I am going to tag more of the article and return in a few more weeks to remove those items that haven't been cited. Sławomir Biały (talk) 12:16, 10 October 2011 (UTC)

Typo
I found a typo in section 3.3 of the article, in the formula immediately after the line "They can be used to derive the four 3-point-rules"

The expression on the right hand side of the equation depends on the variable k, but there is no k on the left hand side. I am assuming that this should be an 'n' on the right instead of a 'k'.

128.132.1.254 (talk) 16:28, 14 March 2016 (UTC)