Since I have thought about this problem for a days, and it turned out that I have no much ideas to solve this by myself, please help me and I would appreciate any comments over the topic.
I've working on the text "Complex Manifolds and Deformation of Complex Structures" written by Kodaira, and stagged at the p.60, right after the equation (2.17) (this specific page number maybe wrong, since I'm reading original Japanese version).
Along this section, he's trying to show how the meromorphic functions over Hopf manifolds of dimension 2 looks like.
First of all, here is how Hopf manifold is defined in this text when dimension is of 2.
Let W be $\mathbb{C}^2-\{0\}$, and $g=(\alpha_1,\alpha_2)\in W\ (\alpha_j\in\mathbb{C},\ |\alpha_j|>1,\ j=1,2)$.
The abelian group generated by the element g, $G:=\{g^m:m\in \mathbb{Z}\}$ acts on W with $g^mz=(\alpha_1^mz_1,\alpha_2^mz_2)$ for all $z=(z_1,z_2)\in W$. Then G can be looked as a subgroup of ${\rm Aut}W$.
Because the action of G on W is free and proper, the quotient space $W/G$ becomes a complex manifold called Hopf manifold.
Let me introduce some variables and notations the author uses over the discussion around this context.
・$\theta_\lambda\in[0,2\pi)$: differs each for all $\lambda$
・$P_\lambda(z), Q_\lambda(z)$: complex coefficients polynomials with 2 variables (the author shows only finitely many terms remain after the application of limit $m\to \infty$ with the action $g^{-m}$)
・$m,\ 1\leq \lambda\leq\nu$: integers
Now for a meromorphic function over Hopf manifold f(z), in a sufficiently small neighborhood around 0, following equation holds.
$$ \displaystyle{ f(z)=\lim_{m\to\infty} \frac{\sum_{\lambda=1}^\nu e^{-im\theta_\lambda} P_\lambda(z)}{\sum_{\lambda=1}^\nu e^{-im\theta_\lambda} Q_\lambda(z)}}\ \ldots\ (2.17) $$
The author states that "with the assumption that $Q_1(z)$ is not identically 0 (around the neighbor around 0),
$$ f(z)=P_1(z)/Q_1(z) $$
holds". This is where I could not figure out.
Why the other terms vanish but the first after the limit applied?
Note 1: As I draw in the diagram below, the meromorphic function f here denotes the one which actually is extended twice in the following sense. For the first step, since the natural quotient map $\pi$ is an unbranched holomorphic covering over W, every meromorphic function over Hopf manifold W/G can be extended uniquely (for each fixed point in the fiber) to a fiber-preserving meromorphic function over W. And secondly, with the use of Levi's extension theorem, the meromorphic function over W can be uniquely extended to that of $\mathbb{C^2}$, thus $f$ in the entire question here really denotes $\hat{f}$ in the diagram. I apologize for the confusion.
Note 2: If my understanding is not in totally wrong direction, the limit ($\lim_{m\to\infty}$) used here would indicate "the neighbor of zero cannot be determined uniquely, yet if you choose one sufficiently small, then for any other neighbors, the limit function meets in the intersection of those neighbors", as it appears in the discussion over stalks of sheaf of function germs.
Note 3:
@PaulSinclair The author actually employed other conditions than just I mentioned here above, so the distribution of indexes $\lambda$ be restricted to some extent. And as you mentioned, I feel also reasonable that those conditions shall affect the terms to be vanished (but unfortunately, I still don't see the reason..).
I will explain those conditions to be precise for the question I asked. First the author defined $f(z)$ as $f(z)=\lim_{m\to \infty}\phi(g^{-m}z)/\psi(g^{-m}z)$, where $\phi(z),\ \psi(z)$ are holomorphic functions on some neighbor of zero, $U_\epsilon(0)$. So this expression can be regarded as a local representation of $f$ over $U_\epsilon(0)$. And then put
$$ \phi(z)=\sum^{\infty}_{h,k=0}b_{hk}z_1^hz_2^k,\ \psi(z)=\sum^{\infty}_{h,k=0}c_{hk}z_1^hz_2^k $$
as the convergent power series. Becase the $f$ is unchanged by the action of G, it can be written as
$$ \displaystyle{f(z)=\lim_{m\to \infty}\frac{\sum (\alpha_1^h\alpha_2^k)^{-m}b_{hk}z_1^hz_2^k}{\sum (\alpha_1^h\alpha_2^k)^{-m}c_{hk}z_1^hz_2^k}} $$
Let $\mu:={\rm min}\{|\alpha_1^h\alpha_2^k| : (b_{hk},c_{hk})\neq (0,0) \}$ (The tuple (h,k) go over all the indices of coefficients that won't vanish simultaneously) and replace $(\alpha_1^h\alpha_2^k)^{-m}$ with $(\alpha_1^h\alpha_2^k/\mu)^{-m}$ from above expression (this replacement do not change the original function since both denominator and numerator are uniformally convergent). Taking the limit, all the terms with $|\alpha_1^h\alpha_2^k|>\mu$ vaish except those for $|\alpha_1^h\alpha_2^k|=\mu$ (which are only finitely many terms).
Thus, for the remains conditioned by $|\alpha_1^h\alpha_2^k|=\mu$, we can only take some $\theta_\lambda\in [0,2\pi)$ such that $e_\lambda:=e^{i\theta_\lambda}=\alpha_1^h\alpha_2^k/\mu$ of arguments $\theta_\lambda$ differs for each $1\leq \lambda\leq \nu$.
And when we put
$$ P_\lambda=\sum_{\alpha_1^h\alpha_2^k=\mu e_\lambda} b_{hk}z_1^h z_2^k,\ Q_\lambda=\sum_{\alpha_1^h\alpha_2^k=\mu e_\lambda} c_{hk}z_1^h z_2^k $$
$f(z)$ can be written in the form of (2.17).