TITLE: Why is the Gaussian so pervasive in mathematics?
QUESTION [56 upvotes]: 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 Atiyah-Singer 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?

REPLY [5 votes]: Edit 2/15/2022{
The utility of the Gaussian $e^{\frac{t^2}{2}}$--its numerous properties--derives 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 n-simplices / 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, Heisenberg-Weyl ladder ops, orthogonality, heat/diffusion evolution equation, normal-ordering (A344678), quantum physics, and the Gaussian distribution, and, conversely.
(Revamped 2/25/21)
The Heisenberg-Weyl 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 Graves-Pincherle commutator/derivative as in this MO-Q.) 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,
$$D_x \; H_n(x) = D_x \; (h.+x)^n = n \; (h.+x)^{n-1} = n \; H_{n-1}(x).$$
In fact, any polynomial sequence with $p_0(x)=1$ such that $D \; p_n(x) = n \; p_{n-1}(x)$ 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 mechanics--the ops of the Graves-Lie-Heisenberg-Weyl 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 two-sided 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 two-sided LPT of the Gaussian is
$$LPT[e^{-t^2}] = \sqrt{\pi} \; e^{(\frac{x}{2})^2}.$$
Because the Gaussian is essentially self-reciprocal 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
$$f(t,x) = e^{tD^2_x} \; f(0,x) = \int_{-\infty}^{\infty} f(0,u) \; G(t,u-x) \; du.$$
This integral is a Segal-Bargmann 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 B-transform 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 MO-Q).
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 MO-Q1, MO-Q2, and MSE-Q).
$\; \; \; \; \;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
$$\hat{h}_{-s} = (\hat{h}.)^{-s} = \int_0^{\infty} e^{-\hat{h}. t} \; \frac{t^{s-1}}{(s-1)!} \; dt$$
$$ =  \int_0^{\infty} e^{-t^2} \; \frac{t^{s-1}}{(s-1)!} \; dt   =  \cos(\pi\frac{ s}{2}) \; \frac{(-s)!}{(-\frac{s}{2})!} = \frac{1}{2}\frac{(\frac{s}{2}-1)!}{(s-1)!} .$$
The Mellin transform gives the general formula for the one-sided Gaussian integral transform of the monomials $x^n$ as
$$n! \; \hat{h}_{-n-1} = \int_0^{\infty} e^{-x^2} \; x^n \; dx = \frac{1}{2} \; (\frac{n-1}{2})! \; . $$
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 $n! \; \hat{h}_{-n-1}$ is $(\frac{1}{2}, \;\frac{1}{2}, \;1, \;3, \;12, \;60, ...)$ with the numerators OEIS A001710.
For $n=0,2,4,6,..$, the sequence $n! \; \hat{h}_{-n-1}$ 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 two-sided 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 Graves-Lie-Heisenberg-Weyl 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 Calogero-Moser systems" by Corrigan and Sasaki.

(Edit 5/14/21)
Generalization: Gaussian functions, moments, and the Heisenberg-Weyl 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 ops--lowering and raising ops $L$ and $R$--such that
$ L \; p_n(x) = n \; p_{n-1}(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
$$\bar{H}_n(x) = (\bar{h}.+x)^n = \sum_{k=0}^n \; \binom{n}{k} \; \bar{h}_k \; x^{n-k},$$
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
$$\bar{H}_{n+1}(x) = x  \; \bar{H}_n(x) - n \; \bar{H}_{n-1}(x)$$
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
$e^{t \; S.(c_1,...)} =  \exp[-\ln(1-c.t)] = \exp[\sum_{k > 0} c_k \; \frac{t^k}{k}].$
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 (n-1)-dimensional simplices (hypertriangles, or hypertetrahedrons).
$(x+D_x)^{n}$ as a normal-ordered 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 objects--vertices labelled with either $x$ or $y$, or perfect (pair) matchings of the unlabeled vertices for the n vertices of the (n-1)-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 1-D 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.