How to find $f_{p}$ such that $\sum a_{i} x_{i} = 1 \implies \sum f_{p} (a_{i}) (x_{i})^{p} = 1 $?

Question

Suppose we have the equation $$\sum_{i=0}^{\infty} a_{i} x_{i} = 1 $$ for some coefficients $(a_{i})_{i=0}^{\infty} $. I wonder how to find corresponding $f_{p}(\cdot)$ such that $$ \sum_{i=0}^{\infty} f_{p} (a_{i}) (x_{i})^{p} = 1 $$ when $p \in \mathbb{Z}_{\geq 2} $. Are there any results on this? Perhaps from the (non)linear algebra, complex analysis or some other area?