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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 $$\lim_{n\rightarrow \infty}\frac{F(n,n)}{F(n-1,n)} =\lim_{n\rightarrow \infty}\frac{F(n+1,n)}{F(n,n)} $$

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    $\begingroup$ Probably won't help. Generating function $$\sum _{m=0}^{\infty } \sum_{n=0}^{\infty }{\it F} \left( n,m \right) {x}^{n}{y}^{m} =-{\frac {\alpha\,xy+\beta\,xy-\alpha\,x-\beta\,y-xy+1}{ \left( y-1 \right) \left( x-1 \right) \left( \gamma\,xy+\alpha\,x+\beta\,y-1 \right) }}$$ $\endgroup$
    – Gerald Edgar
    Commented Apr 10, 2021 at 11:58
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    $\begingroup$ Actually that's how one would start. Then use the contour-integral formula to get algebraic generating functions for the diagonal series with $m=n$ and $m = n \pm 1$, and use the singularities nearest the origin to recover asymptotic formulas for those series. $\endgroup$
    – Noam D. Elkies
    Commented Apr 17, 2021 at 16:02

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