I have the following recurrence relation:
$(n + 1) a_{n + 2} = (w (n + 1) - c) a_{n + 1} - z (n + 1)*a_{n}$
that I would like to either solve, or to get the $n$ goes to Infinity limit of the ratio $a_n/b_n$ where $a_n$ and $b_n$ satisfy that same recurrence relation above, but with different initial conditions: $a_0=0$, $a_1=z$, $b_0=1$, $b_1=w-c/2$.
Problem is that I am a physicist and never learned how to do this sort of thing. I have been reading posts on the recurrence equations for the past few days, tried the Hyper algorithm from A=B book (no solutions found), and looked through lists of special functions in the hopes of finding one that looks right (nothing so far). Can anyone please tell me if there are any tricks that will give me the limit of $a_n/b_n$ that I need without solving for $a_n$ and $b_n$? Or point me in the right direction as far as solving that recurrence relation goes?