Talk:Laguerre polynomials

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[edit] 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)

[edit] Definition of Laguerre polynomials

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)

[edit] 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)

[edit] 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 Wikipedia:Manual of Style and Wikipedia: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)

[edit] 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)