Can $\sum_{n=0}^\infty a_nx_i^n = \sum_{n=0}^\infty b_nx_i^n$ for distinct $(x_i)_{i\in \mathbb{N}}$ from interval $(0,1)$?

analysis power-series analytic-functions

Consider the power series of the following two functions around the origin $$ f(z)= (1-z)^{-1} \cos( (1-z)^{-1}), g(z)=42 f(z).$$ The only singularity of $f,g$ is at $z=1$, hence the associated power series around the origin has radius of convergence equal to $1$. Furthermore, the functions are different, but have the same zeros, which accumulate at $z=1$, thus we get a counterexample.

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