Is there a solution for this non-linear recurrence relation?

Question

One statistics course of mine has the following recurrence.

$$p(n)=\frac{0.16p(n-1)+0.8p(n-2)+0.72p(n-3)}{n}$$

Assuming I know $p(0),$ is there a general formula for getting a solution?

Answer 1

You can solve it only asymptotically. Notice that no matter what starting values are $p(n)$ tends to $0$ as $n$ tends to infinity.

Therefore you can write

$$p_\sim(n)=\frac{1.68p_\sim(n-1)}{n}$$

which is giving

$$p_\sim(n)=\frac{c}{n!}1.68^{n}$$

where $c$ depends on the initial conditions.