Recursive algorithm to compute a sum of product like function
Assuming we are given the $x_{ij}$'s and $i,j,m$, your formula already is a recursive algorithm to compute $F_{i,j}(m)$.  If you use dynamic programming (memoization) to avoid re-computing $F_{i',j'}(m')$ more than once for any given tuple $(i',j',m')$, the running time will be $O(m^3)$, since you only evaluate $F_{i',j'}(m')$ for tuples $(i',j',m')$ where $0 \le i',j',m' \le m$.

If the $x_{ij}$'s are not given, your question is not well-defined.  If that's what you meant, it's possible you might be asking for us to compute the distribution of $F_{i,j}(m)$, but if so, the question needs to be edited accordingly.

Either way, I suspect that there must be some additional base cases you have not provided.  As it currently stands, the solution $F_{i,j}(m)=0$ for all $i,j,m$ satisfies your equation.  I suggest you edit your question to specify the missing base cases.