I have a system of equations of the following

$$z(n)=c+deltabig(p_1z(n+1)+p_2z(n-1)+(1-p_1-p_2)z(n)big)~~if~~ngeq 0$$

and

$$z(n)=deltabig(p_1z(n+1)+p_2z(n-1)+(1-p_1-p_2)z(n)big)~~if~~n< 0$$

One interpretation of the system is like… You receive a reward of $c$ today if and only if your state variable $n$ is non-negative. Tomorrow’s reward is discounted with a factor $delta$, which is $deltain(0,1)$.

And the state $n$ evolves as follows: it becomes $n+1$ with probability $p_1$, $n-1$ with probability $p_2$, and stays the same with probability $1-p_1-p_2$. $c$ is a fixed scalar.

How can I solve for $z(cdot)$ in this case? It is a system of simple linear equations but with infinitely many variables.

If this is not solvable, would it be solvable if I add some boundary conditions and make it a finite problem? For example, restricting to $nin{-100,-99,cdots,99,100}$ and with some proper modification of the equations for $z(-100)$ and $z(100)$?