This is a heuristic question that I think was once asked by Serge Lang. The gaussian: $e^{x^2}$ appears as the fixed point to the Fourier transform, in the punchline to the central limit theorem, as the solution to the heat equation, in a very nice proof of the AtiyahSinger index theorem etc. Is this an artifact of the techniques (such as the Fourier Transform) that people like use to deal with certain problems or is this the tip of some deeper platonic iceberg?

6$\begingroup$ I think the appearances you mention can be sensibly grouped in two broad classes: Fourieranalytic (including the CLT, see e.g. terrytao.wordpress.com/2010/01/05/…) and heatkernelish (incl. AtiyahSinger). Perhaps microlocal analysis is the best bridge between these. $\endgroup$– Steve HuntsmanSep 28, 2010 at 5:35

25$\begingroup$ I just want to point out that the appearance of the Gaussian in AtiyahSinger isn't so mysterious. In the McKeanSinger formula  $Index(D) = Tr_s (e^{tD^2})$  the Gaussian can be replaced with any smooth function which rapidly decays at infinity. The Gaussian is used because $e^{tD^2}$ is the solution operator for a very well understood differential equation, namely the heat equation, and thus we can look up the relevant asymptotic analysis in old PDE textbooks. $\endgroup$– Paul SiegelSep 28, 2010 at 12:01

17$\begingroup$ There is a book "The Normal Distribution: Characterizations with Applications" by Bryc which could be taken as a large collection of probabilistic examples to add to your list. One of my favorites: the normal distribution in $\mathbb{R}^n$ is the unique (up to scaling) rotationinvariant probability measure with independent components. $\endgroup$– Mark MeckesSep 28, 2010 at 12:11

11$\begingroup$ The central limit theorem and the heat equation make no mention of the gaussian in their basic setups; it turns out to be the answer to very natural questions. So I think those two (on the surface completely unrelated) examples show that this phenomenon is definitely not just an artifact of techniques. In particular, there are other methods of proving the CLT or solving the heat equation besides Fourier transforms. $\endgroup$– Mark MeckesSep 28, 2010 at 13:01

8$\begingroup$ The factoring that appears in that volume calculation happens precisely because the standard normal distribution has independent components; the reason you can relate the normal distribution to the sphere is its rotation invariance. So that use of the Gaussian is an application of my favorite property which I mentioned above. As for the CLT and the heat equation, there certainly are deep relationships, but I don't know about "obvious" or "less heavy handed". $\endgroup$– Mark MeckesSep 28, 2010 at 19:34
7 Answers
Quadratic (or bilinear) forms appear naturally throughout mathematics, for instance via inner product structures, or via dualisation of a linear transformation, or via Taylor expansion around the linearisation of a nonlinear operator. The LaplaceBeltrami operator and similar secondorder operators can be viewed as differential quadratic forms, for instance.
A Gaussian is basically the multiplicative or exponentiated version of a quadratic form, so it is quite natural that it comes up in multiplicative contexts, especially on spaces (such as Euclidean space) in which a natural bilinear or quadratic structure is already present.
Perhaps the one minor miracle, though, is that the Fourier transform of a Gaussian is again a Gaussian, although once one realises that the Fourier kernel is also an exponentiated bilinear form, this is not so surprising. But it does amplify the previous paragraph: thanks to Fourier duality, Gaussians not only come up in the context of spatial multiplication, but also frequency multiplication (e.g. convolutions, and hence CLT, or heat kernels).
One can also take an adelic viewpoint. When studying nonarchimedean fields such as the padics $Q_p$, compact subgroups such as $Z_p$ play a pivotal role. On the reals, it seems the natural analogue of these compact subgroups are the Gaussians (cf. Tate's thesis). One can sort of justify the existence and central role of Gaussians on the grounds that the real number system "needs" something like the compact subgroups that its nonarchimedean siblings enjoy, though this doesn't fully explain why Gaussians would then be exponentiated quadratic in nature.

7$\begingroup$ How exactly do you see a Gaussian as a subgroup of $\mathbb{R}$? $\endgroup$ Aug 17, 2017 at 16:32

20$\begingroup$ When you convolve Haar probability measure on a (compact) subgroup with itself, you get back the same measure, and this can in fact be used as a definition of such subgroups. If you convolve a Gaussian probability measure with itself, you almost get back the same Gaussian measure, but it has spread out by a factor of $\sqrt{2}$. So Gaussians are in some sense a "$\sqrt{2}$approximate group". $\endgroup$ Aug 18, 2017 at 1:43
(The sort of obvious answer from teaching statistics several times:)
The sum of two independent normal random variables is again normal, i.e., the shape of the distribution is unchanged under addition except for stretching and scaling.
Moreover, the normal distribution is unique among distributions with finite variance in having this property.
Many phenomena in nature come from adding together various independent or almost independent terms. Therefore, we would expect the normal distribution to show up a lot in natureinspired mathematics.

14$\begingroup$ The normal distribution is not unique in this property. The same is true of all stable distributions. $\endgroup$ Sep 28, 2010 at 12:06

3$\begingroup$ @Mark: Thanks, I have added the assumption of finite variance. $\endgroup$ Sep 28, 2010 at 17:12

$\begingroup$ Fat thumb error on the downvote. Sorry. $\endgroup$ Feb 8, 2021 at 18:51

$\begingroup$ @TomCopeland no worries, I improved my answer now :) $\endgroup$ Feb 8, 2021 at 18:57
One of the reasons for the ubiquity of the Gaussian is displayed in what is probably the most electrifying half page of scientific prose ever writtenMaxwell's argument that the distribution of the velocities of molecules in the ideal gas is Gaussian (now known as the MaxwellBoltzmann distribution). The only physical assumptions used are that the density function depends only on the absolute value of the velocity (and not the direction) and that the components in the directions of the coordinate axes are statistically independent. Mathematically, ths means that the only functions in $3$space which depend only on the distance $r$ from the origin and which split as the product of three functions of one variable are those of the form $ae^{br^2}$. Maxwell does this by inspection but it is easy to give a rigorous proof (under very weak smoothness conditions) and the result holds, of course, in any dimension greater then or equal to $2$. Maxwell's reasoning can be found in his collected papers, or, more accesssibly, in Hawking's anthology "On the Shoulders of Giants".

4$\begingroup$ And this is yet another manifestation of the property I mentioned in the comments on the OP... $\endgroup$ Nov 14, 2012 at 14:19

$\begingroup$ And of course the kinetic energy of a mass $m$ is $m \cdot v^2/2 = p^2/2m$. $\endgroup$ Feb 11, 2021 at 23:16
This is just a minor amplification of one of Terry Tao's points. For any prime $p$, the ring $\mathbb{Z}_p$ of $p$adic integers forms an open compact additive subgroup of $\mathbb{Q}_p$, the completion of $\mathbb{Q}$ under the padic metric, and its characteristic function should be viewed as a padic analogue of the Gaussian. It displays many analogues of the nice properties we see from the Gaussian:
 It is smooth (in the sense that the smooth functions on totally disconnected spaces are defined as the locally constant functions, but this isn't completely tautological, since this class of functions turns out to be useful).
 It is taken to itself under the $p$adic Fourier transform when normalizations are chosen appropriately.
 It obeys something like a central limit theorem. For example, if you flip lots of coins, and ask for the number of heads mod $p^n$, you will, for sufficiently long trials, get a distribution that is close to uniform. It sounds like there could be a way to interpret this sort of convolution in terms of heat flowing, but I don't know a precise statement.
The situation with the real line is more complicated because it is connected but not compact, and therefore has no open compact subgroups. There is a maximal compact multiplicative monoid (occasionally called the "ring of integers of $\mathbb{R}$" informally), given by the closed interval $[1,1]$. You can think of the Gaussian as a smoothing of the uniform distribution on $[1,1]$, but it is not clear to me that this particular analogy is very fruitful.

1$\begingroup$ Do you know if eigenvectors of the Fourier transform (Hermite functions mentioned by Tom's new answer, over $\mathbb{R}$) are written up somewhere, in the $\mathbb{Q}_p$ case? $\endgroup$ Feb 11, 2021 at 22:45

$\begingroup$ BTW I asked a related question mathoverflow.net/questions/382952/… $\endgroup$ Feb 11, 2021 at 22:52
I recently came across a strange and beautiful connection between the Gaussian $e^{x^2}$ and the method of least squares. It turns out that the square in $e^{x^2}$ and the square in ``least squares'' is the same square.
Let $(x_i,y_i)$ (with $1\leq i \leq n$) be the data set, and assume that for each $x$, the $y$'s are normally distributed with mean $\mu(x)=\alpha x+\beta$ and variance $\sigma^2$. Then, the likelihood of generating our data (assuming that the data points are independent) is $$\prod_{i=1}^n \frac{1}{\sqrt{2\pi \sigma^2}} \exp\left(\frac{(y_i\mu(x_i))^2}{2\sigma^2}\right) =\left(\frac{1}{\sqrt{2\pi \sigma^2}}\right)^n \exp\left( \frac{1}{2\sigma^2} \sum_{i=1}^n (y_i  \alpha x+\beta)^2 \right)$$ We would obviously want to choose the parameters $\alpha,\beta$ so that the likelihood is maximized, and this is accomplished by minimizing $$\sum_{i=1}^n (y_i  \alpha x+\beta)^2.$$ In other words, the least squares approximation is the one that makes the data set most likely to happen.

1$\begingroup$ The analogous thing happens for the Laplace distribution and absolute loss, etc. There is certainly a relationship between loss criteria and distributions under which that loss minimizer is the MLE, but I'm not sure it goes any deeper. $\endgroup$– R HahnSep 28, 2010 at 19:35

1$\begingroup$ This is yet another manifestation of my favorite property, mentioned in the comments... $\endgroup$ Sep 28, 2010 at 19:36

$\begingroup$ @R Hahn: See my comment below the question, (2nd from top). The relation does indeed go deeper. $\endgroup$– SuvritSep 29, 2010 at 13:30

3$\begingroup$ Isn't this connection the reason why the Gaussian is called Gaussian? If I remember correctly, Gauss introduced "least squares" as a regression technique not because its derivative gives rise to a linear problem but because the square fits the "normal error model", i.e. corresponds to the Gaussian distribution. $\endgroup$– DirkNov 14, 2012 at 12:51

3$\begingroup$ Sorry if this is a stupid question, but you assume that the data are normally distributed so why is it surprising that the PDF for the normal distribution come up? $\endgroup$– user69208Jun 24, 2018 at 14:04
I'm not an expert, but I believe Stein's method gives a more satisfying connection between the CLT and the heat equation, in particular one that does not involve the Fourier transform. Stein's characterization of the normal distribution and convergence to normality involves an operator closely related to the generator of the OrnsteinUhlenbeck process. On the other hand, the fact that latter has the Gaussian as its invariant measure can be obtained by a trivial transformation of the fundamental solution of the heat equation.

$\begingroup$ Relations among the Hermite polynomials, their raising op, and the OrnsteinUhlenbeck operator are given in mathoverflow.net/questions/184978/…. $\endgroup$ Apr 1, 2021 at 20:00
Edit 2/15/2022{
The utility of the Gaussian $e^{\frac{t^2}{2}}$its numerous propertiesderives from the nature of the coefficients of its Taylor series expansion, naturally. The coefficients, which are aerated OEIS A001147, the odd double factorials $n!!$, enumerate the number of perfect matchings of the vertices of the nsimplices / hypertetrahedra {and complete graphs}. In "Gaussian processes and Feynman diagrams", Faris succinctly sketches the relationship between this characterization and Feynman graphs. In "Graphs on surfaces and their applications", Lando and Zvonkin go into more detail on this and present further associations. The associated Appell sequence, one family of Hermite polynomials, $He_n(z)$ with the e.g.f. $e^{\frac{t^2}{2}} e^{zt}$ has two complementary generators: the raising op $z + \partial_z$ and the binomial transform $e^\frac{{\partial_z^2}}{2} \; z^n = He_n(z)$, from which the properties of the Gaussian, important in so many applications in analysis, algebra, probability, and physics, can be easily derived. When I see a family of Hermite polynomials, I think pair matchings, ribbon graphs, HeisenbergWeyl ladder ops, orthogonality, heat/diffusion evolution equation, normalordering (A344678), quantum physics, and the Gaussian distribution, and, conversely.
(Revamped 2/25/21)
The HeisenbergWeyl algebra associated with the Appell polynomial calculus of the Hermite polynomials provides a way to quickly derive and collate diverse properties of the Gaussian $e^{ \frac{t^2}{\sigma}}$ and connect these to important constructs in math and physics, such as those mentioned in the other responses. The quadratic argument suggests easy extensions to higher dimensions and metrics.
The Hermite polynomials and the Gaussian
The Gaussian is the exponential generating function (e.g.f) for the basic moment sequence for families of Hermite polynomials. One family, $H_n(x)= (H.(x))^n$, in umbral notation, can be characterized several equivalent ways. Three are by the
1) e.g.f.
$$e^{H.(x)t} = e^{t^2} \; e^{xt} = e^{h.t} \; e^{xt} = e^{(h.+x)t},$$
giving
$$H_n(x) = (h.+x)^n = (H.(0)+x)^n$$
with
$$h_n = H_n(0) = \cos(\frac{\pi n}{2}) \; \frac{n!}{(\frac{n}{2})!} ,$$
2) binomial generating operator (BGO) and inverse (the heat operator in this case at unit time)
$$ e^{D^2} \; x^n = e^{h.D} \; x^n = (x+h.)^n = H_n(x)$$
with $h_n$ defined as above as the Taylor series coefficients of the Gaussian, and with the inverse
$$e^{D^2} \; H_n(x) = x^n,$$
($D_{\omega} = \partial_{\omega} = \frac{\partial}{\partial \omega}$ can be considered a partial derivative throughout these notes with the variable often suppressed when obvious for readability),
3) raising (creation) operator
$$ R \; H_n(x) = H_{n+1}(x) $$
with
$$R = e^{D_x^2} \; x \; e^{D_x^2} = x + D_{t=D_x} \; \ln[e^{t^2}] = x + 2 D_x.$$
The first equality in this last string is easily demonstrated using the action of the BGO and its inverse.
$$ R \; H_n(x) = e^{D^2} \; x \; e^{D^2} \; H_n(x) = e^{D^2} \; x \; x^n = e^{D^2} \; x^{n+1} = H_{n+1}(x).$$
To relate a central concept in the Sheffer polynomial calculus, let $e^{D^2} x^n = \hat{H}_n(x)$, the umbral compositional inverse (UCI) to the family $H_n(x)$. Then this last equation reads umbrally as
$$R \; H_n(x) = e^{D^2} \; x \; e^{D^2} \; H_n(x) = e^{D^2} \; x \; H_n(\hat{H}.(x)) = e^{D^2} \; x \; x^n $$
$$ = e^{D^2} \; x^{n+1} = H_{n+1}(x).$$
(The relation to the log can be shown via the GravesPincherle commutator/derivative as in this MOQ.) Note this is an operator conjugation of the basic raising op for the fundamental Appell polynomial sequence $p_n(x) = x^n$.
4) lowering (annihilation / destruction) operator
The lowering op for all Appell sequences is the derivative since they all have the same binomial expansion as the Hermite polynomials, just with different moments,
$$D_x \; H_n(x) = D_x \; (h.+x)^n = n \; (h.+x)^{n1} = n \; H_{n1}(x).$$
In fact, any polynomial sequence with $p_0(x)=1$ such that $D \; p_n(x) = n \; p_{n1}(x)$ is an Appell sequence.
Already, we have, in 1), a form, the e.g.f., related to the integrand for characteristic functions for the Gaussian, used in dealing with the moments and cumulants of random variables; in 2), the heat operator $e^{D_x^2}$ and its inverse, evaluated at unit time; and, in 3) and 4), forms of the ladder ops of quantum mechanicsthe ops of the GravesLieHeisenbergWeyl group/algebra.
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A) Integral transforms of the Gaussian
These characterizations can be used to show that the Gaussian gives a Gaussian under a Fourier transformation (FT) and a twosided Laplace transform (LPT). The solution to the heat equation (or Cauchy problem) can be derived using the FT, and the orthogonality of Hermite polynomials derived from the LPT.
The raising op gives
$$e^{tR} \; 1 = e^{H.(x)t}= e^{t^2} \; e^{xt},$$
so
$$D_t \; e^{t^2} \; e^{xt} = D_t \; e^{tR} \; 1 = R \; e^{t^2} \; e^{xt}.$$
Changing the variable $t \to it$ gives
$$ iD_t \; e^{t^2} \; e^{ixt} = R \; e^{t^2} \; e^{ixt} \;, $$
and integrating gives
$$ i \; \int_{\infty}^{\infty} \; D_t \; e^{t^2} \; e^{ixt} \; dt = R \; \int_{\infty}^{\infty} \; e^{t^2} \; e^{ixt} \; dt \;, $$
so
$$ 0 = (x + 2D_x) \; FT^{1}[e^{t^2}].$$
But also
$$ (x + 2D_x) \; e^{{(\frac{x}{2})}^2} = 0,$$
so the complex conjugate gives the standard Fourier transform
$$ FT[e^{t^2}] = C \; e^{{(\frac{x}{2})}^2},$$
where $C=\sqrt{\pi}$. (The normalization factor can be read off from the Mellin transform of the Gaussian as shown below, or it can be determined the usual way by squaring the integral with $x=0$ and converting to polar coordinates.)
Substituting $ix$ for $x$ in the FT or $ix$ in the inverse FT gives
$$ i \; \int_{\infty}^{\infty} \; D_t \; e^{t^2} \; e^{xt} \; dt = i \;(x  2D_x) \; \int_{\infty}^{\infty} \; e^{t^2} \; e^{xt} \; dt ,$$
(note $\hat{R} = x  2D_x$ is the raising op for $\hat{H}(n(x)$),
and since
$t^2  xt = (t + \frac{x}{2})^2 + (\frac{x}{2})^2,$
$$0 = (x  2D_x) \; \int_{\infty}^{\infty} \; e^{t^2} \; e^{xt} \; dt .$$
Also
$$(x 2D_x) \; e^{(\frac{x}{2})^2} =0,$$
so the twosided LPT of the Gaussian is
$$LPT[e^{t^2}] = \sqrt{\pi} \; e^{(\frac{x}{2})^2}.$$
Because the Gaussian is essentially selfreciprocal under the Fourier transform, the equality in the quantum mechanical uncertainty relation is achieved only for freely moving particles whose probability amplitudes are characterized as Gaussian wave packets.
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B) The heat equation (Cauchy problem)
Now with $t$ time and $x$ spatial displacement in this section,
$$ D_t \; f(t,x) = D_x^2 \; f(t,x)$$
is satisfied by
$$ e^{tD_x^2} \; f(0,x) = f(t,x).$$
The Green's function solution satisfies
$$ G(0^{+},x) = \delta(x) = \int_{\infty}^{\infty} e^{i2\pi \omega x} \; d\omega$$
and
$$G(1,x) = e^{D^2} \; \delta(x) = \int_{\infty}^{\infty} e^{i2\pi \omega H.(x) } \; d\omega = \int_{\infty}^{\infty} e^{(2\pi \omega)^2} e^{i2\pi x \omega} \; d\omega = \frac{1}{2 \sqrt{\pi}} \; e^{(\frac{x}{2})^2} .$$
With $x = \sqrt{t} \; u$, then
$$G(t,x) = e^{tD_x^2} \; \delta(x) = e^{D_u^2} \; \delta(\sqrt{t} \; u) = \frac{1}{\sqrt{t}} e^{D_u^2} \; \delta(u)$$
$$ = \frac{1}{ \sqrt{t}} \frac{1}{2 \sqrt{\pi}} \; e^{(\frac{u}{2})^2} = \frac{1}{\sqrt{4\pi t}} \; e^{\frac{x^2}{4t}} ,$$
and the general solution is the convolution
$$f(t,x) = e^{tD^2_x} \; f(0,x) = \int_{\infty}^{\infty} f(0,u) \; G(t,ux) \; du.$$
This integral is a SegalBargmann transform discussed by Cartier in "Mathemagics" and by Hall in "Holomorphic methods in analysis and mathematical physics."
This is consistent with the BGO when $f(0,x) = x^n$, for which $f(1,x) = e^{D^2} \; x^n = H_n(x).$ And, that any Appell sequence/coefficient matrix can be transformed into any other Appell sequence/coefficient matrix suggests more general 'heat equations' associated to these Appell sequences can be transformed into the Hermite heat equation.
The time can be rescaled to include a diffusion constant (or adjustable variance in the probabilist's jargon).
For some of the physical import of the heat equation, see "Random Walk and the Theory of Brownian Motion" by Mark Kac and "Brownian motion and potential theory" by Hersh and Griego.
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C) The quantum harmonic oscillator
To dovetail with Cartier's arguments, best to jump to another family of Hermite polynomials, that of the probabilists or statisticians $SH_n(x)$, with the
1) e.g.f.
$$ e^{SH.(x)t} = e^{\frac{t^2}{2}} \; e^{xt} \; ,$$
2) BGO and the inverse (the Btransform in Cartier)
$$ e^{\frac{D^2}{2}} \; x^n = SH_n(x) $$
and
$$ e^{\frac{D^2}{2}} \; SH_n(x) = x^n \; ,$$
3) raising op
$$R = e^{\frac{D^2}{2}} \; x \; e^{\frac{D^2}{2}} = x  D \; .$$
Conjugations transform the between the spaces described and listed in the table on page 46 by Cartier. From the Fock $\mathfrak{F(c)}$ to $\mathfrak{L^2(d\gamma)}$ space, for the raising ops,
$$ e^{\frac{D_z^2}{2}} \; z^n \; e^{\frac{D_z^2}{2}} = z  D_z,$$
the lowering op under this conjugation remains invariant as the derivative $D_z$, and for the bases
$$ e^{\frac{D_z^2}{2}} \; z^n = SH_n(z).$$
From $\mathfrak{L^2(d\gamma)}$ to $\mathfrak{L^2(R)}$, for the raising and lowering ops,
$$ e^{\frac{x^2}{4}} \; (x D_x) \; e^{\frac{x^2}{4}} = \frac{x}{2}  D_x,$$
$$ e^{\frac{x^2}{4}} \; D_x \; e^{\frac{x^2}{4}} = \frac{x}{2} + D_x,$$
and, for the bases,
$$ e^{\frac{x^2}{4}} \; SH_n(x) = QH_n(x).$$
The number / state / Euler op for a basis set $p_n(x)$ characterized by lowering, $L$, and raising, $R$, ops is $E = RL$. For example, for $p_n(x) = x^n$, the Euler op is $xD_x$ and $xD_x \; p_n(x) = n \; p_n(x)$. For $\mathfrak{L^2(R)}$ space, action of the Euler op on the basis set gives essentially the Hamiltonian for the quantum harmonic oscillator (mod scaling factors)
$$E = (\frac{x}{2}  D_x) \;(\frac{x}{2} + D_x) = D_x^2 + \frac{x^2}{4}  \frac{1}{2} ,$$
so
$$ (D_x^2 + \frac{x^2}{4}) \; QH_n(x) = (n + \frac{1}{2}) \; QH_n(x).$$
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D) The Gaussian and orthogonality of Hermite polynomials
As noted above the sequence of Hermite polynomials $\hat{H}_n(x)$ that is the UCI to the family $H_n(x)$ has the e.g.f.
1) e.g.f.
$$e^{\hat{H}.(x)t} = e^{t^2} e^{xt}$$
with
$$\hat{h}_n = \cos(\frac{\pi n}{2}) \; \frac{n!}{(\frac{n}{2})!},$$
2) BGO and inverse
$$e^{D_x^2} \; x^n = \hat{H}_n(x),$$
with inverse
$$e^{D_x^2} \hat{H}_n(x) = \; x^n ,$$
3) raising op
$$\hat{R} = e^{D^2} \; x \; e^{D^2}= x 2 \: D_x.$$
From the e.g.f.,
$$ D_{t=0}^n \; e^{t^2} \; e^{2xt} \; D_{s=0}^m \; e^{s^2} \; e^{2xs} \; = \hat{H}_n(2x) \; \hat{H}_m(2x) $$
$$= D_{t=0}^n \; D_{s=0}^m \; e^{(t^2+s^2)} \;e^{2x(t+s)} ,$$
so
$$ D_{t=0}^n \; D_{s=0}^m \; e^{(t^2+s^2)} \; \int_{\infty}^{\infty} \; e^{x^2} \; e^{2x(t+s)} \; dx = \int_{\infty}^{\infty} \; \hat{H}_n(2x) \; \hat{H}_m(2x) \; e^{x^2} \; dx$$
$$ = \; D_{t=0}^n \; D_{s=0}^m \; \sqrt{\pi} \; e^{(t^2+s^2) + (s+t)^2} = \; D_{t=0}^n \; D_{s=0}^m \; \sqrt{\pi} \; e^{2st} = \sqrt{\pi} \; 2^n \; n! \; \delta_{n,m} , $$
or
$$\int_{\infty}^{\infty} \; \hat{H}_n(2x) \; \hat{H}_m(2x) \; e^{x^2} \; dx = \sqrt{\pi} \; 2^n \; n! \; \delta_{n,m},$$
where $\delta_{n,m}$ is the Kronecker delta, which is unity for $n=m$ and vanishes otherwise.
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E) A Gaussian integral transform of Hermite polynomials
$$ D_{t=0}^n \; e^{t^2} \; e^{xt} = H_n(x) $$
A Wick rotation gives
$$ \;(i)^n D_{t=0}^n \; e^{t^2} \; e^{ixt} = H_n(x) .$$
Then
$$(i)^n \; D_{t=0}^n \; e^{t^2} \; \int_{\infty}^{\infty} \; e^{x^2} \; e^{ixt} \; dx = \int_{\infty}^{\infty} \; H_n(x) \; e^{x^2} \; dx$$
$$ =(i)^n \; D_{t=0}^n \sqrt{\pi} \; e^{(1+\frac{1}{4})t^2} = \sqrt{\pi} \; (\frac{5}{4})^{\frac{n}{2}} \cos(\pi\frac{n}{2}) \frac{n!}{(\frac{n}{2})!},$$
and
$$\int_{\infty}^{\infty} \; H_n(x) \; e^{x^2} \; dx = \sqrt{\pi} \; (\frac{5}{4})^{\frac{n}{2}}\; \hat{h}_n = \sqrt{\pi} \; (\frac{5}{4})^{\frac{n}{2}} \; \hat{H}_n(0) .$$
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F) Mellin transform of the Gaussian, Mellin transform interpolation of the coefficients of the Gaussian
The Mellin transform of the Gaussian is central to many discussions in analytic number theory and modular forms, being related to the Jacobi theta functions (e.g., see this MOQ).
The modified Mellin transform can be read off from the coefficients of the e.g.f. of the Gaussian per Ramanujan's master formula as shown in MOQ1, MOQ2, and MSEQ).
$\; \; \; \; \;g(t) = e^{t^2} = e^{\hat{h}. t}= \sum_{n \ge 0} \cos(\frac{\pi n}{2}) \; \frac{n!}{(\frac{n}{2})!} \; \frac{t^n}{n!}, $
$\; \; \; \; \;f(t) = g(t) = e^{t^2} = g(t)= e^{\hat{h}. t},$
and, for $ Re(s) > 0$, the modified Mellin transform gives
$$\hat{h}_{s} = (\hat{h}.)^{s} = \int_0^{\infty} e^{\hat{h}. t} \; \frac{t^{s1}}{(s1)!} \; dt$$
$$ = \int_0^{\infty} e^{t^2} \; \frac{t^{s1}}{(s1)!} \; dt = \cos(\pi\frac{ s}{2}) \; \frac{(s)!}{(\frac{s}{2})!} = \frac{1}{2}\frac{(\frac{s}{2}1)!}{(s1)!} .$$
The Mellin transform gives the general formula for the onesided Gaussian integral transform of the monomials $x^n$ as
$$n! \; \hat{h}_{n1} = \int_0^{\infty} e^{x^2} \; x^n \; dx = \frac{1}{2} \; (\frac{n1}{2})! \; . $$
In particular for $s = 1$, or $n =0$,
$$ \hat{h}_{1} = \int_0^{\infty} e^{x^2} \; dx = \frac{1}{2}\; (\frac{1}{2})! = \frac{\sqrt{\pi}}{2}.$$
For $n=1,3,5,7,9,...$, the sequence $n! \; \hat{h}_{n1}$ is $(\frac{1}{2}, \;\frac{1}{2}, \;1, \;3, \;12, \;60, ...)$ with the numerators OEIS A001710.
For $n=0,2,4,6,..$, the sequence $n! \; \hat{h}_{n1}$ is $\sqrt{\pi} \cdot (\frac{1}{2}, \; \frac{1}{2^2}, \;\frac{3}{2^3}, \;\frac{15}{2^4}, \;\frac{105}{2^5}, ...)$, a sequence with the double factorials of OEIS A001147 (see also A094638) in the numerators.
Both OEIS entries have extensive references.
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G) Cumulants
The coefficients of the differential component of the raising op for an Appell sequence, determined from the natural log of the e.g.f. of the Appell moments, are proportional to the cumulants associated to the moment e.g.f. via the cumulant expansion formula of OEIS A127671. In particular, the raising ops of the Hermite polynomial families are $R = x + D_{t=D_x} \ln[e^{\alpha \; t^2}] = x  \alpha \; D_x$, where $\alpha$ is proportional to the variance, or second cumulant, of the Gaussian probability density function (pdf).
A characteristic function of the normalized Gaussian pdf with $t$ treated as a random variable is, from the twosided LPT above,
$$<e^{ xt}> \; = \int_{\infty}^{\infty} \frac{e^{t^2}}{\sqrt{\pi}} \; e^{ xt} \; dt = \int_{\infty}^{\infty} \frac{1}{\sqrt{\pi}} e^{\hat{H}.(x)t} \; dt = e^{\frac{x^2}{4}} = e^{\frac{h.}{2}x} $$
with the moments given by
$$<t^n > \; = \; D_{x=0}^n \; <e^{xt}> \; = D_{x=0}^n e^{\frac{x^2}{4}} = \frac{h_n}{2^n} = \frac{H_n(0)}{2^n} = \cos(\frac{\pi n}{2}) \; \frac{n!}{(\frac{n}{2})!} \frac{1}{2^{n}},$$
agreeing with the Mellin transform result.
The formal cumulants in terms of formal moments (convergence of the e.g.f.s not required) are given by the cumulant expansion formula (see the associated partition polynomials in the Lang link in A127671)
$$\ln[<e^{xt}>] = \ln[\; e.g.f \; of \; moments \; ] = \; e.g.f. \; of \; cumulants, $$
and, in our case for the Gaussian, reduces to
$$ \ln[e^{\frac{x^2}{4}}] = \frac{1}{2} \frac{x^2}{2} = (<t^2>  <t>^2) \; \frac{x^2}{2} = (<t<t>>^2) \; \frac{x^2}{2}, $$
so the cumulants vanish except for the second cumulant, the variance, which is 1/2. The cumulant partition polynomials are invertible (a graded involution), so the formal cumulants completely define the formal moment e.g.f., and, if this is given by a characteristic function obtained by an invertible transform of a pdf, they define the pdf as well. See "Three lectures on free probability" by Novak and LaCroix for the significance of cumulants in combinatorics and statistics and for an elementary proof of the central limit theorem via cumulants. See also the stat mech refs in OEIS A036040. See the Wick's and Isserlis' theorems for generalizations.
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For more connections of the Gausssian, the Hermite polynomials, and the GravesLieHeisenbergWeyl group/algebra to other areas of math and physics, see "On the role of the Heisenberg group in harmonic analysis" by Roger Howe, "Heisenberg groups, theta functions, and Weil representation" by Yang, "An Exercise(?) in Fourier Analysis on the Heisenberg Group" by Bump, Diaconis, Hicks, and Widom, "Theta and Riemann xi function representations from harmonic oscillator eigensolutions" by Coffey, and "Quantum vs. classical integrability in CalogeroMoser systems" by Corrigan and Sasaki.
(Edit 5/14/21)
Generalization: Gaussian functions, moments, and the HeisenbergWeyl algebra
Many important polynomial sequences $p_n(x)$ (in particular, all Sheffer polynomial sequences, related to invertible lower triangular matrices, including the Bernoulli and families of Laguerre and Hermite polynomials) have ladder opslowering and raising ops $L$ and $R$such that
$ L \; p_n(x) = n \; p_{n1}(x)$ and $R \; p_n(x) = p_{n+1}(x)$
with $L \; p_0(x) = 0.$
The commutator of the ops gives $[L,R] = 1$; i.e., $[L,R] \; p_n(x) = (LR RL) \; p_n(x) = p_n(x).$
For operators that satisfy the commutator relations $[X,[X,Y]] = [Y,[Y,X]]= 0,$ the BCHD expansion reduces to
$$ e^{tX} e^{tY} = e^{t(X+Y) + \frac{t^2}{2}[X,Y]},$$
so the operator disentangling relationship
$$ e^{tR} e^{tL} = e^{t(R+L) + \frac{t^2}{2}[R,L]} = e^{\frac{t^2}{2}} \; e^{t(R+L)}$$
holds, in which a Gaussian function appears.
For $p_0(x) =1$ with the umbral notation and maneuver $p.(x)^n = p_n(x)$,
$$ e^{tR} e^{tL}\; 1 = e^{t R} \; 1= e^{t \; p.(x)} = e^{\frac{t^2}{2}} \; e^{t(R+L)} \; 1 $$
so, with the moments $\bar{h}_n$ of the Gaussian function defined by $e^{ \;\frac{t^2}{2}} = e^{\bar{h}.t}$,
$$e^{\frac{t^2}{2}} \; e^{t \; p.(x)} = e^{t\; \bar{h}.} \; e^{t \; p.(x)} = e^{t\; (\bar{h}.+p.(x))} = e^{t \; \bar{H}.(p.(x))} = e^{t(R+L)} \; 1,$$
with
$$\bar{H}_n(x) = (\bar{h}.+x)^n = \sum_{k=0}^n \; \binom{n}{k} \; \bar{h}_k \; x^{nk},$$
and we can identify
$$(L+R)^n \; 1= (\bar{h}. + p.(x))^n = \bar{H}_n(p.(x));$$
that is,
$L+R$ is the raising op for the sequence $P_n(x) = \bar{H}_n(p.(x))$.
Since the lowering and raising ops for the monomial sequence $p_n(x) = x^n$ are $x$ and $D$, the raising op for the family of modified Hermite polynomials $\bar{H}.(x)$ above (OEIS A099174) is $x+D_x$, and the form of the polynomials gives $D_x$ as the lowering op. In the earlier answer above, you can see that diverse properties of the Gaussian function/distribution follow from this operator relation. Many can be generalized. For example, the raising op formula
$$(x+D) \; \bar{H}_n(x) = \bar{H}_{n+1}(x)$$
gives an instance
$$\bar{H}_{n+1}(x) = x \; \bar{H}_n(x)  n \; \bar{H}_{n1}(x)$$
of the general recurrence relation that is a sufficient and necessary condition for a sequence to be a set of orthonormal polynomials on the real line w.r.t. a weight function/distribution (see the discussion surrounding Favard's theorem in Operator Theory: A Comprehensive Course in Analysis, Part 4 by Barry Simon).
Another example is the evolution equation
$$ D_t \; f(x,t) = (R+L) \; f(x,t),$$
which follows from the above exponential rep with the solution
$$f(x,t) = e^{t \; P.(x)} = e^{t \; \bar{H}_n(p.(x))} = e^{t \; (\bar{h}.+p.(x))}$$
$$ = e^{t \bar{h}.} \; e^{t \; p.(x)} = e^{ \frac{t^2}{2}} \; A(t) \; e^{x \; B(t)} , $$
an e.g.f for a Sheffer sequence, for which $A(t)$ and $B(t)$ are analytic about the origin and $A(0) =1$, $B(0)=0$, and $B'(0) \neq 0$.
From the calculus of Appell Sheffer sequences, a solution of the deformed heat/diffusion equation
$$D_t \; \tilde{g}(x,t) = [ \; \frac{D_x^2}{2} + \sum_{n \geq 2} \; c_n \; \frac{D_x^n}{n} \;] \; \tilde{g}(x,t) $$
is
$$\tilde{g}(x,t) = e^{t\frac{D_x^2}{2}} \; \exp[\; t \; \sum_{n \geq 2} \; c_n \; \frac{D_x^n}{n} \;] \; g(x)$$
$$ = g[ H.[S.(y,\gamma_2,\gamma_3,...)] \sqrt{t}] = g[ S.(H.(y),\gamma_2,\gamma_3,...) \sqrt{t}] = g[ S.(x,t\bar{c}_2,tc_3,...)]$$
with
$ \gamma_n = \; t^{1\frac{n}{2}} \; c_n$ for $n > 1$,
$y =x/ \sqrt{t} $,
$\bar{c}_2 = 1 + c_2$, and
$S_n(c_1,c_2,...,c_n) $ are the Stirling partition polynomials of the first kind (OEIS A036039), a.k.a. the cycle index polynomials of the symmetric groups, defined by
$e^{t \; S.(c_1,...)} = \exp[\ln(1c.t)] = \exp[\sum_{k > 0} c_k \; \frac{t^k}{k}].$
Since these umbral compositions can be represented by multiplication of invertible lower triangular matrices, conjugations (or 'axes rotations') can be used to map between the nondeformed and deformed reps.
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Edit (5/30/21):
A Combinatorial Perspective
Underlying all of the relationships above is the fact $e^{t^2/2} = e^{\bar{h}. t}$ is the e.g.f. for the aerated odd double factorials ($\bar{h}_0=1,\;\bar{h}_1=0,\;\bar{h}_2=1,\;\bar{h}_3=0, \;\bar{h_4}=3,...$), which enumerate the number of perfect matchings of the n vertices of the regular (n1)dimensional simplices (hypertriangles, or hypertetrahedrons).
$(x+D_x)^{n}$ as a normalordered operator, i.e., expanded and expressed with all derivatives to the right of the variable $x$, has the same coefficients as the polynomial $\bar{H}_{n}(x+y) = (\bar{h}.+x+y)^{n}$ in the commutative variables $x$ and $y$ with the multinomial enumerating permutations of three families of objectsvertices labelled with either $x$ or $y$, or perfect (pair) matchings of the unlabeled vertices for the n vertices of the (n1)dimensional simplex. For example, $(x+D)^2 = xx +xD+Dx + D^2 = x^2 + 2xD + D^2+1$ corresponds to $H_n(x+y) = (h.+x+y)^2 = h_0 \dot (x+y)^2 + 2 h_1 \cdot (x+y) + h_2 = x^2 +2xy +y^2+1$ which, in turn, corresponds to a line segment, the 1D simplex, with both vertices labeled with $x$'s; or one with an $x$, the other a $y$; or both vertices labeled with $y$'s; or one unlabeled matched pair.

2$\begingroup$ Also good to know: the Hermite functions give a Schauder basis for the Schwartz space $\mathscr{S}(\mathbb{R})$ and therefore, and more importantly, also for the space of tempered distributions $\mathscr{S}'(\mathbb{R})$. They essentially trivialize lots (but not all) of functional analytic questions about these spaces. $\endgroup$ Feb 11, 2021 at 22:49

$\begingroup$ @AbdelmalekAbdesselam, refs? Connected to the use of the Gaussian as a nascent Dirac delta and the Cauchy problem, similar to Sato's theory of hyperfunctions? $\endgroup$ Feb 11, 2021 at 23:29

1$\begingroup$ Don't know about nascent delta. For the Schauder basis, I have some notes from a course I taught. Otherwise there is aip.scitation.org/doi/abs/10.1063/… but not well written, it's also in Reed Simon Vol 1, not better, most of the proof punted in the exercises. Simon's new book on analysis is much better. $\endgroup$ Feb 11, 2021 at 23:37

$\begingroup$ BTW, Treves in his book on distributions does not mention this. Other authors credit ReedSimon, but this was well known to Schwartz and Grothendieck (mentioned in Schwartz's book on distributions and A.G. in his thesis in Memoirs of AMS). I don't know who first discovered the result. Wiener? $\endgroup$ Feb 11, 2021 at 23:43

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