See the blog post: Hadamard’s gap theorem at Uniformly at Random.
It contains a proof of the result by Fatou:
A function whose power series expansion has integer coefficients and radius of convergence 1 is either rational (in $\mathbb{Q}(x)$) or transcendental (over $\mathbb{Q}(x)$).
If $r$ is rational, then the decimal expansion will be eventually periodic. So we have rational function. (Indeed this can be done explicitly.)
Otherwise, when $r$ is irrational, then the resulting function cannot be rational (plug in $1/10$, then you get irrational number). Thus, we have transcendence of $f$.
In particular, your functions $F$ in the beginning are transcendental. However, getting closed form will be extremely hard for those examples.
