Let $\alpha,\beta, \gamma \in \mathbb{R}^+$ be and the function

$$ F(m,n)= \begin{cases} 1, & \text{if $m n=0$ }; \\ \alpha F(m ,n-1)+ \beta F(m-1,n )+ \gamma F(m-1,n-1), & \text{ if $m n>0$. }% \end{cases} $$ Please, a proof for $$\lim_{n\rightarrow \infty}\frac{F(n,n)}{F(n-1,n)} =\lim_{n\rightarrow \infty}\frac{F(n+1,n)}{F(n,n)} $$