Is there a general approach for finding a generating function for a recurrence which has a dependence between its indices?
As an example consider the paths with north- and east-steps in a 2d-lattice contained in a band of width $T$ going from (0,0) to (m,n), the recurrence relation is then (I hope)
\begin{align*} C(m,n) = \begin{cases} 1\quad &\text{if }n= m = 0\\ 0 & \text{if } |m-n| > T\text{ or }m<0\text{ or }n<0 \\ C(m-1,n) + C(m,n-1)\quad& \text{else.} \end{cases} \end{align*}
I know that there are combinatorial approaches to solve this problem but was wondering whether there is a general method using generating functions for this case directly (comparable to the case without the 2nd condition)? Where can I learn about it?
Thanks in advance!