By employing Frobinious or Power Series approach, we my come up with a recurrence relation that is only solvable if we set any constant lower than $a_0$ or higher than $a_n$ vanish.
For example, in the case of the reccurence relation for the Bessel function
$$a_n = \frac{-1}{(n+\lambda)^2-m^2}a_{n-2}$$
Such is only solvable if could solve for $\lambda$ and set $a_{n-2}$ to vanish - since it is lower than $a_n$ at $n=0$. Sometimes, we could deal with a recurrence relation were in it requires us to terminate all other highest $a_n$ to solve it.
In case we encountered a recurrence relation:
$$a_{n+1} = \frac{n^2-(m-2\lambda)^2}{(n+\lambda)^2-m^2}a_{n}$$
To solve it, we should first determine $\lambda$ by setting either the highest or the lowest order term to vanish.
My question is, How do we determine which terms should go to zero? (Terms lower than $a_0$ or terms higher than $a_n$?