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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.

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  • $\begingroup$ Thanks for answering, I am looking for the standard form of the generating function of Hermite polynomials $\endgroup$ Commented Dec 11, 2020 at 1:26
  • $\begingroup$ I have edited the question, adding the equation I need $\endgroup$ Commented Dec 11, 2020 at 1:37
  • $\begingroup$ Thank you for answer, but I am looking for examples of the derivation of generating Hermite function, like Legendre Polynomials and electric dipole's problem $\endgroup$ Commented Dec 13, 2020 at 18:34
  • $\begingroup$ I believe this is the shortest, most elegant derivation that exposes the logic of the generating function. $\endgroup$ Commented Dec 13, 2020 at 18:53

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From the definition of Hermite polynomials, $$G(x,t)=\sum_{n=0}^{\infty}H_n(x)\frac{t^n}{n!} = \sum_{n=0}^{\infty}\frac{t^n}{n!} (-)^n e^{x^2} \partial_x^n e^{-x^2} \\ \bbox[yellow]{=e^{x^2} e^{-t\partial_x} e^{-x^2} = e^{x^2}e^{-(x-t)^2} }=e^{-t^2+2tx}. $$ 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
$$ (\partial_x^2 -2x\partial_x +2n )H_n =0 ~~~~\leftrightarrow ~~~~(\partial_x^2 -2x\partial_x +2t\partial_t)G=0. $$

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...
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