Suppose
$X_i=\nu_i+\frac{m-i}{m}X_{i+1}+\frac{i}{m}X_{i-1},\quad 1\le i\le m$
where $X_0=X_{m+1}=0$. I need to find an expression for $X_i$ in terms of $v_i$, $i$, and $m$. I know how to find it like inverse of a matrix, but I need a better presentation for it (like a closed form or a summation). Any help is greatly appreciated! (I am not sure but I think it is like the equation of a non-causal auto-regressive model)
Denote by $X$, the vector $X=(\begin{array}{cccc}X_0&X_1&\cdots&X_{m+1}\end{array})^T$ and by $v$, the vector $v=(\begin{array}{cccc}v_1&v_2&\cdots&v_{m}\end{array})^T$. Now your equation turns out to be $$-\frac{i}{m}X_{i-1}+X_i-\frac{m-i}{m}X_{i+1}=v_i,~~~1\leq i\leq m$$ Then $MX=v$ or $X=M^{-1}v$, where $M$ is a $m\times (m+2)$ matrix given by $\left( \begin{array}{ccc} 1 & -\frac{m-1}{m} & 0 & 0 & 0 & 0 & \cdots\\ -\frac{2}{m} & 1 & -\frac{m-2}{m} & 0 & 0 & 0 & \cdots\\ 0 & -\frac{3}{m} & 1 & -\frac{m-3}{m} & 0 & 0& \cdots\\ 0 & 0 & -\frac{4}{m} & 1 & -\frac{m-4}{m} & 0 & \cdots\\ \cdots& \cdots& \cdots& \cdots& \cdots& \cdots& \cdots\end{array} \right)$.