I have a question about the asymptotic behaviour of linear recurrences with polynomial coefficients related to the ones with constant coefficients.
If we have a relation
$$n^ku_{n+k}+P_1(n)u_{n+k-1}+P_2(n)u_{n+k-2}+\cdots+P_k(n)u_n=0 \hspace{10mm}(1)$$
where $P_m$ are polynomials of degree $k$ and their leading coefficients are $p_1, p_2, \cdots,p_k$, how is this relation related to this other one with constant coefficients $$u_{n+k}+p_1u_{n+k-1}+p_2u_{n+k-2}+\cdots+p_ku_n=0\hspace{10mm}(2)$$
The question is motivated by answers to some questions like this one and also because I'm trying to solve, and maybe generalize this other question.
My idea is that if we divide relation (1) by $n^k$ then as $n$ gets really big each polynomial is almost equal to their leading coefficient so for this big $n$ relation (1) is almost the same as relation (2) and, I guess that then if $a_n$, $b_n$ are solutions to relations (1) and (2) respectively, we have that $a_n/b_n$ converges to some constant.
If this reasoning is correct, how can we rigurously prove it? Can we say anything about the constant towards it converges?
In case it is not right, where does it fails, can we apply it to any particular cases?
Thanks a lot for the answers.