Consider, for instance, the gambler ruins problem:
https://www1.cmc.edu/pages/faculty/Bhunter/GamblersRuin.pdf
Let $p_k$ be the probability of achieving the maximum $N$ dollars when we have initially $k$ dollars.
The recurrence for $p_k$ is given by
\begin{equation} p_k = a p_{k+1} + bp_{k-1} \end{equation} where $a$ and $b$ are the probabilities or winning and losing 1 dollar, per round, respectively. We have $p_0=0$ and $p_N=1$.
The characteristic root equation is derived from \begin{equation} \sum p_k s^k = \sum a p_{k+1} s^k + \sum bp_{k-1} s^k \end{equation} and \begin{equation} \sum p_k s^k = \sum a p_{k} s^{k-1} + \sum b p_{k} s^{k+1} \end{equation} where \begin{equation} \sum p_k s^{k+1} = \sum a p_{k} s^{k} + \sum b p_{k} s^{k+2} \end{equation} i.e., \begin{equation} s = a + b s^{2} \end{equation}
is there a simple way to relate the solution to the problem using z transform and the solution using the characteristic root equation?
is there an intuitive way to understand that if $r_1$ and $r_2$ are the roots of the characteristic root equation, then $p_k = C_1 r_1^k + C_2 r_2^k$?
can we say that the z transform method and the characteristic root equation method are in reality the same method?
(This question is a follow up to Solving recurrence relations using characteristics root techinque and How to prove the characteristic equation based solution of recurrence relations?)
