Asymptotics of ratios of polynomially recursive sequences A sequence $a_n$ is said to be polynomially recursive (P-recursive) if it satisfies:
$$p^{[r]}(n)a_{n+r}+\cdots+p^{[1]}(n)a_{n+1}+\cdots + p^{[0]}(n)a_n=0$$
where $p^{[i]}(t)\in \mathbb{Q}[t]$ are polynomials with rational coefficients, with $p^{[0]},p^{[r]}$ not identically zero.
For example, $a_n:=n!$ is one such sequence since: $2a_{n+2}-(n+2)a_{n+1}-(n+1)(n+2)a_n=0$, with initial conditions $a_0:=0, a_1:=1$.
Fix a set of polynomials $\{p^{[i]}(t)\}_{i=0}^r$, and suppose $a_n,b_n$ are a pair of sequences that satisfy the same above recurrence, with different initial conditions. Furthermore, suppose that both sequences aren’t ultimately periodic or constant.
Define $L :=\lim_{n\rightarrow\infty} \frac{a_n}{b_n}$.

Main Question:  Is it obvious when $L\in (0,\infty)$? In other words, when does $L$ exist, is non-zero and non-infinite? Aside from numerically evaluating the limit for large enough $n$, are there any algorithmic methods for deducing that $L \in (0,\infty)$?

There are many known (non-trivial) results about the growth rates of such sequences, for example results due to Poincaré, Birkhoff and Trjitzinsky, Wimp and Zeilberger, and Mezzarobba and Salvy. However, I'm unable to find good references related to my question, especially as a function of initial conditions. The main difficulty I have is that I’m not sure how to find good lower bounds on the growth rates of such sequences.
 A: This is a difficult question in general; see for instance

*

*https://people.mpi-sws.org/~joel/publications/positivity_and_minimality_holonomic21abs.html

*https://people.mpi-sws.org/~joel/publications/holonomic-second-order21abs.html
for recent work on special cases. (As you noted, the paper by Bruno Salvy and myself that you mention is purely about upper bounds and thus not terribly relevant.)
However, there are sufficient conditions for $L$ to be nonzero and finite that can be verified algorithmically and cover a fair number of (“easy”) cases. In particular, by passing to the differential equation on the generating series, one may be able to get “asymptotic expansions with error bounds” of the form, say, $|a_n - α f(n)| ≤ g(n)$, $|b_n - β f(n)| ≤ h(n)$ where $f$, $g$, $h$ are explicit functions with $g, h = o(g)$ and $α$, $β$ are constants that can be bounded from both sides. For more details on this kind of ideas, see for example

*

*https://arxiv.org/abs/2011.08155

*https://hal.archives-ouvertes.fr/hal-03291372/
