Consider the following recurrence relation:
\begin{equation} \gamma C_{m,n}+n\alpha C_{n,m}+ \beta \{C_{n+1,m}+ n C_{n-1,m}\}=EC_{n,m} \end{equation}
where $\gamma, \alpha$ and $\beta$ are constants. The question is to find $C_{n,m}$ and $E$.
If $\gamma=0$ I know how to solve it. One can ignore index $m$ and the problem would be a one variable recurrence relation. In this case, one set of solution for example is $E=-\frac{\beta^{2}}{\alpha}$ and $C_{n}=(\frac{-\beta}{\alpha})^{n}$. There are other sets of solutions as well.
The problem gets complicated when $\gamma$ is not zero. I was wondering if a solution exists.