Discounted gambler's ruin hitting probability ($u_k={\mathbb{E}}_k[s^{\tau }{1}_{\{X_{\tau }=0\}}]$)

For a nearest-neighbour random walk on $\{0,\dots ,N\}$ with right-step Probability $p$, left-step Probability $q=1-p$, and Absorption time $\tau$, first-step conditioning gives $u_k=s(p{u}_{k+1}+q{u}_{k-1})$, $u_0=1$, $u_N=0$. For $0<s<1$, the two positive distinct characteristic roots are $r_{\pm }=(1\pm \sqrt{1-4pq{s}^2})/(2ps)$. Solving the boundary equations gives $u_k=(r_{+}^N{r}_{-}^k-r_{-}^N{r}_{+}^k)/(r_{+}^N-r_{-}^N)$. The maximum principle $\Vert u-v{\Vert }_{\infty }\le s\Vert u-v{\Vert }_{\infty }$ proves uniqueness for equal boundary data. This is the joint generating function of the Absorption time and the lower-end absorption event.