Let $r\in \mathbb N$ and $f$ be an entire function on $\mathbb C$. One assumes that for every $R\in\mathbb C[z]$ there exists polynomials $P_{i,R}\in\mathbb C[z]$ ($0\le i\le r$) not all zero such that for all $z\in\mathbb C$, one has $$\sum_{i=0}^rP_{i,R}(z)(f+R)^{(i)}(z)=0.$$ Then, $f$ is a polynomial.

Any clue to prove that?

Thanks in advance.

Joaopa