For the example above, seems to be the operation "differentiate, divide by x, integrate" (with some minor technical tweaks), but it's probably more complicated than that in general...

Actually, this seems to be it (modulo some minor technical stuff). My previous post was getting a bit lengthy, so I'll post some calculations in here instead.
We have that
so
Consequently, by integrating both sides of the equation above and dropping the constant of integration, and by assuming that the order of integration and summation is interchangeable, we see that
In other words, a differential
operator (in the fractional calculus sense, where integration is considered differentiation of order 1) such that
is
I can't say that I've seen it previously, but I like its symmetry...
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