Consider a simple symmetric random walk on $\mathbb{Z}$, $(S_t)_{t \geq 0}$, with $S_0=0$. Let $k$ and $j$ be two positive integers. Let $P_{k,j}$ be the probability that the walker hits the vertex $k$ for the first time in $j$ steps, conditioning that this happens before returning to the origin.
Namely, let $\tau_k$ be the hitting time of the vertex $k$ and let $\tau^+$ be the return time to the origin, $P_{k,j} := P(\tau_k=j \, | \, \tau_k<\tau^+)$.
Question: Let $c$ be a positive constant $c<1$. Is there a way to rewrite the following expression in a nicer form? $$\sum\limits_{k \geq 1} \sum\limits_{j\geq 1} P_{k,j} \, c^{j-1}$$
Observe that $ P_{k,j}>0$ iff $j \geq k$.