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?



Questions about expansion of $f(x)=\sum_{i=1}^{\infty} a_i x^i$
Consider the sequence $1,0,0,1,0,0,0,0,1,\dots$ given by $a\_n=1$ if $n$ is a square, otherwise $a\_n=0$. I claim that no nonzero sequence $b\_n$ of non-negative integers with $b\_n\le a\_n$ for all $n$ can satisfy a constant coefficient homogeneous linear recurrence relation. For any sequence satisfying such a relation of order $d$ with $d$ consecutive terms equal to zero must be the zero sequence, and any sequence of non-negative integers dominated by $a\_n$ must have arbitrarily long runs of zeros. 


Now if $r(x)$ is a rational function, then the coefficients of its power series expansion must satisfy a constant coefficient linear recurrence. So if $f(x)=x+x^4+x^9+\cdots$, it can't be written as a sum, finite or infinite, of rational functions with poles and with power series with non-negative integer coefficients. 


EDIT. More generally, a sum of a finite number of rational functions is a rational function, and a power series represents a rational function if and only if its coefficients satisfy a constant coefficient homogenous linear recurrence, so the question of representing $f$ as a finite sum boils down to the question of determining whether the coefficients of $f$ satisfy such a relation. How to do this will depend on how $f$ is known. 


I don't know what to say about infinite sums, other than to give an example. If we write $d(n)$ for the number of divisors of $n$, then $$f(x)=\sum d(n)x^n={x\over1-x}+{x^2\over1-x^2}+{x^3\over1-x^3}+\cdots$$ while $f$ is certainly not a rational function. 


