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Which formal power series $$A(x)=\sum_{n\geq 0} a_n x^n=a_0+a_1 x+ a_2x^2+\dots$$ around the center $c=0$ can be expressed as a rational fraction, i.e. $$A(x)=\frac{P(x)}{Q(x)},$$ where $P(x)$ and $Q(x)$ are polynomials?

Note that I am talking about formal power series and formal polynomials, that is, we consider them to be syntactic objects rather than discussing function-theoretic/analytic properties (in particular, it does not matter whether our formal power series $A(x)$ converges).

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