Talk:Hermite polynomials

message: This is an oh so minor edit, but the page said it was listing the first ten polynomials when in fact it listed eleven.  Mbset (talk) 23:53, 17 June 2008 (UTC)
thread_title: Indexing of the polynomials

message: Can anyone provide more concrete examples of when the Hermite polynomials arise in real-life science or engineering applications? 171.64.133.56 22:50, 24 February 2006 (UTC)
thread_title: Applications of the Hermite polynomials

message: This article needs to have some discussion of the relationship of Hermite polynomials to the eigenfunctions of Schrodinger's equation for the quantum harmonic oscillator, since that is one of the major problems in which they arise.  (I didn't want to insert it offhand...have to do some checking to make sure the conventions match.)
More generally, I would recommend discussing the history and applications of the Hermite polynomials before giving their mathematical definition, to make this article more broadly accessible.
—Steven G. Johnson 03:35, 14 October 2005 (UTC)
replies: text: Which came first? Study of the polynomials, or study of quantum mechanics? --HappyCamper 00:19, 15 October 2005 (UTC)
replies: text: Well, I thought Charles Hermite himself introduced the Hermite polynomials, and he died in 1901, when quantum mechanics was just getting started.  I wouldn't be surprised if he never heard of quantum physics. Michael Hardy 00:26, 15 October 2005 (UTC)
replies: text: Ah, yes you are right. The solutions to the harmonic oscillator do use Hermite polynomials, but perhaps a mentioning in passing is sufficient. --HappyCamper 01:18, 15 October 2005 (UTC)

text: Hi Micheal, you may recall our earlier conversation, why, at the very top of this very talk page!  Do I need to report you to Wikipedia Esperanza as having too high a stress level? What can I do to ease your stress and make you feel better about things?  linas 02:14, 16 October 2005 (UTC)

text: Yup. See this rich paper:  --HappyCamper 01:18, 15 October 2005 (UTC)
replies: text: Ohh, I really like the structure constants c_{klm} defined on page two of that ref. Care to add that to this article? Not too often that one sees such nice, cleanly defined coeffs outside of Lie algebras. linas 02:25, 16 October 2005 (UTC)
thread_title:  Harmonic oscillator? 

message: I'm going to follow Abramowitz & Stegun conventions in this comment, since I agree with Linas:
it's as close to a standard as exists for special functions. It looks to me like the short section
Eigenfunctions of the Fourier transform is incorrect. The functions H_n(x)e^{-\frac{x^2}{2}}
are indeed eigenfunctions of the Fourier transform when the physicist's definition is used.
However the eigenvalues are -(-i)^n for the forward transform and -i^n for the
inverse transform. (The product of the forward and inverse eigenvalues must be unity; if both eigenvalues
were -i^n, as the entry states, their product would be (-1)^n).
Additionally, the probabilist's Hermite polynomials do not appear to me to yield eigenfunctions 
of the Fourier transform when multiplied by e^{-\frac{x^2}{2}}. 
Since H_{e_n}(x) = 2^{-\frac{n}{2}} H_n \left( \frac{x}{\sqrt{2}} \right), 
the scaling theorem can be used to show that the functions H_{e_n}(x) e^{-\frac{x^2}{4}}
are eigenfunctions of the 'Fourier-like' transform
T(f(x)) = \frac{1}{2\sqrt{\pi}} \int_{-\infty}^{\infty}f(x) e^{-\frac{itx}{2}} dx and it's inverse.
The eigenvalues are once again -(-i)^n and -i^n respectively.  --jmh 01:13, 15 January 2006 (UTC)
thread_title: Eigenfunctions

message: Disclaimer:  I do not speak or read French.
I have just reverted the French grammatical change made by
Eskimbot.  I assume it was enforcing a WP style guideline that
says things in this context should be singular.  But orthogonal
polynomials are special.  There's no such thing as an
orthogonal polynomial -- it has nothing to be orthogonal
to.  I believe this issue has been discussed in the past
by one of the regular gang, probably Michael Hardy.
If there is a style guideline that is enforced by bots
and that we need an exception for, there may be an
ongoing problem.  Is this going to continue to happen?
Can we flag this so that it doesn't?
William Ackerman 04:07, 21 January 2006 (UTC)
thread_title:  Plurals 

message: 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. (this may be coincidence but the user name Rea coincides with the initials of the author).
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)
thread_title:  Bibliography: Self-references 

message: User:Genjix has changed the definitions to match
the "physicists" polynomials, from the previous use of the
"probabilists" polynomials.  This is a very controversial
point, being discussed at length on this page.  I happen
to be in the "physicist" camp and sympathize with the change,
but, unfortunately, that makes a number of things on
the page (weight function, graphs, etc.) incorrect.
So I have temporarily patched things by showing both conventions
in some of the most prominent places, and labeled the graph
as showing the probabilists' polynomials.  This issue needs to
be cleared up, and I'm not sure how to proceed.
William Ackerman 22:43, 27 February 2006 (UTC)
replies: text: Including everything on both conventions, and the context within which each is preferable, would seem to be the ideal way to write this.  I'll probably be back. Michael Hardy 23:48, 27 February 2006 (UTC)
thread_title:  physicists vs. probabilists redux; partial revert 

message: Thanks very much to User:MaciejDems, we now have two SVG versions of images (see :File:Herm5.svg and :File:Herm50.svg). I am not familiar with this topic, so I am presenting them to whoever is watching this page so that they can be included in the article where necessary. Also, if anyone would like to help me out, can you go to the commons and categorize these images. I really don't know what they are or how they should be categorized. Thanks!-Andrew c [talk] 15:29, 11 February 2009 (UTC)
thread_title:  New images! 

message: I noticed Cubic_Hermite_Interpolations redirects here. I think the redirect should better point to the relevant page here: Hermite_interpolation or maybe Cubic_Hermite_spline but not here... 82.130.71.230 (talk) 17:26, 1 February 2010 (UTC)
thread_title:  Wrong redirect 

message: Is it possible that the wikipedia page be brought into line and use the same notation as Wolfram. ie, so called Physicists polynomials be denoted H_n(x) and so called probabalists polynomials be denoted He_n(x). The Wolfram page, with many references, also states the the probabalists polynomials are rarely used, so why is it that the wikipedia article focuses on them so much? Especially seeing as certain characteristics are lost (H_n(0) no longer generates Hermite numbers, and the relation to generalised hermite polynomials (which wikipedia doesnt seem to have an article for))  —Preceding unsigned comment added by 92.9.145.232 (talk) 14:24, 20 March 2010 (UTC)
replies: text: Abramowitz and Stegun used that notation years before Wolfram :) http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP?Res=150&Page=775&Submit=Go. -- Avi (talk) 21:31, 10 June 2010 (UTC)
replies: text: Rosenhead & Fletcher's Mathematical Tables recommended the use of He_n(x) for the physicists polynomials and he_n(x) for the probabilist's polynomials. Quite useful it seems to me. Concerning the last but one comment, it may be of interest to know that probabilists polynomials are commonly used in communication engineering in connexion with noise transmission.JFB80 (talk) 08:04, 27 February 2011 (UTC)
thread_title:  Wolfram 

message: Ought we change n to \alpha in this section, and replace the factorial sign with a gamma function?  Part of the point of writing something as a contour integral (aside from asymptotic estimations using steepest descent etc.) is that it generalises to any complex n.  I hesitate to do this since I can't see off the top of my head what this does for the orthogonality relationship between the functions.  7daysahead (talk) 13:44, 14 October 2010 (UTC)
thread_title:  Contour integrals 

message: Where did the asymptotic expansions come from? the Abramowitz & Stegun reference is bogus, there's nothing on that page that even resembles the results stated. Anyone? This result needs a good reference, there's precious little information on either asymptotic expansions for Hermite polynomials or their zeros on the web, and this needs to be reliable. (Episanty (talk) 06:05, 22 March 2011 (UTC)
replies: text: Did you look in Erdelyi,Magnus & Oberhettinger (cited reference)? That is rather encyclopaedic.JFB80 (talk) 11:57, 22 March 2011 (UTC)
replies: text: I found a more accurate path within Abramowitz and Stegun. I don't know how the previously cited formula, 13.3.8, could be used to obtain the asymptotic expansion claimed, but I changed it to a new path through the Tricomi function U(a,b,z), in which the limit process is clearly described in the reference.  Episanty (talk) 05:39, 18 April 2011 (UTC)
thread_title:  Asymptotic expansion 

message: The multiplitcation theorem for He(gamma x) is wrong. For the correct form one has to either replace both times He() by H() or add a factor (1/2)^k in the sum.
Furthermore the upper bound of i is missing. (It is n/2 rounded down)  — Preceding unsigned comment added by 138.246.47.39 (talk) 12:55, 20 July 2011 (UTC)
thread_title:  Multiplication theorem 

message: The notation He_n\,\! is being using for the probabilists' Hermite polynomials, and H_n\,\! for the physicists ones. This notation is incorrect: He_n\,\! written like this always means the product of two variables H\,\! and e_n\,\!. Multi-character function names are always written in Roman instead, like \sin\,\!, \exp\,\!, or (as this article should be) \operatorname{He}\,\!.
I know this looks a little odd since H is not Roman, but it's still better than looking like a product. (To be honest, I think this is because He is terrible notation, but it looks like this was consensus.)
Any objections if I fix the article? In the above paragraph I've forced math images to make my point, but in the article I would just use  (i.e. ) when not already in an equation. Quietbritishjim (talk) 17:13, 3 December 2011 (UTC)
replies: text: Alternatively, we could use a non-standard (but less stupid) notation like \tilde{H} (H̃ in text). Quietbritishjim (talk) 17:20, 3 December 2011 (UTC)

text: As evidence, I notice that the MathWorld page on Hermite polynomials uses . Quietbritishjim (talk) 17:39, 3 December 2011 (UTC)
thread_title:  Notation for probabilists' Hermite polynomials 

message: The IP user User:74.108.147.80 added the below comment to the article. I don't know whether it is correct or not, but it belongs on the talk page rather than the article, so I am moving it here.
These first two expressions for even and odd values of n do not find Hermite polynomials for even and odd values of n. For instance this can be seen clearly from the summation for Hermite polynomials with even n, when n=4, (n/2 - l)! goes to infinity.  For the summation for odd n, when n=5, ((n-1)/(2 - l))!also goes to infinity, rendering these first two expressions useless, However this third summation that finds Hermite polynomials with even and odd n works perfectly for all relevant n.
It looks a bit like the author doesn't realise that 0!=1 (rather than 0), so that it is fine to divide by 0!
Quietbritishjim (talk) 00:38, 12 April 2012 (UTC)
replies: text: The comment is clearly not correct.  There's no way any term of this will "[go] to infinity" in any way that will make things undefined.  Sławomir Biały (talk) 02:53, 12 April 2012 (UTC)
thread_title:  Comment on explicit expressions by 74.108.147.80 

message: I think the normalization of the Hermite Function is incorrect. As written the integral from -infinity to infinity evaluates to pi^(1/4). I think sqrt(pi) in the first set of parenthesis should just be pi, in which case the integral equals 1. I can't find a reference outside wikipedia for this though. 128.100.76.24 (talk) 17:48, 4 September 2012 (UTC)
thread_title:  Hermite function normalization 

message: In mathematics "normalization" does not always mean "having unit length", contrary to recent edit summaries.  Often things are "normalized so that the integral is one", but things can be normalized to other convenient values as well (as is the case with the Hermite polynomials).  See, for instance, Arfken and Weber "Methods of mathematical physics".  Sławomir Biały (talk) 12:18, 9 October 2012 (UTC)
replies: text: I have to agree with the other editor that, in my experience, normalization means "having unit norm". However there can still be different normalizations depending on what norm is involved. Very often L1, L2 and L∞ are the three main choices. Could that be the case here? (With Hermite functions, not Hermite polynomials?) If so, there would be multiple "normalizations" regardless of what you mean by that word. Quietbritishjim (talk) 14:49, 9 October 2012 (UTC)

text: By the way, a better word to use here would probably be "convention". Quietbritishjim (talk) 14:56, 9 October 2012 (UTC)
replies: text: I agree with the now-used word "standardization".  But, to reply to your comment, if the norm is up for grabs, I don't see how this is different from what I already said.  Certainly "different normalizations" (in my sense) will be "normalized" (in your sense) with respect to different norms.  In fact, for orthogonal polynomials, there's no need to leave weighted L^2 spaces for that.  But when people refer to different normalizations, this is seldom what they actually mean (even if it is true). I'm just saying, from my own experience, having read the books that I have, and communicated effectively with the other individuals that I have met in my career, that this seems to be the case.  (I have at times been active in several different areas of applied mathematics and mathematical physics, and so I consider myself somewhat experienced with the use of terms like these in different areas, although obviously people focused in different areas of research don't always agree about everything.) Sławomir Biały (talk) 01:25, 10 October 2012 (UTC)
thread_title:  Normalization 

message: I apologise in advance for reopening this whole "probabilists vs physicists" debate, but the current compromise (having two different definitions H_n and He_n which are really just rescalings of each other) seems to me confusing at best. I would be very tempted to use the more general notation H_n(x,c), where H_2(x,c) = x^2 - c, H_3(x,c) = x^3 - 3cx, H_4(x,c) = x^4 - 6cx^2 + 3c^2, etc. In this way, the "probabilists" version is H_n(x,1), while the "physicists" version (which is really the "study of the quantum harmonic oscillator" version; when using Hermite polynomials for Wick ordering, no physicist would dream of using that convention) is proportional to H_n(x,1/2). The reason why the H_n(x,c) are natural is of course that they are orthogonal for the Gaussian measure with variance c. In particular, H_n(x,c) is the same as the Wick power :x^n: with respect to the Gaussian measure with variance c. Hairer (talk) 15:55, 26 April 2014 (UTC)
replies: text: Why is reduplicative parallel exposition confusing? They have different names, their scaled connection is stated clearly up front, all physicists learn them through their trusty paradigm of the quantum harmonic oscillator just fine, and nobody in his right mind could fail to scale by another, general, c the expressions given, if the need arose. Why does the quick user, the proverbial undergraduate,  checking conventions in a hurry on a cell phone, have to genuflect to a spuriously "general" expression and substitute N=2 for a bicycle on a vehicle of N wheels? I would be alarmed at further abstraction in the exposition: I immediately form a bad impression on a paper I'd be refereeing, seeing this type of stuff in its preamble. This is not a treatise, but an accessible, practical tool: the generalization proposed is dealt with adequately here.  A mathematically inclined reader should delve right away into  the general section and leave Hermites alone, to start with. A signal processor/information scientist would rush to  the Fractional FT section, and would shake his fist at further complications. I would let sleeping dogs lie--it was so tricky reaching an equilibrium on this.  Cuzkatzimhut (talk) 16:26, 26 April 2014 (UTC)
replies: text: In connection with the first comment it may be of interest to note that polynomials of this form with c as variance σ² have already been used in communication theory for general Gaussian processes. JFB80 (talk) 09:10, 29 April 2014 (UTC)
replies: text: I fear you are being unclear: Are talking about these in this article, and what exactly are you proposing to do to them? Of course generic variance can be of interest; that is why they are here. What is the specific proposal?   Cuzkatzimhut (talk) 14:24, 29 April 2014 (UTC)
replies: text: I checked the article and see that the polynomials for general variance are mentioned at the end (but in a strange notation). Their properties are exactly similar to the probabilists polynomials but made homogeneous using the variance. They also have a multidimensional version related to known multidimensional Hermite polynomials (not mentioned in the article) I have made a lot of use of these polynomials in my work and have denoted them in a similar way to that described above. My proposal: change the notation in the article to that used in the literature on these polynomials.JFB80 (talk) 16:07, 29 April 2014 (UTC)
thread_title:  A different compromise? 

message: "Surprisingly simple", albeit wrong: the left hand side is a monomial of order , while the r.h.sides involve polynomials of order floor(n/2) or lower.   Cuzkatzimhut (talk) 22:53, 27 January 2015 (UTC)
thread_title: wrong  Inverse Explicit Expression