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The exponential of a derivative is the shift operator $e^{a\partial} f(x)=f(a+x)$. I wander if there exists a compact expression for the action of the operator \begin{equation} e^{g(x)+a\partial} f(x)=? \end{equation}

Possible ways that I thought to find the answer are 1) Explicitely by expanding the Taylor series and resumming different classes of terms, or 2) Via a generalization of the solution provided in this post Exponential of a function times derivative (see the last comment there). Thanks a lot!

Francesco
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  • Thank you Tyma. The first step in your proposed solution doesn't wor because the derivativa and $g(x)$ do not commute. – Francesco Jul 21 '21 at 07:33

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After an explicit resummation, guesses and checks, I found the following compact relation: \begin{equation} f^{g(x)+a\partial}f(x)=e^{G(x+a)-G(x)}f(x+a)\,, \end{equation} where, \begin{equation} G(x)=\int g(x) dx \,. \end{equation} However I was not able to prove this more abstractly.

Francesco
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