I have the following pair of recursion relations for $f_{m,n}$, featuring two indices/variables $(m,n)$:
$$\sin{\left(\frac{n \hbar}{2} \right)} f_{m+1,n} - \sin{\left(\frac{m \hbar}{2} \right)} f_{m,n+1} + \sin{\left(\frac{(m-n) \hbar}{2} \right)} f_{m-1,n-1} = 0 ,$$ $$\cos{\left(\frac{n \hbar}{2} \right)} f_{m+1,n} + \cos{\left(\frac{m \hbar}{2} \right)} f_{m,n+1} + \cos{\left(\frac{(m-n) \hbar}{2} \right)} f_{m-1,n-1} = E f_{m,n} . $$ In the above, $m$ and $n$ are integers, while $E$ and $\hbar$ are real constants. I am looking to use/generate the $f_{m,n}$ for $m,n \geq 0$ in the work I am doing, but generally $m,n$ can be positive and negative.
Given the trigonometric nature of the coefficients of the $f$'s, I expect that it may be difficult to find an analytic solution to these relations. I have seen that generating functions can potentially be used to solve multivariable equations like these, but what I would really like to know first is, is there an algorithmic way of counting how many initial conditions are required to generate all possible $f_{m,n}$ from these relations? Additionally, are there set values of $(m,n)$ which must be used as initial conditions/data? For example, must $f_{1,1} = c$, where $c$ is a constant, form an initial condition for some reason?
I cite these equations from equation (3.5) of this paper. Here the authors use symmetry properties (coming from physics) of the $f_{m,n}$ which reduces complexity and means very few initial data are needed (i.e. they only need an initial condition for $f_{1,1}$). I wanted to do this without any of the additional symmetry properties.
Any ideas are greatly appreciated, thanks!