I have to solve this system:
\begin{equation} \begin{cases} -R x_0 + w_1 x_1 + v_2 x_2 = 0 \\ R x_0 -(v_1 + w_1 + R) x_1 + w_2 x_2 + v_3 x_3 = 0\\ \vdots \\ R x_{i-1} -(v_i + w_i + R) x_i + w_{i+1} x_{i+1} + v_{i+2} x_{i+2} = 0\\ \vdots \\ \end{cases} \label{system_stat_stat} \end{equation}
where $R$, $\beta$ and $\gamma$ are parameters and ( $x_i$; $i\geq 0$ ) an infinite vector $$\begin{cases} v_i=\beta i (i-1) \\ w_i=\gamma i \\ q_i = -R-v_i-w_i \end{cases} $$ Actually I'm working with probability distributions. We have one more equation due to the normalization, $\sum_{i=0}^\infty x_i = 1$.
Does anyone recognize the problem with a similar one?
there exist a closed (not recursive) analytical solution of this system ($x_i=f(i,R,\beta,\gamma)$)? if yes, how can I get it?
also a reference about this kind of problem would be helpful.
thanks a lot in advice.
more info:
The matrix representation of the system is the following pentadiagonal
$$
\bar{x}P = (x_0, x_1, \dots, x_i,\dots)
\begin{pmatrix}
-R & R & 0 & 0 & \dots & 0 & \dots \\
w_{1} & q_1 & R & 0 & \dots & 0 & \dots \\
v_2 & w_{2} & q_2 & R & \dots & 0 & \dots \\
\vdots & \vdots & \vdots & \vdots & \ddots & R & \dots \\
0 & 0 & 0 & v_i & w_{i} & q_i & \dots \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots\\
\end{pmatrix}
=0
$$
where $q_i = - w_i - v_i - R$