There seems to be a PDE for $g(a,b,x)=\sum_{n\ge0}f_n(a,b)x^n$,
which can be thought of as a boundary value problem in the triangle $0\lt a\lt b\lt1$.
$$g_{aab}+g_{abb}+x^3g=0$$
($x$ is a parameter and subscripts are derivatives)
with boundary values $g(0,b,x)=1$, $g_a(a,a,x) = x$, and
$g_{ab}(a,1,x) = x^2$. This comes from iterating the $f_n$ recurrence,
after Pietro's remarks that $(a,b)\to(b-a,1-a)$ has period 3
suggested looking at third derivatives.
Does that determine $g$ uniquely, nicely? I don't know yet.
[Edit: I wrongly wrote $g$ at first using $\frac{x^n}{n!}$.]