# Talk:Hermite polynomials

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## Indexing of the polynomials

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)

## Applications of the Hermite polynomials

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)

I really have absolutely no idea how to edit pages and i am far too busy to learn, but shouldn't there be something on the self-duality of hermite polynomials. (self-duality means that a function is its own fourier transform). This property is inherrently important in the use of Hermite poly's and is one of the physical reasons that they are the solution to the Schrodinger Wave equation for the x^2 potential. A reference that shows this is http://www.math.rutgers.edu/~sdmiller/math574/math574notes.pdf just above equation 4.5. Again sorry about me not knowing how to edit pages. - Chris 21st May 2007

## Conventions

This article does not follow the standard abramowitz & stegun conventions. Grrrrr >:( Most of the article seems to discuss He_n not H_n and this is confusing w.r.t. standard quantum mechanics usage. I'd like to see consistent notation usage, but don't want to embark on this cleanup now. linas 04:56, 26 Mar 2005 (UTC)

Standard?? I'm tempted to say: nobody uses that convention; only physicists use it. The convention used here is standard. Among probabilists. The convention used among physicists seems to be a result of the idiotic practice of defining the "error function" as
${\displaystyle \operatorname {erf} (x)=\int _{0}^{x}e^{-u^{2}}\,du\,}$
rather than as
${\displaystyle \operatorname {erf} (x)=\int _{0}^{x}e^{-u^{2}/2}\,du\,}$
which is rather obviously the right way to do it. Michael Hardy 21:29, 26 Mar 2005 (UTC)
I'm glad that you were only tempted, and didn't actually say any of those things :) A&S is more of a standard than any other text on special functions I can think of. As to "nobody uses that convention"; of the half-dozen books I have that might mention it, I'm pretty sure none of them used the convention in this article. I don't doubt the probabilists use a different convention; the goal here is to give two different functions two different unique names so that we can reliably know which one is being talked about. My pet peeve is that the Gamma function is clearly off by one, but it seems that only number theorists are aware of this fact. There are actually plenty of stories like this in math. What can one do? We now live in a global, connected world, the virtual distance between different topics is much much smaller. linas 01:57, 2 Apr 2005 (UTC)
Well should at least some weight be given to the fact that the convention followed by probabilists and statisticians makes sense and the one followed by physicists and special-functions theorists does not? And also the fact that the convention followed by non-statisticians can be regarded as at least a little bit non-standard, since there aren't very many non-statisticians (except among those who never think about Hermite polynomials at all)? Michael Hardy 02:33, 2 Apr 2005 (UTC)
Uh, well, this bit about one convention making sense, and the other not, is at best debatable and at worst trolling. In the harmonic oscillator (if I recall correctly) using the statistician's definition would lead to ugly factors of square-root two hanging out in ugly places (if I remember correctly). And vice-versa. I presume that this is the reason for the divergent definitions. No-one wants to use the "ugly" form in their work. And as to the suggestion to turn this into a popularity contest, I don't think this solves any problems. Never mind the fact that the statisticians would loose, being vastly outnumbered by undergrads studying the harmonic oscillator. I was fishing for constructive suggestions, not attacks. linas 04:15, 3 Apr 2005 (UTC)
Well, the number of undergrads studying statistics is far larger. But definitely the article should be explicit about divergent conventions. Michael Hardy 23:26, 3 Apr 2005 (UTC)
A constructive suggestion might be: "lets go thorough our collective references and see who uses which definition". linas 04:22, 3 Apr 2005 (UTC)
And it appears that I have only one book (besides A&S) that covers Hermite polys, and the convention that book uses is in the none-of-the-above category. (Its Ugo Fano's Physics of Atoms and Molecules). Oh well. linas 04:35, 3 Apr 2005 (UTC)
Shankar uses the physicists def. Surely all good quantum textbooks cover it with the physicists definition. —Preceding unsigned comment added by 92.22.115.41 (talk) 01:04, 20 March 2010 (UTC)

## Comprise

A usage note at comprise in the American Heritage Dictionary says:

USAGE NOTE: The traditional rule states that the whole comprises the parts and the parts compose the whole. In strict usage: The Union comprises 50 states. Fifty states compose (or constitute or make up) the Union. Even though careful writers often maintain this distinction, comprise is increasingly used in place of compose, especially in the passive: The Union is comprised of 50 states. Our surveys show that opposition to this usage is abating. In the 1960s, 53 percent of the Usage Panel found this usage unacceptable; in 1996, only 35 percent objected. See Usage Note at include.

So I will not order Lethe to feel embarrassed for more than one minute, but at least for now I'm going to be a conservative on this issue, so I've changed it. Michael Hardy 01:12, 9 Jul 2004 (UTC)

D'oh!!! Lethe 14:33, Jul 9, 2004 (UTC)

## Harmonic oscillator?

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)

1. The conventions don't match, theres a factor of 2.
2. I don't understand why this can't be done in the article quantum harmonic oscillator, with this article providing no more than a sentance or two pointing the reader there.
3. Agree about history: this article should get a proper introduction, including a 1-5 sentance history/uses/applications that is a part of the introduction. The formal definition following next. If there is a longer historical commentary, it goes in its own separate section.

linas 00:16, 15 October 2005 (UTC)

Which came first? Study of the polynomials, or study of quantum mechanics? --HappyCamper 00:19, 15 October 2005 (UTC)
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)
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)

Linas, aren't Hermite polynomials used independently of quantum physics, in probability theory? Michael Hardy 00:29, 15 October 2005 (UTC)

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)
Yup. See this rich paper: [1] --HappyCamper 01:18, 15 October 2005 (UTC)
Ohh, I really like the structure constants ${\displaystyle c_{klm}}$ 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)

## Eigenfunctions

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 ${\displaystyle H_{n}(x)e^{-{\frac {x^{2}}{2}}}}$ are indeed eigenfunctions of the Fourier transform when the physicist's definition is used. However the eigenvalues are ${\displaystyle -(-i)^{n}}$ for the forward transform and ${\displaystyle -i^{n}}$ for the inverse transform. (The product of the forward and inverse eigenvalues must be unity; if both eigenvalues were ${\displaystyle -i^{n}}$, as the entry states, their product would be ${\displaystyle (-1)^{n}}$).

Additionally, the probabilist's Hermite polynomials do not appear to me to yield eigenfunctions of the Fourier transform when multiplied by ${\displaystyle e^{-{\frac {x^{2}}{2}}}}$. Since ${\displaystyle H_{e_{n}}(x)=2^{-{\frac {n}{2}}}H_{n}\left({\frac {x}{\sqrt {2}}}\right)}$, the scaling theorem can be used to show that the functions ${\displaystyle H_{e_{n}}(x)e^{-{\frac {x^{2}}{4}}}}$ are eigenfunctions of the 'Fourier-like' transform ${\displaystyle T(f(x))={\frac {1}{2{\sqrt {\pi }}}}\int _{-\infty }^{\infty }f(x)e^{-{\frac {itx}{2}}}dx}$ and it's inverse. The eigenvalues are once again ${\displaystyle -(-i)^{n}}$ and ${\displaystyle -i^{n}}$ respectively. --jmh 01:13, 15 January 2006 (UTC)

The eigenvalues given here and in the external site at wolfrom.com are off by -1. The correct eigenvalues (physicist form) are ${\displaystyle (-i)^{n}}$ for the forward transform and ${\displaystyle i^{n}}$ for the inverse transform. This is readily checked for n=0. This may have been the cause of much confusion. It can be further verified by checking equation 43 of Gaussian integral page of Wolfrom.com. Note that the formula holds for the punctured complex plane. Ustad NY 18:58, 23 September 2007 (UTC)

## Plurals

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)

## 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. (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)

## physicists vs. probabilists redux; partial revert

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)

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)

I'd like to propose that, to help clear up the physicist/probabilist dichotomy, we change the symbol for the probabilists' polynomials to "He". This is the symbol used by Abramowitz and Stegun. I think we should continue to have the tags "(physicist)" and "(probabilist)" next to the equations. Having the same symbol letter for both functions is really awkward. Opinions? William Ackerman 17:02, 21 March 2006 (UTC)

• Agree w/ Will above. linas 15:45, 18 November 2006 (UTC)

## Completeness relation

The "completeness relation" section is not clearly written. Obviously whoever put it there was verbally challenged. I can't tell what it says. Not that I've exerted great effort on the point, but the meaning should be clear without that. Michael Hardy (talk) 18:05, 11 December 2008 (UTC)

The section is looking a bit better, chiefly thanks to the efforts of Bdmy. After reviewing it today, it strikes me as unnecessary though. The ψn are orthonormal in L2, and so are also an orthonormal basis (by the Weierstrass approximation theorem). The "identity" should be stated in a way that is an immediate consequence of this. siℓℓy rabbit (talk) 12:47, 12 December 2008 (UTC)

I don't think your Weierstrass statement is correct (see the standard proof that I added after the computation). I sort of like the algebraic tricks present in the computation, although they are probably very similar to many computations with the generating functional of Hermite's functions. Of course, I would hate that my cleaning was a total waist of time, but this is only my  problem... Bdmy (talk) 15:53, 12 December 2008 (UTC) Bdmy (talk) 16:12, 12 December 2008 (UTC)

Perhaps we can't get linear density of ψn in L2 by Stone-Weierstrass. Actually I hadn't thought much about it. Still, I find stating the result distributionally rather than in terms of L2 seems backwards. One should first get (by whatever method) density of the ψn in L2, and then conclude this identity as a trivial consequence of the Riesz-Fischer theorem. siℓℓy rabbit (talk) 16:19, 12 December 2008 (UTC)
I just found the proof of the result: |funny|nice|refreshing|having some spirit from physics|???. Of course if you know that you have an orthonormal basis, then you don't write this funny distribution-relation: you just write (or rather: you don't even mention) the very-close-to-the-definition equivalent relation

${\displaystyle \langle f,g\rangle ={\Bigl \langle }{\Bigl (}\sum \langle f,\psi _{n}\rangle \,\psi _{n}{\Bigr )},g{\Bigr \rangle }.}$
By the way, I don't understand your insistance to refer to Fischer-Riesz for this expansion, that is today a very consequence of the definition of Hilbert spaces, consequence that probably Hilbert, Schmidt and many others knew as well. Bdmy (talk) 18:14, 12 December 2008 (UTC)
I hardly think that one mention of it qualifies as insistence. Anyway, whatever. I think you at least take my point that the section seems to make a big deal out of something that is more easily understood without using distributions. siℓℓy rabbit (talk) 18:32, 12 December 2008 (UTC)
I agree that it is not a big deal. I just think there could be a little corner somewhere to show these funny four or five lines of computation. There are not really any distributions there: just the fact that for physicists,
${\displaystyle \int \mathrm {e} ^{itx}\,\mathrm {d} t=2\pi \,\delta (x),}$
which they say in the distribution sense, just to be on a safe side. Perhaps after a simple mathematical proof of completeness? Bdmy (talk) 18:50, 12 December 2008 (UTC)
Dear Silly Rabbit,
I did one step in your direction by putting a standard argument for completeness first.
But I did one step to prove my point, by giving a pointwise identity with a parameter u < 1 that implies the distributional relation you don't like. I strongly believe that this relation has some encyclopedic value. I didn't know it before, and I am happy that I learnt it here.
Of course the remodeled section needs checking and cleaning... Bdmy (talk) 08:47, 14 December 2008 (UTC)
One more obvious remark: once all this is cleaned up, it should rather go under "Properties" than wait the end of the article. Bdmy (talk) 12:35, 14 December 2008 (UTC)
Or perhaps: the classical proof goes with "Properties", and the Completeness relation stays where it is, with a link to it. Bdmy (talk) 13:20, 14 December 2008 (UTC)

### "Completeness" seems like the wrong word...

...for what's in that section. The Hermite polynomials are orthogonal with respect to the normal distribution. So one sometimes seeks to expand a function that is quadratically integrable with respect to that distribution as an infinite linear combination of Hermite polynomials. "Completeness" ought to mean that every quadratically integrable function can be written in that way. Michael Hardy (talk) 19:23, 12 December 2008 (UTC)

Yes but that is precisely the content of the relation
${\displaystyle \sum _{n=0}^{\infty }\psi _{n}(x)\psi _{n}(y)=\delta (x-y),}$
except that it expresses the equivalent fact that every function square integrable for Lebesgue on the line can be expressed as an orthogonal series in Hermite functions. I may agree that this point is not stressed so far. Also: the "physicists" Hermite polynomials are orthogonal for the weight exp(−x2), not for the normal distribution, and the normalization in this section is "physicist". Bdmy (talk) 19:37, 12 December 2008 (UTC)

I don't see that that is the content of that relation. Let's say that instead of the whole sequence of Hermite polynomials you had a sequence that consists of all but the first ten Hermite polynomials. Then wouldn't the relation above still hold, and would the sequence then not fail to be "complete" in the sense I described? "Completeness" would mean that the sequence contains enough polynomials to do the job. If you omit some of them, then there would not be enoungh. Michael Hardy (talk) 19:57, 12 December 2008 (UTC)

Well, this is pure algebra... If it was not total non-sense since the beginning, then you get
${\displaystyle \sum _{n=11}^{\infty }\psi _{n}(x)\psi _{n}(y)=\delta (x-y)-\sum _{n=0}^{10}\psi _{n}(x)\psi _{n}(y)}$
which is not δ, and you don't have completeness anymore. If you care to read the "proof" (I would understand that you don't!), you'll see that it is crucial to the computation that all integers are present. If you apply the Dirac distribution ${\displaystyle \delta (x-y)}$ (that is to say: the Lebesgue measure on the diagonal of R2) to a function f(x)g(y) of two variables, you get from the completeness relation
${\displaystyle \int \!\!\int {\Bigl (}\sum \psi _{n}(x)\psi _{n}(y){\Bigr )}f(x)g(y)\,\mathrm {d} x\,\mathrm {d} y=\int f(x)g(x)\,\mathrm {d} x}$
which means that
${\displaystyle \int {\Bigl (}\sum {\Bigl (}\int \psi _{n}(x)f(x)\,\mathrm {d} x{\Bigr )}\psi _{n}(y){\Bigr )}g(y)\,\mathrm {d} y=\langle f,g\rangle }$
and gives, since it is true for every g
${\displaystyle \sum {\Bigl (}\int \psi _{n}(x)f(x)\,\mathrm {d} x{\Bigr )}\psi _{n}(y)=\sum \langle f,\psi _{n}\rangle \psi _{n}(y)=f(y),}$
in other words the system is complete. Bdmy (talk) 20:25, 12 December 2008 (UTC) Bdmy (talk) 09:28, 13 December 2008 (UTC) Bdmy (talk) 09:31, 13 December 2008 (UTC)

OK, I'll look at this more closely. But not right now...... Michael Hardy (talk) 12:54, 13 December 2008 (UTC)

## New images!

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)

## Wrong redirect

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)

## Wolfram

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)

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)
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)

## Contour integrals

Ought we change ${\displaystyle n}$ to ${\displaystyle \alpha }$ 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)

## Asymptotic expansion

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))

Did you look in Erdelyi,Magnus & Oberhettinger (cited reference)? That is rather encyclopaedic.JFB80 (talk) 11:57, 22 March 2011 (UTC)
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 ${\displaystyle U(a,b,z)}$, in which the limit process is clearly described in the reference. Episanty (talk) 05:39, 18 April 2011 (UTC)

## Multiplication theorem

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)

## Notation for probabilists' Hermite polynomials

The notation ${\displaystyle He_{n}\,\!}$ is being using for the probabilists' Hermite polynomials, and ${\displaystyle H_{n}\,\!}$ for the physicists ones. This notation is incorrect: ${\displaystyle He_{n}\,\!}$ written like this always means the product of two variables ${\displaystyle H\,\!}$ and ${\displaystyle e_{n}\,\!}$. Multi-character function names are always written in Roman instead, like ${\displaystyle \sin \,\!}$, ${\displaystyle \exp \,\!}$, or (as this article should be) ${\displaystyle \operatorname {He} \,\!}$.

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 Hen (i.e. {{nowrap|He<sub>''n''</sub>}}) when not already in an equation. Quietbritishjim (talk) 17:13, 3 December 2011 (UTC)

Alternatively, we could use a non-standard (but less stupid) notation like ${\displaystyle {\tilde {H}}}$ ( in text). Quietbritishjim (talk) 17:20, 3 December 2011 (UTC)
As evidence, I notice that the MathWorld page on Hermite polynomials uses Hen. Quietbritishjim (talk) 17:39, 3 December 2011 (UTC)

## Comment on explicit expressions by 74.108.147.80

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)

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)

## Hermite function normalization

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)

## Normalization

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)

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)
By the way, a better word to use here would probably be "convention". Quietbritishjim (talk) 14:56, 9 October 2012 (UTC)
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)

## Hermite functions images

I was working with Hermite functions and found out that my graphics doesn't match those on the page. So I guess there may be some error (or maybe some clarification is necessary about formulas for these graphics). I can provide code for wolframalpha.com plotter if necessary. — Preceding unsigned comment added by Metallo lom (talkcontribs) 17:58, 5 March 2013 (UTC)

Can you be a bit more specific about which things don't match? Note that the two images at the top are Hermite polynomials rather than Hermite functions. The two images lower near the middle/bottom are (meant to be) Hermite functions. The shape in those images looks correct to me, especially n=0,1,2, but I don't know if the scales on the axes are correct. Quietbritishjim (talk) 21:09, 5 March 2013 (UTC)
Checked everything more closely and found out that everything is okay. Sorry for false alarm. — Preceding unsigned comment added by Metallo lom (talkcontribs) 16:31, 22 March 2013‎ (UTC)
Good, thanks! Quietbritishjim (talk) 17:00, 22 March 2013 (UTC)

## A different compromise?

I apologise in advance for reopening this whole "probabilists vs physicists" debate, but the current compromise (having two different definitions ${\displaystyle H_{n}}$ and ${\displaystyle He_{n}}$ 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 ${\displaystyle H_{n}(x,c)}$, where ${\displaystyle H_{2}(x,c)=x^{2}-c}$, ${\displaystyle H_{3}(x,c)=x^{3}-3cx}$, ${\displaystyle H_{4}(x,c)=x^{4}-6cx^{2}+3c^{2}}$, etc. In this way, the "probabilists" version is ${\displaystyle H_{n}(x,1)}$, 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 ${\displaystyle H_{n}(x,1/2)}$. The reason why the ${\displaystyle H_{n}(x,c)}$ are natural is of course that they are orthogonal for the Gaussian measure with variance ${\displaystyle c}$. In particular, ${\displaystyle H_{n}(x,c)}$ is the same as the Wick power ${\displaystyle :x^{n}:}$ with respect to the Gaussian measure with variance ${\displaystyle c}$. Hairer (talk) 15:55, 26 April 2014 (UTC)

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)
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)
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)
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)

## Generalizations

Section Generalizations contains few mathematical errors. First of all, consider the following statement

"They are given by ${\displaystyle {\mathit {He}}_{n}^{[\alpha ]}(x)=\alpha ^{-{\tfrac {n}{2}}}He_{n}^{[1]}\left({\frac {x}{\sqrt {\alpha }}}\right)=(2\alpha )^{-{\tfrac {n}{2}}}H_{n}\left({\frac {x}{\sqrt {2\alpha }}}\right)=e^{-\alpha D^{2}/2}x^{n}.}$"

The last equality is wrong, since polynomials ${\displaystyle e^{-\alpha D^{2}/2}x^{n}=x^{n}+\ldots }$ always have high-order coefficient 1.

If we take ${\displaystyle {\mathit {He}}_{n}^{[\alpha ]}(x)=\alpha ^{-{\tfrac {n}{2}}}He_{n}^{[1]}\left({\frac {x}{\sqrt {\alpha }}}\right)=(2\alpha )^{-{\tfrac {n}{2}}}H_{n}\left({\frac {x}{\sqrt {2\alpha }}}\right)}$ as the definition, then formula for the umbral composition

${\displaystyle \left({\mathit {He}}_{n}^{[\alpha ]}\circ {\mathit {He}}^{[\beta ]}\right)(x)={\mathit {He}}_{n}^{[\alpha +\beta ]}(x)}$

is obviously wrong (see, for instance, n=1).

A possible way to fix the problem is to redefine generalized polynomials ${\displaystyle {\mathit {He}}_{n}^{[\alpha ]}(x)=\alpha ^{\tfrac {n}{2}}He_{n}^{[1]}\left({\frac {x}{\sqrt {\alpha }}}\right)=e^{-\alpha D^{2}/2}x^{n}.}$

Note, that for this family umbral composition holds.

Tretiykon (talk) 23:58, 30 June 2014 (UTC)

Good point! I strongly suspect that is why the "scale normalization" of the polynomials Hen was introduced in the first place, and that the original author blew the sign of the exponents, which I lazily assumed was the right one. But consistency with the physicists' Hns and their normalization might also be paramount, and they do not have 1 as the coefficient of the highest power...I suspect the original author's motivation for the normalization was a simple reduction to the physicists' sequence, which, however is differently normalized. But I agree the purpose of the exercise is, I think, to "deform" the monomials to the Hens, and track down their umbral correspondents. It might take a careful scan to see that consistency with, e.g. section 2.4 is maintained. I go ahead and provide a fix as per your suggestion, making only 2 small changes in this section, but need to check consistency with 2.4 later. Cuzkatzimhut (talk) 15:13, 1 July 2014 (UTC)
Have now checked consistency with 2.4, for α=β=1/2. I desisted providing the 2-liner derivation of the identity from the differential representation, to avoid misunderstandings further confusion. An astute reader cannot help reconstructing it by inspection, completing the binomial (x+y)n and appreciating derivation w.r.t. x and to y amounts to derivation w.r.t. x+y on it. Thanks for your salutary input. Cuzkatzimhut (talk) 19:12, 1 July 2014 (UTC)
The section "Generalization" has some issues.
• There are no references in the section. Wolfram [2] suggests The Umbral Calculus by Roman (1984).
• Scaling is one way to generalize Hermite polynomials. There are generalizations other than scaling.
• The following equation was wrong:
${\displaystyle {\textit {He}}_{n}^{[\alpha ]}(x)=\alpha ^{\tfrac {-n}{2}}{\textit {He}}_{n}^{[1]}\left({\frac {x}{\sqrt {\alpha }}}\right)=\left({\frac {\alpha }{2}}\right)^{\tfrac {-n}{2}}H_{n}\left({\frac {x}{\sqrt {2\alpha }}}\right)=e^{-\alpha D^{2}/2}\left({\frac {x}{\alpha }}\right)^{n}.}$
Orthogonal polynomials obtained by different terms of the equation differs in normalization coefficient. Assuming that the leading coefficient is equal to 1, I have changed the equation to
${\displaystyle {\textit {He}}_{n}^{[\alpha ]}(x)=\alpha ^{\tfrac {n}{2}}{\textit {He}}_{n}\left({\frac {x}{\sqrt {\alpha }}}\right)=\left({\frac {\alpha }{2}}\right)^{\tfrac {n}{2}}H_{n}\left({\frac {x}{\sqrt {2\alpha }}}\right)=e^{-\alpha D^{2}/2}(x^{n}).}$
46.185.69.213 (talk) 17:43, 24 April 2015 (UTC)
This generalization looks to me very much like 'original research'. Polynomials with general variance have been used quite a lot in the literature and the description should be based on that work (with references). The notation used here is quite unsuitable. For alpha read sigma squared.JFB80 (talk) 06:32, 25 April 2015 (UTC)
Roman, above, pp. 87-93, looks like original research to you?? You have a better source for umbral calculus? Cuzkatzimhut (talk) 10:47, 25 April 2015 (UTC)

## wrong Inverse Explicit Expression

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

## Error in the definition

If you by ${\displaystyle \left(x-{\frac {d}{dx}}\right)^{n}\cdot 1}$ mean ${\displaystyle \sum _{k=0}^{n}{n \choose k}{\frac {d^{k}}{dx^{k}}}x^{n-k}(-1)^{k}}$ there is no need to multiply by 1, since ${\displaystyle x^{0}=1}$.

Then it must be wrong, try for example ${\displaystyle n=2}$.

I think it must be

${\displaystyle \left(x-{\frac {1}{2}}{\frac {d}{dx}}\right)^{n}=\sum _{k=0}^{n}{n \choose k}{\frac {d^{k}}{dx^{k}}}x^{n-k}(-{\frac {1}{2}})^{k}}$

for the probabilists polynomial and

${\displaystyle \left(2x-{\frac {1}{2}}{\frac {d}{dx}}\right)^{n}=\sum _{k=0}^{n}{n \choose k}{\frac {d^{k}}{dx^{k}}}(2x)^{n-k}(-{\frac {1}{2}})^{k}}$

for the physicists. Pjlub (talk) 13:57, 4 April 2016 (UTC)

??? That's the point, to avoid wrong combinatoric interpretations and use plain Heaviside symbolic calculus, whose commutativity manipulations are straightforward. 1 indicates the vacuum annihilated by ∂, and both it and x are noncommuting operators. How could you possibly misread it? You are expanding the binomial mindlessly as though ∂ and x commute? Just do due diligence:
${\displaystyle (x-\partial )^{2}\cdot 1=(x-\partial )x\cdot 1=(x^{2}-x-x\partial )\cdot 1=x^{2}-x}$. Likewise, ${\displaystyle (x-\partial )^{3}\cdot 1=(x-\partial )(x^{2}-1)\cdot 1=x^{3}-3x}$, etc. Please consider the Appel sequence structure illuminated thereby and expanded upon below.
Ok. I see. Maybe it should be noted (for mindless people like me ;-))?
And I think you meant
${\displaystyle (x-\partial )^{2}\cdot 1=(x-\partial )x\cdot 1=(x^{2}-\partial x)\cdot 1=x^{2}-1}$.
${\displaystyle (x-\partial )^{2}\cdot 1=(x-\partial )x\cdot 1=(x^{2}-x-x\partial )\cdot 1=x^{2}-x}$.