I am currently studying discrete mathematics and my book covers a theorem for finding a particular solution. The book basically tells me to solve a non homogeneous recurrence relation in two steps. First find the solution for the associated homogeneous recurrence relation, then find the particular solution.
$$ ( a_n^{(h)}+ a_n^{(p)}) $$
I know how to solve the associated homogeneous recurrence relation.
The next step is to solve the particular solution, for which it says the following.
Suppose that ${a_n}$ satisfies the linear nonhomogeneous recurrence relation $$ a_n = c_1a_{n-1} + c_2a_{n-2} + ... + c_k a_{n-k} + F(n) $$ Where $c_1, c_2, ..., c_k$ are real numbers and $$ F(n) = (b_tn^t + > b_{t-1}n^{t-1} + ... + b_1n + b_0)s^n $$ Where $b_0, b_1, ... b_t$ and $s$ are real numbers. When $s$ is not a root of the characteristic equation of the associated linear homogeneous recurrence relation, there is a particular solution of the form $$ (p_tn^t + p_{t-1}n^{t-1} > + ... + p_1n + p_0)s^n $$ When $s$ is a root of the characteristic equation and its multiplicity is $m$, there is a particular solution of the form $$ n^m(p_tn^t + p_{t-1}n^{t-1} + ... + p_1n + p_0)s^n $$
I do understand that I have to use a different formula depending on if my $s$ is equal to one of my roots, found when calculating the associated homogeneous recurrence relation.
What I don't quite understand is what the $s$ stands for.
Using an example $s(n) = 8*s^{n-2} - 16*s^{n-4} + n^3$ with $s(0)=0$, $s(1)=1$, $s(2)=2$, $s(2)=2$
This gives me roots $r = 2$ with multiplicity $2$ and $r = -2$ with multiplicity $2$ as well.
What would be the $s$ and why would it be the $s$? How do I determine the $s$?
Once I have determined the $s$, do I just fill in the formula and find values for $p$ to determine the entire particular solution?
If someone could elaborate on this piece of theory and explain it a bit more step by step I hope to understand this theorem better, the book kind of fails on explaining it in smaller steps.