Pair of functions $F(x)$ (transcendental),$A(x)$ (algebraic) with expanded series of positive integer coefficient linked by derivative

Question

$$F(x)=\sum_0^{\infty}b_k x^k,b_k\in \mathcal{N} \bigcup 0,\exists M \space b_k \leq M^k$$. $$A(x)=\sum_0^{\infty}a_k x^k,a_k\in \mathcal{N} \bigcup 0,\exists M \space a_k \leq L^k$$

where $F(x)$ is transcendental function of of Riemann surface,$A(x)$ is algebraic function of Riemann surface.

Question:are there pair of functions $F(x),A(x)$ such that $$A(x)= x \frac{dF(x)}{dx}$$

Further,$F(x)$ with what property ,or $A(x)$ with what property,is $$A(x)= x \frac{dF(x)}{dx}$$ valid?

Answer 1

If your question is on the generalisation of the following result by Polya (J fur die reine angwt math, 1921): " Let $F\in \mathbb{Z}[[x]]$ with non zero radius of convergence and suppose that $xF^{\prime}(x)$ is a rational function, then $F$ is a rational function" obtained by replacing the word "rational" by "algebraic", then I think that this is proved in Yves Andr\'e's book "G-functions and Geometry".