I have the following relation:

$$ \sum_{d|n} (1+1/x)^{d-1} F_{n/d}(x^d)=L_n(x) $$

where the right hand side is (for every $n$) a polynomial in $x$, which I have an expression for, but it's not extremely beautiful. The family of polynomials $F_k(x)$ is unknown, and is what I'm looking for.

Since this is close to Dirichlet convolution, I have not quite given up hope that there is something similar to Möbius inversion, that would give me $F_k(x)$ explicitely. Is this possible? Related instances of such a problem may also be interesting.

A possibly weaker, but still sufficient solution would be an expression in terms of $L_k$ and $R_k$ of the expression

$$ \sum_{d|n} R_d(x) F_{n/d}(x^d) $$

where $R_k(x)$ is another family of polynomials, which is also unknown.