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Consider the recurrence equation:

$x_{n+1} = \frac{ a_1 n + a_0}{b_2 n^2 + b_1 n + b_0} x_n $.

(or for that matter any ratio of two polynomial with the denominator having a higher defree)

Are there known results on characterizing the behavior of the solution like whether it increases then increases and has one local maxima?

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Obviously for large $n$, $$x_{n+1} \approx \frac {a_1}{b_2n}x_n$$ so $\lim_n x_n = 0$. The approximation will start to hold soon after $n$ becomes larger in magnitude than all the coefficients other than the leading ones. Before then, $x_n$ may or may not jump around, depending on the coefficients. Asuuming that all coefficients and $x_0$ are positive. If the leading coefficients are much larger than the remaining ones and $b_2 > a_1$, then $x_n$ will start moving towards $0$ immediately. if $a_1 > b_2$, then $x_n$ will rise until $n$ is larger than their ratio, after that it will start decreasing.

When the leading coefficients are not larger than the others, one can expect some jumping around at first. But no matter what, once $n$ becomes large enough, $x_n$ is heading uniformly towards $0$.

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If $x_{n+1} = \frac{ a_1 n + a_0}{b_2 n^2 + b_1 n + b_0} x_n $, let $f(t) =\sum_{n=0}^{\infty} x_n t^n $.

Factor out $a_1$ and $b_2$ to write, where $r = \frac{a_1}{b_2}$, $x_{n+1} = r\frac{ n + c}{ n^2 + d_1 n + d_0} x_n = r\frac{ n + c}{ (n+u)(n+v)} x_n $.

Then $x_{n} = x_0\prod_{k=0}^{n-1} r\frac{ k + c}{ (k+u)(k+v)} = x_0r^n\prod_{k=0}^{n-1} \frac{ k + c}{ (k+u)(k+v)} $ so that

$\begin{array}\\ f(t) &=\sum_{n=0}^{\infty} x_n t^n\\ &=x_0\sum_{n=0}^{\infty} t^nr^n\prod_{k=0}^{n-1} \frac{ k + c}{ (k+u)(k+v)}\\ &=x_0\sum_{n=0}^{\infty} (tr)^n\prod_{k=0}^{n-1} \frac{ k + c}{ (k+u)(k+v)}\\ &=x_0\,_2F_2(c, 1;u, v; tr)\\ \end{array} $

where $\,_2F_2$ is a hypergeometric function (see, for example, https://en.wikipedia.org/wiki/Generalized_hypergeometric_function).

In this case, $x_n \to 0$ and the series for $f(t)$ converges for all $t$.

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