In complex field, assume $f(x)=\sum_{i=1}^{\infty} a_i x^i$ where $a_i \in {\bf N}$ or $a_i = 0$, and $f(x)$ converges in an area.

**Question 1:** are there $$f(x)=p(x)+\sum_{i=1}^{\infty}r_i(x), $$
or $$f(x)=p(x)+\sum_{i=1}^{n}r_i(x),$$
where $p(x)$ is a polynomial with all coefficients which are natural numbers, and $r_i(x)$ is a quotient of polynomials with at least one pole(which means denominator is polynomial with at least one zero point)，$r_i(x)$ can be expanded as $$r_i(x) =\sum_{j=1}^{\infty} b_{ij} x^j, \text{ }b_{ij} \in N\text{ or } b_{ij} = 0 ?$$

**Question 2:** If they do exist, how to compute them?

**Question 3:** Under what condition $$f(x)=p(x)+\sum_{i=1}^{n}r_i(x),n <\infty?$$

**Question 4:** Are there finite number of $$T_i(x) =\sum_{j=1}^{\infty} b_{ij} x^j, \text{ }b_{ij} \in N\text{ or } b_{ij} = 0 $$,where
$$r_i(x) =\sum_{j=1}^{\infty} b_{ij} x^j, \text{ }b_{ij} \in N\text{ or } b_{ij} = 0 $$,
are all "finitely generated" by $T_i(x)$ by multiply and addition?

isa quotient of polynomials. It is the quotient of $a_ix^i$ and $1$, for example. – Mariano Suárez-Alvarez♦ Jul 21 '13 at 1:55