Let $\phi(z)=\sum_{m=1}^\infty (\phi_{m}z^m-\psi_{-m}z^{-m})$ where $\phi_m, \psi_{-m}\in \mathbb{C}$.
If we can find one polynomial $P(z)\in \mathbb{C}[z]$ such that $P(z)\phi(z)=0$, how can I get some properties about $\sum_{m=1}^\infty \phi_{m}z^m$ and $\sum_{m=1}^\infty \psi_{-m}z^{-m}$ from this equation $P(z)\phi(z)=0$?
For example:
$(1):$Can I get that $\phi(z)$ is a rational function?
$(2):$ Can I get $\sum_{m=1}^\infty \phi_{m}z^m=\sum_{m=1}^\infty \psi_{-m}z^{-m}$?
Here, the left hand and the right hand can be viewed as the same rational function expanding at $0$ and $\infty$?
Why I have this question
I am reading a paper Asymptotic representations and Drinfeld rational fractions, the doi of this paper is $10.1112/S0010437X12000267$.
In this paper's lemma 3.9, if we define $\Psi(z)= \sum_{k=0}^\infty \Psi_{i,k}z^k$, the author proves that $$\sum_{m=0}^Na_mz^{N-m}\bigg(\Psi_i(z)-\sum_{p=0}^m\Psi_{i,p}z^p\bigg)=0.$$
Then he conclude that $\phi(z)= \sum_{k=0}^\infty \phi_{i,k}z^k$ is rational.
I want to know if the author means that $\Psi_i(z)= \sum_{p=0}^m\Psi_{i,p}z^p$ is the true rational function?
Any help and references are greatly appreciated.
Thanks!
This is a Laurent series. I presume it is supposed to converge to $\phi(z)$, at least in some annulus $A = \{z: r < |z| < R\}$ with $r < R$.
If $P(z) \phi(z) = 0$ where $P$ is a polynomial not identically $0$, then $\phi(z) = 0$ everywhere in $A$. And then all the coefficients are $0$ by the Cauchy integral formula.