Heh, sure.
Promote from a function to a linear operator on the space of functions, . The action of this operator is just “multiply by ”. We’ll similarly define meaning to multiply by the first, second integral of , etc.
Observe:
Now we can calculate what we get when applying times. The calculation simplifies when we note that all terms are of the form . Result:
Now we apply the above operator to :
The sum terminates because a polynomial can only have finitely many derivatives.
Oh, very cool, thanks! Spoiler tag in markdown is: