Let $(u_n)$ and $(v_n)$ be two linear recurrent sequences, satisfying the same recurrence relation. Suppose that $v_n$ is nonzero for all large enough $n$, then we can set $w_n=\frac{u_n}{v_n}$. I have two questions about $(w_n)$ :
-Is $(w_n)$ still linear recurrent (not necessarily with the same recurrence relation, but with a new relation that may be computed directly from the relation ruling $(u_n)$ and $(v_n)$, independently of the initial values) ? My guess is that the answer is no.
-Is $(w_n)$ still recurrent (i.e. satisfies a not necessarily linear recurrence formula that may be computed directly from the relation ruling $(u_n)$ and $(v_n)$, independently of the initial values) ? My guess is that the answer is yes.