In the context of linear differential equations, I've stumbled upon the following identity for an arbitrary pair of polynomials $P$ and $Q$ with real or complex coefficients: $$ P\left(\frac{d}{dx}\right)\bigl(e^{xy}Q(x)\bigr) =\sum_{n=0}^\infty\frac{P^{(n)}(y)e^{xy}Q^{(n)}(x)}{n!} = Q\left(\frac{d}{dy}\right)\bigl(e^{xy}P(y)\bigr). $$ This can be more or less easily checked by using Taylor expansions of $P\bigl(\frac{d}{dx}\bigr)$ at $y$ and of $Q\bigl(\frac{d}{dy}\bigr)$ at $x$: $$ P\left(\frac{d}{dx}\right) =\sum_{n=0}^\infty\frac{P^{(n)}(y)}{n!}\left(\frac{d}{dx} - y\right)^n, \quad Q\left(\frac{d}{dy}\right) =\sum_{n=0}^\infty\frac{Q^{(n)}(x)}{n!}\left(\frac{d}{dy} - x\right)^n. $$

Is there any easy way to "see" that $P\bigl(\frac{d}{dx}\bigr)\bigl(e^{xy}Q(x)) = Q\bigl(\frac{d}{dy}\bigr)\bigl(e^{xy}P(y)\bigr)$ without "getting hands dirty"?

Is this identity a part of some general theory? It makes me think of Fourier analysis, but I do not know much about it.


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