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$$\bar{H}_{n+1}(x) = x \; \bar{H}_n(x) - n \; \bar{H}_{n-1}(x)$$

$$\bar{H}_{n+1}(x) = x \; \bar{H}_n(x) + n \; \bar{H}_{n-1}(x)$$

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.

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.

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.

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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.