Ruin Probability via Probability Generating Functions

Consider a random walk on $\{0, 1, 2, \ldots, N\}$ with absorbing barriers at $0$ and $N$. At each step, the walker moves $+1$ with probability $p$ and $-1$ with probability $q = 1 - p$. Let $G_k(s) = E[s^{\tau} \mid X_0 = k]$ be the probability generating function of the absorption time $\tau$, for $0 < k < N$.

(a) Derive a recurrence relation for $G_k(s)$. (b) For $N = 4$, $k = 2$, and $p = q = 1/2$, find $G_2(s)$ explicitly and use it to compute $E[\tau]$ and $\text{Var}(\tau)$.