I consider for example the following function of two variables given by $$f(x,y)=\sum_{n=0}^{+\infty}\frac{\delta_{x}^{-m} \exp(\delta_{x})}{\delta_{y}^{-m}\exp(\delta_{y})}\left(\frac{x}{y}\right)^{n}$$ Where $\delta_{x}=\frac{d}{dx}$ and $\delta_{y}=\frac{d}{dy}$ and $m$ is a positive integer. My question is: when $x < y$ can we write f as $$f(x,y)=\frac{\delta_{x}^{-m} \exp(\delta_{x})}{\delta_{y}^{-m}\exp(\delta_{y})}\sum_{n=0}^{+\infty} \left(\frac{x}{y}\right)^{n}=\frac{\delta_{x}^{-m} \exp(\delta_{x})}{\delta_{y}^{-m}\exp(\delta_{y})}g(x,y)?$$ such that $g(x,y)=\frac{1}{1-\frac{x}{y}}=\frac{y}{y-x}$. If the formula above is correct can we prove it more explicitly. Precisely, constructing a corollary for which $\frac{\delta_{x}^{-m} \exp(\delta_{x})}{\delta_{y}^{-m}\exp(\delta_{y})}$ is a well-defined operator denoted by $A_{x,y}^{m}$. I need a response, if someone have an idea.

Best regards and thank you.

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