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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.


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This is not true. Take $r=2$, $f(z)=e^z$, $P_0=R'-R^{\prime\prime}$, $P_1=R^{\prime\prime}-R$, $P_2=R-R'$.

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I see. I was mistaken by the case $r=1$. For the case $r=2$, can we describe the functions $f$ for which it is true? – joaopa Apr 15 '13 at 6:13
joaopa: You should vote my answer up if you want me to answer further questions:-) – Alexandre Eremenko Apr 15 '13 at 13:43

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