I was trying to solve the following recursion problem. It seems like because of the different nature of the boundary conditions it is getting strange although I know the solution exists. The problem is as follows:
$a(n,k) = pf a(n-1,k) + p(1-f) a(n,k-1) + (1-p) a(n+1,k)$ where $n,\,k >0$
$a(0,k) = p(1-f) a(0,k-1) + (1-p) a(0,k+1) + (1-p) a(1,k)$
$a(n,0) = pf a(n-1,0) + (1-p) a(n+1,0)$ and
$a(0,0)=(1-p)(a(1,0)+a(0,1))$
How can we solve it?