I've come across this funny problem while messing around with integration by parts. Probably made a mistake somewhere.

In the integration of parts expression, it's possible to expand it further.

Plugging the second expression into the first, we get

I don't think this is standard notation, but it's better than writing out all the "violin holes".

Expanding ad infinitum we get

The problem is that it seems like we will get an infinite string of polynomials on the right hand side, but not on the left.