Proof with polynomial differential operators

Heyya,

Got a problem that I have no idea how to approach.

The question is a follows:

is a polynomial operator of degree . Suppose is a fold root of , but not a fold root.

Verify that where is the derivative of .

I can see some use for the exponential shift formula. That would make the left hand side equal .

From here, I can't see where to go.

I'm thinking maybe there is a typo in the question. Maybe is supposed to be . However, the above is copied directly from the source. I don't know what to make of it.

Any input is greatly appreciated.

Thanks in advance.