Is there a general solution for multi-variable recurrences of the following form which come from Markov chain analysis?

$f(i,j,k) = p_1 f(i-1, j,k) + p_2 f(i, j-1,k) + p_3 f(i,j,k-1) + p_4 f(i-1,j-1,k) + \dots$ $\dots +p_7f(i-1,j-1,j-1)$

where $\sum p_\ell = 1$ and the base cases which are for $f(i,j,0), f(i,0,k)$, $f(0,j,k$) are all either $1$ or $0$. The $p_\ell$ can themselves be linear functions of $i,j,k$.