Suppose we have a collection of functional power series with invariant sum $0$,
i.e, assume $$F_k(x)=\sum_{n=0}^{\infty} n f_k(n,x)x^n=0, \ x \in \Bbb Q$$ where $f_k(n,x)$ is a polynomial of degree $k$ in $n$ and $x$.
$(1)$ What does these kind of $0$-sum power series mean in mathematics?
$(2)$ What does the functional power series with $\text{zero sum}$ represent in mathematics and physical world?
$(3)$ Can I consider it as a rational invariant or giving rational sum for all rational $x$ ?
$(4)$ Do these kind functional power series with $0$-sum represent something in p-adic analysis?
Please help me at least with short hints.
I just want to connect or want to understand the meanings of such series.
Please share anything about it.