Let $E_P$ be a polynomial operator, defined by: $$E_P(x) := \frac{p_0x^0}{0!}+\frac{p_1x^1}{1!}+\frac{p_2x^2}{2!}+\cdots$$ (where $P(x) = p_0x^0+p_1x^1+p_2x^2+\cdots$)
Is there any conventional way to express this in terms of $P(x)$?
What if we know $P(x)=\ln'(Q(x))$?