$$\begin{equation} S(r,k) = f(r)S(0,k-1) + g(r)S(r+1,k-1) \end{equation}\ ,$$ where $r=0,1,2,\dots$ and $k=1,2,3,\dots$ . Also, $0<f(r)<1$ is an increasing function and $0<g(r)<1$ is a decreasing function, while $f(r)+g(r)<1$.

The base case is $S(r,1) = f(r) + g(r)\ \forall\ r$.

Do you think a closed-form expression of $S(r,k)$ is achievable?