Recently, i was trying to solve this recurrence relation $$ a_{n+4} = \frac{-\alpha(x)}{(n+4)} \cdot a_{n+3} +\frac{-\beta(x)}{(n+3)\cdot (n+4)} \cdot a_{n+2} $$
But i can't solve for $a_n$
I've tried to solve using Wolfram|Alpha and i got this result.
How do i get this result and how to solve it.
Edit: n = 0,1,2,3....
Hint Multiplying by $(n+4)!$ you get $$(n+4)!a_{n+4} = -\alpha(x) *(n+3)! a_{n+3} -\beta(x) *(n+2)!a_{n+2}$$
Let $b_n=n! a_n$ then, your recurrence is $$b_{n+4}= -\alpha(x) \cdot b_{n+3} -\beta(x) \cdot b_{n+2}$$ which is a standard second order recurrence.
Solve it, and then $$a_n=\frac{b_n}{n!}$$