Is there a way to get the generating function of Hermite polynomials?

Question

I would like to know if there is any physical model in which the generating function of the Hermite polynomials arises, I know the problem of the quantum harmonic oscillator but I have not found the generating function there.
In my notes, the function I am referring to is the following
begin{equation}
G(x,t)=e^{-t^2+2tx}=sum_{n=0}^{infty}H_n(x)frac{t^n}{n!}
end{equation}
where $H_n(x)$ are the Hermite polynomials, I am interested on the exponential form.

Cosmas Zachos · Accepted Answer

From the definition of Hermite polynomials,
$$G(x,t)=sum_{n=0}^{infty}H_n(x)frac{t^n}{n!}  
= sum_{n=0}^{infty}frac{t^n}{n!} (-)^n e^{x^2} partial_x^n e^{-x^2} 
bbox[yellow]{=e^{x^2} e^{-tpartial_x} e^{-x^2} = e^{x^2}e^{-(x-t)^2}  }=e^{-t^2+2tx}.
$$
The operator series has been summed to a formal exponential, which is precisely Lagrange's shift operator, yielding the standard expression directly (yellow).
Note that
$$
(partial_x^2 -2xpartial_x +2n )H_n =0 ~~~~leftrightarrow ~~~~(partial_x^2 -2xpartial_x +2tpartial_t)G=0.
$$
Is this what you want?

If you were comfortable with the above, you could segue to the celebrated Mehler kernel that your textbook probably covers if it's any good...

Is there a way to get the generating function of Hermite polynomials?

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