Consider the recurrence relation one can obtain from the radial Schrödinger equation for the hydrogen atom, where $\psi(r)=\sum_{n=0}^{\infty} a_nr^n$:
$$(n+3)(n+2)a_{n+2}+2a_{n+1}=2|E|a_n, n\geq0$$
Subject to $a_{n} = 0$ for $n \lt 0$, and $\lim_{n\rightarrow\infty} \sup |a_n|^{\frac{1}{n}}=0$. The solutions to this equation are well-known (these are just the hydrogen s-orbitals), so I am not interested in solving it, or obtaining the quantised energies. What I am interested in is how one can connect the rate of decay of the absolute value of the coefficients to the form of the recurrence relation, when one cannot provide a nice closed-form solution of the above equation.
Given that the above is an eigenvalue equation, it is natural to assume a solution of the form (of course, it is a separate question entirely how generally applicable this assumption is - it certainly does not hold when $a_n=0$, but it does hold for the solutions to the hydrogen atom):
$$a_{n+1} = -f(n+1)a_n$$
for some $f(n)$ that is strictly positive for all $n>0$. It is clear from the form of the recurrence relation that we want $f(n)\sim O(\frac{1}{n})$, so for large enough $n$, we can say that $f(n)$ will be a strictly decreasing function. However, when one studies the actual solutions to the above recurrence relation, it seems that $f(n+1) < f(n)$ for all $n>0$, not just at the large $n$ limit. Therefore, I am wondering if one can relate the form and/or the coefficients in the above recurrence relation to some sort of upper bound of coefficient decay that is true for all $n$, not just at the asymptotic limit. Or put another way: how can we enforce quantisation with a limited number of terms without necessarily having a closed-form solution that we can confidently say decays to 0 at infinite $n$.
I also suspect the answer to this would be related to putting an upper bound to the value of $2|E|$, which of course happens to be 1. I imagine there must be a deeper reason why $2|E|$ cannot be e.g. $1000$, which would give coefficients with oscillatory behaviour that will ultimately not converge to the desired limits.
I realise there is a lot to this question, but any insight would be appreciated, as it seems that recurrence relations are not as well-studied as some other branches of mathematics.
