Let $\alpha$ be an algebraic number and $\beta$ be one of its algebraic conjugate. Suppose $|\alpha|, |\beta| > 1$.
Let $P_i(x)\in \mathbb{Q}[x]$ be a sequence of polynomials with rational coefficients.
Suppose $\sum_{i=1}^{\infty} \dfrac{P_i(\alpha)}{\alpha^i} \in \mathbb{Q}[\alpha]$ so that we can write $\sum_{i=1}^{\infty} \dfrac{P_i(\alpha)}{\alpha^i} = Q(\alpha)$ for some $Q(x)\in\mathbb{Q}[x]$.
Then depending on the choice of $\alpha, \beta, P_i(x)$, the following would or would not hold:
$$\sum_{i=1}^{\infty} \dfrac{P_i(\beta)}{\beta^i} = Q(\beta).$$
I know that if the sum only has finitely many terms, then applying the "obvious" field isomorphism $(\alpha \rightarrow \beta)$ on the both sides of $\sum_{i=1}^{N} \dfrac{P_i(\alpha)}{\alpha^i} = Q(\alpha)$ we can have the result. And I also find some examples so that the equality does not hold.
Is there any results/reference that is related to this problem? Thanks!