Actually, I will be asking two, but related, questions.

Question 1. Is there some theory about recurrent relations with several indices. For example, If we have a relation on $A_{i;j}$: $$A_{i+\alpha; j+\beta} = F(A_{i+\alpha-1;j+\beta -1}, \ldots, A_{i;j}).$$

And related

Question 2. Let $m, n$, $Q_{m}^ {n}\in \mathbb{N}$. Then solve given 2-recurrent relation: $$Q_{m}^{n+1} = 2Q_{m}^{n} + Q_{m-1}^{n-1}$$ with the following conditions on $Q$: \begin{align}Q_{0}^{n} &= 2^n\\ Q_{j}^{j} &= 1,\\ Q_{k}^{l} &= 0,\quad \text{for } k>l. \end{align}

I have solved the last problem by pattern recognition followed by induction , two ways that I do not like to prove something. I would like to get more rough method.

I do not want to reveal the motivation of this problem (though it is not so covered) now in order to save your impartiality.