The following question arised in my research, and I was unable to settle it after playing with it for sometime. Let $\{a^k_i\}_{i\geq 1}$ (for $k\in \{1,2,3,4\}$) be four sequences of real numbers. For each $k\in \{1,2,3,4\}$, consider the function $f_k:]-r_k,r_k[\rightarrow \mathbb{R}$ given by $x\mapsto\sum_{i=1}^{\infty}a_i^kx^i$ (Where $r_k$ is defined as $\frac{1}{\limsup_{i\rightarrow \infty}|a_i^k|^{\frac{1}{i}}}$).

It is given that for all $x$ sufficiently close to $0$, we have $f_2(f_1(x))=f_4(f_3(x))$. It is also given that for some $x_0\in\mathbb{R}$, we have that both $f_2(f_1(x_0)), f_4(f_3(x_0))$ exist. Must $f_2(f_1(x_0))= f_4(f_3(x_0))$ ? I would be very grateful if you can help, thank you.

I know that the answer is yes in some special cases like $f_1$ equals identity map, or $f_1,f_2,f_3,f_4$ are rational functions (Which raises the question what if the $f_k$(s) are algebraic functions ?). I also considered extending the domain of the $f_k$(s) to an open subset of the complex plane and possibly open the route for analytic continuations and complex analysis machinery but was not successful with that.